Relative Divergence Measures \\ and Information Inequalities
Inder Jeet Taneja
There are many information and divergence measures exist in the literature on information theory and statistics. The most famous among them are Kullback-Leiber's \cite{kul} relative information and Jeffreys \cite{jef} J-divergence, Information radius or Jensen difference divergence measure due to Sibson \cite{sib}. Burbea and Rao \cite{bur1, bur2} has also found its applications in the literature. Taneja \cite{tan2} studied another kind of divergence measure based on arithmetic and geometric means. These three divergence measures bear a good relationship among each other. But there are another measures arising due to J-divergence, JS-divergence and AG-divergence. These measures we call here relative divergence measures or non-symmetric divergence measures. Here our aim is to obtain bounds on symmetric and non-symmetric divergence measures in terms of relative information of type s using properties of Csisz\'{a}r's f-divergence.
math0505204.tex
\documentclass[12pt]{amsart}
\usepackage{amscd,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{hyperref}

\newtheorem{definition}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{result}{Result}[section]
\newtheorem{example}{Example}[section]
\newtheorem{proposition}{Proposition}[section]
\numberwithin{equation}{section}

\setlength{\paperwidth}{215mm} \setlength{\paperheight}{280mm}
\setlength{\oddsidemargin}{10mm} \setlength{\evensidemargin}{10mm}
\setlength{\topmargin}{0mm} \setlength{\headheight}{10mm}
\setlength{\headsep}{13mm} \setlength{\textwidth}{140mm}
\setlength{\textheight}{200mm} \setlength{\footskip}{15mm}
\setlength{\marginparwidth}{0mm} \setlength{\marginparsep}{0mm}

\title[To appear in: \textit{Inequality Theory and Applications},
\textbf{4(}2004)]{Relative Divergence Measures \\ and Information
Inequalities}

\author{Inder Jeet Taneja}
\address{Departamento de Matem\'{a}tica\\ Universidade Federal
de Santa Catarina\\
88.040-900 Florian\'{o}polis, SC, Brazil}

\email{taneja@mtm.ufsc.br}

\urladdr{http://www.mtm.ufsc.br/$\sim $taneja}

\thanks{To appear in: \textit{Inequality Theory and Applications},
\textbf{4(}2004)}

\keywords{Relative information of type s; J-divergence; Relative
J-divergence; JS-divergence; Relative JS-divergence;
AG-divergence; Relative AG-divergence; Csisz\'{a}r f-divergence.}

\subjclass[2000]{94A17; 26D15}

\begin{document}

\begin{abstract}
There are many information and divergence measures exist in the
literature on information theory and statistics. The most famous
among them are Kullback-Leiber's \cite{kul} \textit{relative
information} and Jeffreys \cite{jef} \textit{J-divergence},
\textit{Information radius} or \textit{Jensen difference
divergence measure} due to Sibson \cite{sib}. Burbea and Rao
\cite{bur1, bur2} has also found its applications in the
literature. Taneja \cite{tan2} studied another kind of divergence
measure based on \textit{arithmetic and geometric means}. These
three divergence measures bear a good relationship among each
other. But there are another measures arising due to
\textit{J-divergence}, \textit{JS-divergence} and
\textit{AG-divergence}. These measures we call here
\textit{relative divergence measures} or \textit{non-symmetric
divergence measures}. Here our aim is to obtain bounds on
\textit{symmetric }and \textit{non-symmetric divergence measures}
in terms of \textit{relative information of type s} using
properties of Csisz\'{a}r's \textit{f-divergence}.
\end{abstract}

\maketitle

\section{Introduction}

Let
\[
\Gamma _n = \left\{ {P = (p_1 ,p_2 ,...,p_n )\left| {p_i >
0,\sum\limits_{i = 1}^n {p_i = 1} } \right.} \right\}, \,\, n
\geqslant 2,
\]

\noindent be the set of all complete finite discrete probability
distributions.

The Kullback Leibler's (1951) \textit{relative information} is given by
\begin{equation}
\label{eq1}
K(P\vert \vert Q) = \sum\limits_{i = 1}^n {p_i \ln (\frac{p_i }{q_i })} ,
\end{equation}

\noindent for all $P,Q \in \Gamma _n $.

We observe that the measure (\ref{eq1}) is not symmetric in $P$
and $Q$. Its \textit{symmetric version} famous as
\textit{J-divergence} (Jeffreys \cite{jef}; Kullback and Leiber
\cite{kul}) is given by
\begin{equation}
\label{eq2}
J(P\vert \vert Q) = K(P\vert \vert Q) + K(Q\vert \vert P)
 = \sum\limits_{i = 1}^n {(p_i - q_i )\ln (\frac{p_i }{q_i })} .
\end{equation}

For simplicity, we call here the measure $K(Q\vert \vert P)$ the
\textit{adjoint} of $K(P\vert \vert Q)$ and vice-versa.

Alternatively, the measure $J(P\vert \vert Q)$ can also be written
in the following way:
\begin{equation}
\label{eq3}
J(P\vert \vert Q) = D(P\vert \vert Q) + D(Q\vert \vert P),
\end{equation}

\noindent where
\begin{equation}
\label{eq4}
D(P\vert \vert Q) = \sum\limits_{i = 1}^n {(p_i - q_i )\ln \left( {\frac{p_i
+ q_i }{2q_i }} \right)}
\end{equation}

\noindent and
\begin{equation}
\label{eq5}
D(Q\vert \vert P) = \sum\limits_{i = 1}^n {(q_i - p_i )\ln \left( {\frac{p_i
+ q_i }{2p_i }} \right)} .
\end{equation}

Let us consider the following two measures:
\begin{equation}
\label{eq6}
F(P\vert \vert Q) = K\left( {P\vert \vert \frac{P + Q}{2}} \right) =
\sum\limits_{i = 1}^n {p_i \ln \left( {\frac{2p_i }{p_i + q_i }} \right)}
\end{equation}

\noindent and
\begin{equation}
\label{eq7}
G(P\vert \vert Q) = K\left( {\frac{P + Q}{2}\vert \vert P} \right) =
\sum\limits_{i = 1}^n {\left( {\frac{p_i + q_i }{2}} \right)\ln \left(
{\frac{p_i + q_i }{2p_i }} \right)} .
\end{equation}

The \textit{adjoint} forms of the measures (\ref{eq6}) and
(\ref{eq7}) are given by
\begin{equation}
\label{eq8}
F(Q\vert \vert P) = K\left( {P\vert \vert \frac{P + Q}{2}} \right) =
\sum\limits_{i = 1}^n {q_i \ln \left( {\frac{2q_i }{p_i + q_i }} \right)}
\end{equation}

\noindent and
\begin{equation}
\label{eq9}
G(Q\vert \vert P) = K\left( {\frac{P + Q}{2}\vert \vert Q} \right) =
\sum\limits_{i = 1}^n {\left( {\frac{p_i + q_i }{2}} \right)\ln \left(
{\frac{p_i + q_i }{2q_i }} \right)} .
\end{equation}

\noindent respectively, and the \textit{symmetric forms} are given
by
\begin{equation}
\label{eq10}
I(P\vert \vert Q) = \frac{1}{2}\left[ {F(P\vert \vert Q) + F(Q\vert \vert
P)} \right]
\end{equation}

\noindent and
\begin{equation}
\label{eq11}
T(P\vert \vert Q) = \frac{1}{2}\left[ {G(P\vert \vert Q) + G(Q\vert \vert
P)} \right],
\end{equation}

\noindent respectively.

The three measures $J(P\vert \vert Q)$, $I(P\vert \vert Q)$ and
$T(P\vert \vert Q)$ are related with each other as
\begin{equation}
\label{eq12} J(P\vert \vert Q) = 4\left[ {I(P\vert \vert Q) +
T(P\vert \vert Q)} \right].
\end{equation}

Moreover, the measures (\ref{eq4}) can also be written as
\begin{equation}
\label{eq13}
D(P\vert \vert Q) = 2\left[ {F(Q\vert \vert P) + G(Q\vert \vert P)} \right]
\end{equation}

The measure $I(P\vert \vert Q)$ is famous in the literature as
\textit{information radius} (Sibson \cite{sib}) or \textit{Jensen
difference divergence measure}, or simply, \textit{JS-divergence}
(Burbea and Rao \cite{bur1, bur2}). The measure (\ref{eq9}) is new
in the literature and is studied for the first time by Taneja
\cite{tan2}, called \textit{arithmetic and geometric mean
divergence measure} or simply, \textit{AG-divergence.} More
details on these divergence measures can be seen in Taneja
\cite{tan1, tan3}.

For simplicity, we shall call the measure $D(P\vert \vert Q)$ by
\textit{relative J-divergence}, the measure $F(P\vert \vert Q)$ by
\textit{relative JS-divergence }and the measure $G(P\vert \vert
Q)$ by \textit{relative AG- divergence}. The measure $D(P\vert
\vert Q)$ is due to Dragomir et al. \cite{dgp}. The measure
$F(P\vert \vert Q)$ has been studied by many authors (Shioya and
Da-te \cite{shd}; Barnet et al. \cite{bcd} ; Lin \cite{lin}; Lin
and Wong \cite{liw}). The measure $G(P\vert \vert Q)$ we have
considered here for the first time and is a part of the measure
$T(P\vert \vert Q)$.

The one parametric generalization of the Kullback-Leibler
\cite{kul} \textit{relative information} studied in a different
way by Cressie and Read \cite{crr} is given by
\begin{equation}
\label{eq14} \Phi _s (P\vert \vert Q) = \begin{cases}
 {K_s (P\vert \vert Q) = \left[ {s(s - 1)} \right]^{ - 1}\left[
{\sum\limits_{i = 1}^n {p_i^s q_i^{1 - s} } - 1} \right],} & {s \ne 0,1} \\
 {K(Q\vert \vert P) = \sum\limits_{i = 1}^n {q_i \ln \left( {\frac{q_i }{p_i
}} \right)} ,} & {s = 0} \\
 {K(P\vert \vert Q) = \sum\limits_{i = 1}^n {p_i \ln \left( {\frac{p_i }{q_i
}} \right)} ,} & {s = 1} \\
\end{cases},
\end{equation}

\noindent for all $P,Q \in \Gamma _n $ and $s \in (-\infty, \infty)$.\\

The measure (\ref{eq14}) admits the following particular cases:
\begin{itemize}
\item[(i)] $\Phi _{ - 1} (P\vert \vert Q) = \frac{1}{2}\chi
^2(Q\vert \vert P).$

\item[(ii)] $ \Phi _0 (P\vert \vert Q) = K(Q\vert \vert P).$

\item[(iii)] $\Phi _{1 / 2} (P\vert \vert Q) = 4\left[ {1 -
B(P\vert \vert Q)} \right] = 4\,h(P\vert \vert Q)$

\item[(iv)] $\Phi _1 (P\vert \vert Q) = K(P\vert \vert Q).$

\item[(v)] $\Phi _2 (P\vert \vert Q) = \frac{1}{2}\chi ^2(P\vert
\vert Q).$
\end{itemize}

\bigskip
The measures $B(P\vert \vert Q)$, $h(P\vert \vert Q)$ and $\chi
^2(P\vert \vert Q)$ appearing in parts (iii) and (v) above, are
given by
\begin{equation}
\label{eq15}
B(P\vert \vert Q) = \sqrt {p_i q_i } ,
\end{equation}
\begin{equation}
\label{eq16}
h(P\vert \vert Q) = 1 - B(P\vert \vert Q) = \frac{1}{2}\sum\limits_{i = 1}^n
{(\sqrt {p_i } - \sqrt {q_i } )^2}
\end{equation}

\noindent and
\begin{equation}
\label{eq17}
\chi ^2(P\vert \vert Q) = \sum\limits_{i = 1}^n {\frac{(p_i - q_i )^2}{q_i
}} = \sum\limits_{i = 1}^n {\frac{p_i^2 }{q_i } - 1}
\end{equation}

\noindent respectively.

The measure $B(P\vert \vert Q)$ is famous as Bhattacharya
\cite{bha} \textit{coefficient}, the measure $h(P\vert \vert Q)$
is famous as Hellinger \cite{hel} \textit{discrimination} and the
measures $\chi ^2(P\vert \vert Q)$ is known by \textit{Chi-square}
\cite{pea} \textit{divergence}.

For more studies on the measure (\ref{eq14}) refer to Liese and
Vajda \cite{liv}, Taneja \cite{tan4}, Taneja and Kumar \cite{tak}
and Cerone et al. \cite{cdo}.

Our aim in this paper is to obtain bounds on the \textit{relative
divergence measures} that we shall classify as
\textit{non-symmetric divergence measures} given by
(\ref{eq4})-(\ref{eq7}) and on the \textit{divergence measures
}classifying as \textit{symmetric divergence measures} given by
(\ref{eq2}), (\ref{eq10}) and (\ref{eq11}) in terms of
\textit{generalized relative information} or \textit{relative
information of type s} given by (\ref{eq14}). These bounds are
studied by use of some properties of Csisz\'{a}r \cite{csi1}
\textit{f-divergence}.

\section{Csisz\'{a}r $f-$Divergence }

Given a convex function $f:[0,\infty ) \to \mathbb{R}$, the $f -
$divergence measure introduced by Csisz\'{a}r \cite{csi1} is given
by
\begin{equation}
\label{eq18}
C_f (P\vert \vert Q) =
\sum\limits_{i = 1}^n {q_i f\left( {\frac{p_i }{q_i }} \right)} ,
\end{equation}

\noindent where $P,Q \in \Gamma _n $.

It is well known in the literature \cite{csi1, csi2} that
\textit{if } $f$ \textit{ is convex and normalized, i.e., }$f(1) =
0,$ \textit{then the Csisz\'{a}r function }$C_f (P\vert \vert
Q)$\textit{ is nonnegative and convex in the pair of probability
distribution }$(P,Q) \in \Gamma _n \times \Gamma _n .$

Now we shall prove the convexity and nonnegativity of the measures
given in Section 1.

\begin{example} \label{exa21} (\textit{Relative J-divergence}). Let us
consider
\begin{equation}
\label{eq19} f_{D_1 } (x) = (x - 1)\ln \left( {\frac{x + 1}{2}}
\right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets $C_f (P\vert \vert Q) =
D(P\vert \vert Q): = D_1 $, where $D\left( {P\vert \vert Q}
\right)$ is as given by (\ref{eq4}).

Moreover,
\begin{equation}
\label{eq20}
{f}'_{D_1 } (x) = \frac{x - 1}{x + 1} + \ln \left( {\frac{x + 1}{2}}
\right)
\end{equation}

\noindent and
\begin{equation}
\label{eq21}
{f}''_{D_1 } (x) = \frac{x + 3}{(x + 1)^2}
\end{equation}

Thus from (\ref{eq21}) we see that $f_{D_1 } ^{\prime \prime }(x)
> 0$ for all $x > 0$, and hence, $f_{D_1 } (x)$ is \textit{convex}
for all $x > 0$. Also, we have $f_{D_1 } (1) = 0$. In view of this
we can say that the \textit{relative J-divergence} is
\textit{nonnegative} and \textit{convex} in the pair of
probability distributions $(P,Q) \in \Gamma _n \times \Gamma _n $.
\end{example}

\begin{example} \label{exa22} (\textit{Adjoint of relative J-divergence}).
Let us consider
\begin{equation}
\label{eq22} f_{D_2 } (x) = (1 - x)\ln \left( {\frac{x + 1}{2x}}
\right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
D(Q\vert \vert P): = D_2 ,$ where $D(Q\vert \vert P)$ is as given
by (\ref{eq5}).

Moreover,
\begin{equation}
\label{eq23}
{f}'_{D_2 } (x) = \frac{x - 1}{x(x + 1)} - \ln \left( {\frac{x + 1}{2x}}
\right),
\end{equation}

\noindent and
\begin{equation}
\label{eq24}
{f}''_{D_2 } (x) = \frac{3x + 1}{x^2(x + 1)^2}
\end{equation}

Thus from (\ref{eq24}) we see that ${f}''_{D_2 } (x) > 0$ for all
$x > 0$, and hence, $f_{D_2 } (x)$ is \textit{convex} for all $x >
0$. Also, we have $f_{D_2 } (1) = 0$. In view of this we can say
that the  \textit{adjoiont of relative J-divergence} is
\textit{nonnegative} and \textit{convex} in the pair of
probability distributions $(P,Q) \in \Gamma _n \times \Gamma _n $.
\end{example}

\begin{example} \label{exa23} (\textit{Relative JS-divergence}). Let us
consider
\begin{equation}
\label{eq25} f_{F_1 } (x) = \frac{1 - x}{2} - x\ln \left( {\frac{x
+ 1}{2x}} \right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
F(P\vert \vert Q): = F_1 $, where $F\left( {P\vert \vert Q}
\right)$ is as given by (\ref{eq6}).

Moreover,
\begin{equation}
\label{eq26}
{f}'_{F_1 } (x) = \frac{1 - x}{2(x + 1)} - \ln \left( {\frac{x + 1}{2x}}
\right),
\end{equation}

\noindent and
\begin{equation}
\label{eq27}
{f}''_{F_1 } (x) = \frac{1}{x(x + 1)^2}
\end{equation}

Thus from (\ref{eq27}) we see that ${f}''_{F_1 } (x) > 0$ for all
$x > 0$, and hence, $f_{F_1 } (x)$ is \textit{convex} for all $x >
0$. Also, we have $f_{F_1 } (1) = 0$. In view of this we can say
that the  \textit{relative JS-divergence} is \textit{nonnegative}
and \textit{convex} in the pair of probability distributions
$(P,Q) \in \Gamma _n \times \Gamma _n $.
\end{example}

\begin{example} \label{exa24} (\textit{Adjoint of relative
JS-divergence}). Let us consider
\begin{equation}
\label{eq28} f_{F_2 } (x) = \frac{x - 1}{2} - \ln \left( {\frac{x
+ 1}{2}} \right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
F(Q\vert \vert P): = F_2 ,$ where $F(Q\vert \vert P)$ is as given
by (\ref{eq8}).

Moreover,
\begin{equation}
\label{eq29}
{f}'_{F_2 } (x) = \frac{x - 1}{2(x + 1)}
\end{equation}

\noindent and
\begin{equation}
\label{eq30}
{f}''_{F_2 } (x) = \frac{1}{(x + 1)^2}
\end{equation}

Thus from (\ref{eq30}) we see that ${f}''_{F_2 } (x) > 0$ for all
$x > 0$, and hence, $f_{F_2 } (x)$ is \textit{convex} for all $x >
0$. Also, we have $f_{F_2 } (1) = 0$. In view of this we can say
that the  \textit{adjoint of relative JS-divergence }is
\textit{nonnegative} and \textit{convex} in the pair of
probability distributions $(P,Q) \in \Gamma _n \times \Gamma _n $.
\end{example}

\begin{example} \label{exa25} (\textit{Relative AG-divergence}). Let us
consider
\begin{equation}
\label{eq31} f_{G_1 } (x) = \frac{x - 1}{2} + \frac{x + 1}{2}\ln
\left( {\frac{x + 1}{2x}} \right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
G(P\vert \vert Q): = G_1 $, where $G\left( {P\vert \vert Q}
\right)$ is as given by (\ref{eq7}).

Moreover,
\begin{equation}
\label{eq32}
{f}'_{G_1 } (x) = \frac{1}{2}\left[ {\frac{x - 1}{x} + \ln \left( {\frac{x +
1}{2x}} \right)} \right],
\end{equation}

\noindent and
\begin{equation}
\label{eq33}
{f}''_{G_1 } (x) = \frac{1}{2x^2(x + 1)}
\end{equation}

Thus from (\ref{eq33}) we see that ${f}''_{G_1 } (x) > 0$ for all
$x > 0$, and hence, $f_{G_1 } (x)$ is \textit{convex} for all $x >
0$. Also, we have $f_{G_1 } (1) = 0$. In view of this we can say
that the  \textit{relative AG-divergence} is \textit{nonnegative}
and \textit{convex} in the pair of probability distributions
$(P,Q) \in \Gamma _n \times \Gamma _n $.
\end{example}

\begin{example} \label{exa26} (\textit{Adjoint of relative
AG-divergence}). Let us consider
\begin{equation}
\label{eq34} f_{G_2 } (x) = \frac{1 - x}{2} + \frac{x + 1}{2}\ln
\left( {\frac{x + 1}{2}} \right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
G(Q\vert \vert P): = G_2 $, where $G\left( {Q\vert \vert P}
\right)$ is as given by (\ref{eq9}).

Moreover,
\begin{equation}
\label{eq35}
{f}'_{G_2 } (x) = \frac{1}{2}\ln \left( {\frac{x + 1}{2}} \right),
\end{equation}

\noindent and
\begin{equation}
\label{eq36}
{f}''_{G_2 } (x) = \frac{1}{2(x + 1)}
\end{equation}

Thus from (\ref{eq36}) we see that ${f}''_{G_2 } (x) > 0$ for all
$x > 0$, and hence, $f_{G_2 } (x)$ is \textit{convex} for all $x >
0$. Also, we have $f_{G_2 } (1) = 0$. In view of this we can say
that the  \textit{adjoint of relative AG-divergence} is
\textit{nonnegative} and \textit{convex} in the pair of
probability distributions $(P,Q) \in \Gamma _n \times \Gamma _n $.
\end{example}

\begin{example} \label{exa27} (\textit{J-divergence}). Let us consider
\begin{equation}
\label{eq37} f_J (x) = (x - 1)\ln x, \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
J(P\vert \vert Q)$, where $J\left( {P\vert \vert Q} \right)$ is as
given by (\ref{eq2}).

Moreover,
\begin{equation}
\label{eq38}
{f}'_J (x) = 1 - x^{ - 1} + \ln x,
\end{equation}

\noindent and
\begin{equation}
\label{eq39}
{f}''_J (x) = \frac{x + 1}{x^2}
\end{equation}

Thus from (\ref{eq39}) we see that ${f}''_J (x) > 0$ for all $x >
0$, and hence, $f_J (x)$ is \textit{convex} for all $x > 0$. Also,
we have $f_J (1) = 0$. In view of this we can say that the
\textit{J-divergence }is \textit{nonnegative} and \textit{convex}
in the pair of probability distributions $(P,Q) \in \Gamma _n
\times \Gamma _n $.
\end{example}

\begin{example}\label{exa28} (\textit{JS-divergence}). Let us consider
\begin{equation}
\label{eq40} f_I (x) = \frac{x}{2}\ln x - \frac{x + 1}{2}\ln
\left( {\frac{x + 1}{2}} \right), \,\, x \in (0,\infty )
\end{equation}

\noindent in (\ref{eq18}), then one gets  $C_f (P\vert \vert Q) =
I(P\vert \vert Q)$, where $I\left( {P\vert \vert Q} \right)$ is as
given by (\ref{eq10}).

Moreover,
\begin{equation}
\label{eq41}
{f}'_I (x) = - \frac{1}{2}\ln \left( {\frac{x + 1}{2x}} \right),
\end{equation}

\noindent and
\begin{equation}
\label{eq42}
{f}''_I (x) = \frac{1}{2x(x + 1)}
\end{equation}

Thus from (\ref{eq42}) we see that ${f}''_I (x) > 0$ for all $x >
0$, and hence, $f_I (x)$ is \textit{convex} for all $x > 0$. Also,
we have $f_I (1) = 0$. In view of this we can say that the
\textit{JS-divergence }is \textit{nonnegative} and \textit{convex}
in the pair of probability distributions $(P,Q) \in \Gamma _n
\times \Gamma _n $.
\end{example}

\begin{example} \label{exa29} (\textit{AG-divergence}). Let us consider
\begin{equation}
\label{eq43} f_T (x) = \left( {\frac{x + 1}{2}} \right)\ln \left(
{\frac{x + 1}{2\sqrt x }} \right), \,\, x \in (0,\infty ),
\end{equation}

\noindent in (\ref{eq18}), then one gets $C_f (P\vert \vert Q) =
T(P\vert \vert Q),$ where $T(P\vert \vert Q)$ is as given by
(\ref{eq11}).

Moreover,
\begin{equation}
\label{eq44}
{f}'_T (x) = \frac{1}{4}\left[ {1 - x^{ - 1} + 2\ln \left( {\frac{x +
1}{2\sqrt x }} \right)} \right],
\end{equation}

\noindent and
\begin{equation}
\label{eq45}
{f}''_T (x) = \frac{1}{4}\left( {\frac{x^2 + 1}{x^3 + x^2}} \right).
\end{equation}

Thus from (\ref{eq45}) we see that ${f}''_T (x) > 0$ for all $x >
0$, and hence, $f_T (x)$ is \textit{convex} for all $x > 0$. Also,
we have $f_T (1) = 0$. In view of this we can say that the
\textit{AG-divergence} is \textit{nonnegative} and \textit{convex}
in the pair of probability distributions $(P,Q) \in \Gamma _n
\times \Gamma _n $.
\end{example}

The above examples give only the \textit{nonnegativity} and
\textit{convexity} of the \textit{symmetric} and
\textit{non-symmetric divergence measures}. Here we shall make use
of this property to get bounds in terms of \textit{relative
information of type s}. For more properties of these measures
refer to Taneja \cite{tan7}.

\section{Csisz\'{a}r $f-$Divergence and Relative
Information of Type $s$}

The following two theorems are due to Taneja \cite{tan4} and
Taneja and Kumar \cite{tak}.

\begin{theorem} \label{the31} Let $P,Q \in \Gamma _n $ and $s \in \mathbb{R}: = ( -
\infty ,\infty )$, then we have
\begin{equation}
\label{eq46} 0 \leqslant \Phi _s (P\vert \vert Q) \leqslant
E_{\Phi _s } (P\vert \vert Q),
\end{equation}

\noindent where
\begin{equation}
\label{eq47} E_{\Phi _s} (P\vert \vert Q) = \begin{cases}
 {(s - 1)^{ - 1}\sum\limits_{i = 1}^n {(p_i - q_i )\left( {\frac{p_i }{q_i
}} \right)^{s - 1},} } & {s \ne 1} \\
 {\sum\limits_{i = 1}^n {(p_i - q_i )\ln \left( {\frac{p_i }{q_i }}
\right),} } & {s = 1} \\
\end{cases}.
\end{equation}

Let $P,Q \in \Gamma _n $ be such that there exists $r,R$ with $0 <
r \leqslant \frac{p_i }{q_i } \leqslant R < \infty $, $\forall i
\in \{1,2,...,n\}$, then
\begin{equation}
\label{eq48} 0 \leqslant \Phi _s (P\vert \vert Q) \leqslant
A_{\Phi _s } (r,R)
\end{equation}

\noindent where
\begin{equation}
\label{eq49} A_{\Phi _s } (r,R) = \frac{1}{4}(R - r)^2\mbox{
}\begin{cases}
 {\frac{R^{s - 1} - r^{s - 1}}{(R - r)(s - 1)},} & {s \ne 1} \\
 {\frac{\ln R - \ln r}{R - r},} & {s = 1} \\
\end{cases}
\end{equation}

Futher, if we suppose that $0 < r \leqslant 1 \leqslant R < \infty
$, $r \ne R$, then
\begin{equation}
\label{eq50} 0 \leqslant \Phi _s (P\vert \vert Q) \leqslant
B_{\Phi _s} (r,R)
\end{equation}

\noindent where
\begin{equation}
\label{eq51} B_{\Phi _s } (r,R) = \begin{cases}
 {\frac{(R - 1)(r^s - 1) + (1 - r)(R^s - 1)}{(R - r)s(s - 1)},} & {s \ne
0,1} \\
 {\frac{(R - 1)\ln \frac{1}{r} + (1 - r)\ln \frac{1}{R}}{(R - r)},} & {s =
0} \\
 {\frac{(R - 1)r\ln r + (1 - r)R\ln R}{(R - r)},} & {s = 1} \\
\end{cases}
\end{equation}

Moreover, the following inequalities hold:
\begin{equation}
\label{eq52} E_{\Phi _s } (P\vert \vert Q) \leqslant A_{\Phi _s }
(r,R),
\end{equation}
\begin{equation}
\label{eq53} B_{\Phi _s } (r,R) \leqslant A_{\Phi _s } (r,R)
\end{equation}

\noindent and
\begin{equation}
\label{eq54} B_{\Phi _s } (r,R) - \Phi _s (P\vert \vert Q)
\leqslant A_{\Phi _s } (r,R).
\end{equation}
\end{theorem}

\begin{theorem} \label{the32}  Let $f:I \subset \mathbb{R}_ + \to
\mathbb{R}$ the generating mapping is normalized, i.e., $f(1) = 0$
and satisfy the assumptions:
\begin{itemize}

\item[(i)] $f$is twice differentiable on $(r,R)$;

\item[(ii)] there exists real constants $m,M$ such that $0 < m <
M$ and
\end{itemize}
\begin{equation}
\label{eq55} m \leqslant x^{2 - s}{f}''(x) \leqslant M, \quad
\forall x \in (r,R), \quad
 - \infty < s < \infty .
\end{equation}

\noindent then, we have
\begin{equation}
\label{eq56} m \Phi _s (P\vert \vert Q)
 \leqslant C_f (P\vert \vert Q) \leqslant M
\Phi _s (P\vert \vert Q)
\end{equation}

\noindent and
\begin{align}
\label{eq57} m\left[ {E_{\Phi _s } (P\vert \vert Q) - \Phi _s
(P\vert \vert Q)} \right] & \leqslant E_{C_f } (P\vert \vert Q) -
C_f (P\vert \vert Q)\\
& \leqslant M\left[ {E_{\Phi _s } (P\vert \vert Q) - \Phi _s
(P\vert \vert Q)} \right].\notag
\end{align}

Let $P,Q \in \Gamma _n $ be such that there exists $r,R$ with $0 <
r \leqslant \frac{p_i }{q_i } \leqslant R < \infty $, $\forall i
\in \{1,2,...,n\}$, then
\begin{align}
\label{eq58} m\left[ {A_{\Phi _s } (r,R) - \Phi _s (P\vert \vert
Q)} \right] & \leqslant A_{C_f } (r,R) - C_f (P\vert \vert Q)\\
& \leqslant M\left[ {A_{\Phi _s } (r,R) - \Phi _s (P\vert \vert
Q)} \right]\notag
\end{align}

Further, if we suppose that $0 < r \leqslant 1 \leqslant R <
\infty $, $r \ne R$, then
\begin{align}
\label{eq59} m\left[ {B_{\Phi _s } (r,R) - \Phi _s (P\vert \vert
Q)} \right] & \leqslant B_{C_f } (r,R) - C_f (P\vert \vert Q)\\
& \leqslant M\left[ {B_{\Phi _s } (r,R) - \Phi _s (P\vert \vert
Q)} \right].\notag
\end{align}
\end{theorem}

The Theorem \ref{the31} is obtained by applying some of the
results due to Dragomir \cite{dra1, dra2}. The Theorem \ref{the32}
unifies some of the results studied by Dragomir \cite{dra3, dra4,
dra5}. For an improved version of Theorem \ref{the32} refers to
Taneja \cite{tan5}.

The aim here is to apply Theorem \ref{the32} by taking different
values of $f$ given by examples \ref{exa21}-\ref{exa29}. This we
have done only applying the inequalities (\ref{eq56}), while the
results for the inequalities (\ref{eq57})-(\ref{eq59}) can be done
on similar lines. These details are omitted here.

\section{Bounds On Non-Symmetric Divergence Measures}

In this section, we have applied the inequalities (\ref{eq56}) and
used the condition (\ref{eq55}) to obtain bounds for the measures
given in (\ref{eq4})-(\ref{eq9}).

\begin{theorem} \label{the41} The following bounds on
\textit{relative J-divergence} hold:
\begin{equation}
\label{eq60}
\frac{r^{2 - s}(r + 3)}{(r + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant D(P\vert \vert Q) \leqslant
\frac{R^{2 - s}(R + 3)}{(R + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\leqslant \frac{3}{4}
\end{equation}

\noindent and
\begin{equation}
\label{eq61}
\frac{R^{2 - s}(R + 3)}{(R + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant D(P\vert \vert Q) \leqslant
\frac{r^{2 - s}(r + 3)}{(r + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\geqslant 2.
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq62} g_{D_1 } (x) = x^{2 - s}{f}''_{D_1 } (x) = \frac{x^{2
- s}(x + 3)}{(x + 1)^2}, \,\, x \in (0,\infty ),
\end{equation}

\noindent where ${f}''_{D_1 } (x)$ is as given by (\ref{eq21}).

From (\ref{eq62}), one can get
\begin{equation}
\label{eq63}
{g}'_{D_1 } (x) = - \frac{x^{1 - s}\left[ {(s - 1)x^2 + (4s - 3)x + 3(s -
2)} \right]}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {s \leqslant \tfrac{3}{4}} \\
 { \leqslant 0,} & {s \geqslant 2} \\
\end{cases}.
\end{equation}

In view of (\ref{eq63}), we conclude that
\begin{equation}
\label{eq64} m = \mathop {\inf }\limits_{x \in [r,R]} g_{D_1 } (x)
= \begin{cases}
 {\frac{r^{2 - s}(r + 3)}{(r + 1)^2},} & {s \leqslant \tfrac{3}{4}} \\
 {\frac{R^{2 - s}(R + 3)}{(R + 1)^2},} & {s \geqslant 2} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq65} M = \mathop {\sup }\limits_{x \in [r,R]} g_{D_1 } (x)
= \begin{cases}
 {\frac{R^{2 - s}(R + 3)}{(R + 1)^2},} & {s \leqslant \tfrac{3}{4}} \\
 {\frac{r^{2 - s}(r + 3)}{(r + 1)^2},} & {s \geqslant 2} \\
\end{cases}.
\end{equation}

In view of (\ref{eq64}), (\ref{eq65}) and (\ref{eq56}), we get the
inequalities (\ref{eq60}) and (\ref{eq61}).
\end{proof}

Some particular cases of the Theorem \ref{the41} are summarized in
the following corollary.

\begin{corollary} \label{cor41} The following bounds hold:
\begin{equation}
\label{eq66}
\frac{r^3(r + 3)}{2(r + 1)^2}\chi ^2(Q\vert \vert P) \leqslant D(P\vert
\vert Q) \leqslant \frac{R^3(R + 3)}{2(R + 1)^2}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq67}
\frac{r^2(r + 3)}{(r + 1)^2}K(Q\vert \vert P) \leqslant D(P\vert \vert Q)
\leqslant \frac{R^2(R + 3)}{(R + 1)^2}K(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq68}
\frac{4r^{3 / 2}(r + 3)}{(r + 1)^2}h(P\vert \vert Q) \leqslant D(P\vert
\vert Q) \leqslant \frac{4R^{3 / 2}(R + 3)}{(R + 1)^2}h(P\vert \vert Q)
\end{equation}

\noindent and
\begin{equation}
\label{eq69}
\frac{R + 3}{2(R + 1)^2}\chi ^2(P\vert \vert Q) \leqslant D(P\vert \vert Q)
\leqslant \frac{r + 3}{2(r + 1)^2}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq66}), (\ref{eq67}) and
(\ref{eq68}) follows from (\ref{eq60}) by taking $s = - 1$, $s =
0$ and $s = \frac{1}{2}$ respectively. The inequalities
(\ref{eq69}) follow form (\ref{eq61}) by taking $s = 2$.
\end{proof}

The case $s = 1$ is not included in the inequalities (\ref{eq60})
and (\ref{eq61}). This we shall do separately in the following
proposition.

\begin{proposition} \label{pro41} The following inequality hold:
\begin{equation} \label{eq70} D(P\vert \vert Q) \leqslant
\frac{9}{8}K(P\vert \vert Q).
\end{equation}
\end{proposition}

\begin{proof} For $s = 1$ in (\ref{eq62}), we have
\begin{equation}
\label{eq71}
g_D (x) = \frac{x(x + 3)}{(x + 1)^2}.
\end{equation}

This gives
\begin{equation}
\label{eq72}
{g}'_D (x) = - \frac{x - 3}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {x \leqslant 3} \\
 { \leqslant 0,} & {x \geqslant 3} \\
\end{cases}.
\end{equation}

Thus we conclude from (\ref{eq72}) that the function $g_D (x)$
given by (\ref{eq71}) is increasing in $x \in (0,3)$ and
decreasing in $x \in (3,\infty )$, and hence
\begin{equation}
\label{eq73} M = \mathop {\sup }\limits_{x \in (0,\infty )} g_D
(x) = \frac{9}{8}.
\end{equation}

Now (\ref{eq73}) together with (\ref{eq56}) give the required
result.
\end{proof}

\begin{theorem} \label{the42} The following bounds on \textit{adjoint
}of\textit{ relative J-divergence} holds:
\begin{equation}
\label{eq74}
\frac{r^{ - s}(3r + 1)}{(r + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant D(Q\vert \vert P) \leqslant
\frac{R^{ - s}(3R + 1)}{(R + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\leqslant - 1
\end{equation}

\noindent and
\begin{equation}
\label{eq75}
\frac{R^{ - s}(3R + 1)}{(R + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant D(Q\vert \vert P) \leqslant
\frac{r^{ - s}(3r + 1)}{(r + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\geqslant \frac{1}{4}
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq76} g_{D_2 } (x) = x^{2 - s}{f}''_{D_2 } (x) = \frac{x^{
- s}(3x + 1)}{(x + 1)^2}, \,\, x \in (0,\infty ),
\end{equation}

\noindent where ${f}''_{D_2 } (x)$ is as given by (\ref{eq24}).

From (\ref{eq76}), one can get
\begin{equation}
\label{eq77}
{g}'_D (x) = - \frac{x^{ - s}\left[ {3(s + 1)x^2 + (4s - 1)x + s}
\right]}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {s \leqslant - 1} \\
 { \leqslant 0,} & {s \geqslant \frac{1}{4}} \\
\end{cases}.
\end{equation}

In view of (\ref{eq77}), we conclude that
\begin{equation}
\label{eq78} m = \mathop {\inf }\limits_{x \in [r,R]} g_D (x) =
\mathop {\min }\limits_{x \in [r,R]} g_D (x) = \begin{cases}
 {\frac{r^{ - s}(3r + 1)}{(r + 1)^2},} & {s \leqslant - 1} \\
 {\frac{R^{ - s}(3R + 1)}{(R + 1)^2},} & {s \geqslant \frac{1}{4}} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq79} M = \mathop {\sup }\limits_{x \in [r,R]} g_D (x) =
\begin{cases}
 {\frac{R^{ - s}(3R + 1)}{(R + 1)^2},} & {s \leqslant - 1} \\
 {\frac{r^{ - s}(3r + 1)}{(r + 1)^2},} & {s \geqslant \frac{1}{4}} \\
\end{cases}.
\end{equation}

Now from (\ref{eq78}), (\ref{eq79}) and (\ref{eq56}), we get the
inequalities (\ref{eq74}) and (\ref{eq75}).
\end{proof}

Some particular cases of the Theorem \ref{the42} are summarized in
the following corollary.

\begin{corollary} \label{cor42} The following bounds hold:
\begin{equation}
\label{eq80}
\frac{r(3r + 1)}{(r + 1)^2}\chi ^2(Q\vert \vert P) \leqslant D(Q\vert \vert
P) \leqslant \frac{R(3R + 1)}{(R + 1)^2}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq81}
\frac{4(3R + 1)}{\sqrt R (R + 1)^2}h(P\vert \vert Q) \leqslant D(Q\vert
\vert P) \leqslant \frac{4(3r + 1)}{\sqrt r (r + 1)^2}h(P\vert \vert Q),
\end{equation}
\begin{equation}
\label{eq82}
\frac{3R + 1}{R(R + 1)^2}K(P\vert \vert Q) \leqslant D(Q\vert \vert P)
\leqslant \frac{3R + 1}{R(R + 1)^2}K(P\vert \vert Q)
\end{equation}

\noindent and
\begin{equation}
\label{eq83}
\frac{3R + 1}{R^2(R + 1)^2}\chi ^2(P\vert \vert Q) \leqslant D(Q\vert \vert
P) \leqslant \frac{3r + 1}{r^2(r + 1)^2}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq80}) follows from
(\ref{eq74}) by taking $s = - 1$. The inequalities (\ref{eq81}),
(\ref{eq82}) and (\ref{eq83}) follows from (\ref{eq75}) by taking
$s = \frac{1}{2}$, $s = 1$ and $s = 2$ respectively.
\end{proof}

The case $s = 0$ is not included in the inequalities (\ref{eq74})
and (\ref{eq75}). This we shall do separately in the following
proposition.

\begin{proposition} \label{pro42} The following bound hold:
\begin{equation}
\label{eq84} D(Q\vert \vert P) \leqslant \frac{9}{8}K(Q\vert \vert
P),
\end{equation}
\end{proposition}

\begin{proof} For $s = 0$ in (\ref{eq76}), we have
\begin{equation}
\label{eq85}
g_{D_2 } (x) = \frac{3x + 1}{(x + 1)^2}.
\end{equation}

This gives
\begin{equation}
\label{eq86}
{g}'_{D_2 } (x) = - \frac{3x - 1}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {x \leqslant \frac{1}{3}} \\
 { \leqslant 0,} & {x \geqslant \frac{1}{3}} \\
\end{cases}.
\end{equation}

Thus from (\ref{eq86}) we conclude that the function $g_{D_2 }
(x)$ given by (\ref{eq85}) is increasing in $x \in
(0,\frac{1}{3})$ and decreasing in $x \in (\frac{1}{3},\infty )$,
and hence
\begin{equation}
\label{eq87}
M = \mathop {\sup }\limits_{x \in (0,\infty )} g_{D_2 } (x) = g_{D_2 }
(\frac{1}{3}) = \frac{9}{8}.
\end{equation}

Now (\ref{eq87}) together with (\ref{eq56}) give the required
result.
\end{proof}

\begin{theorem} \label{the43} The following bounds on \textit{relative
JS-divergence} hold:
\begin{equation}
\label{eq88}
\frac{r^{1 - s}}{(r + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant F(P\vert \vert Q) \leqslant
\frac{R^{1 - s}}{(R + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\leqslant - 1
\end{equation}

\noindent and
\begin{equation}
\label{eq89}
\frac{R^{1 - s}}{(R + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant F(P\vert \vert Q) \leqslant
\frac{r^{1 - s}}{(r + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\geqslant 1.
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq90} g_{F_1 } (x) = x^{2 - s}{f}''_{F_1 } (x) = \frac{x^{1
- s}}{(x + 1)^2}, \,\, x \in (0,\infty )
\end{equation}

\noindent where ${f}''_{F_1 } (x)$ is as given by (\ref{eq27}).

From (\ref{eq90}) one can get
\begin{equation}
\label{eq91}
{g}'_{F_1 } (x) = - \frac{x^{ - s}\left[ {(s + 1)x + (s - 1)} \right]}{(x +
1)^3}
\begin{cases}
 { \geqslant 0,} & {s \leqslant - 1} \\
 { \leqslant 0,} & {s \geqslant 1} \\
\end{cases}.
\end{equation}

In view of (\ref{eq91}), we conclude that
\begin{equation}
\label{eq92} m = \mathop {\inf }\limits_{x \in [r,R]} g_{F_1 } (x)
= \begin{cases}
 {\frac{r^{1 - s}}{(r + 1)^2},} & {s \leqslant - 1} \\
 {\frac{R^{1 - s}}{(R + 1)^2},} & {s \geqslant 1} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq93} M = \mathop {\sup }\limits_{x \in [r,R]} g_{F_1 } (x)
= \begin{cases}
 {\frac{R^{1 - s}}{(R + 1)^2},} & {s \leqslant - 1} \\
 {\frac{r^{1 - s}}{(r + 1)^2},} & {s \geqslant 1} \\
\end{cases}.
\end{equation}

Now (\ref{eq92}) and (\ref{eq93}) together with (\ref{eq56}) give
the required result.
\end{proof}

Some particular cases of the Theorem \ref{the43} are summarized in
the following corollary.

\begin{corollary} \label{cor43} The following bounds hold:
\begin{equation}
\label{eq94}
\frac{r^2}{2(r + 1)^2}\chi ^2(Q\vert \vert P) \leqslant F(P\vert \vert Q)
\leqslant \frac{R^2}{2(R + 1)^2}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq95}
\frac{1}{(R + 1)^2}K(P\vert \vert Q) \leqslant F(P\vert \vert Q) \leqslant
\frac{1}{(r + 1)^2}K(P\vert \vert Q)
\end{equation}

\noindent and
\begin{equation}
\label{eq96}
\frac{1}{2R(R + 1)^2}\chi ^2(P\vert \vert Q) \leqslant F(P\vert \vert Q)
\leqslant \frac{1}{2r(r + 1)^2}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq94}) follows from
(\ref{eq88}) by taking $s = - 1$. The inequalities (\ref{eq95})
and (\ref{eq96}) follows from (\ref{eq89}) by taking $s = 1$ and
$s = 2$ respectively.
\end{proof}

The cases $s = 0$ and $s = \frac{1}{2}$ are not included in the
inequalities (\ref{eq88}) and (\ref{eq89}). This we shall do
separately in the following proposition.

\begin{proposition} \label{pro43} The following bounds hold:
\begin{equation}
\label{eq97}
F(P\vert \vert Q) \leqslant \frac{1}{4}K(Q\vert \vert P)
\end{equation}

\noindent and
\begin{equation}
\label{eq98}
F(P\vert \vert Q) \leqslant \frac{3\sqrt 3 }{4}h(P\vert \vert Q).
\end{equation}
\end{proposition}

\begin{proof} For $s = 0$ in (\ref{eq90}), we have
\begin{equation}
\label{eq99}
g_{F_1 } (x) = \frac{x}{(x + 1)^2}.
\end{equation}

This gives
\begin{equation}
\label{eq100}
{g}'_{F_1 } (x) = - \frac{x - 1}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {x \leqslant 1} \\
 { \leqslant 0,} & {x \geqslant 1} \\
\end{cases}.
\end{equation}

From (\ref{eq100}) we conclude that the function $g_{F_1 } (x)$
given by (\ref{eq100}) is increasing in $x \in (0,1)$ and
decreasing in $x \in (1,\infty )$, and hence
\begin{equation}
\label{eq101}
M = \mathop {\sup }\limits_{x \in (0,\infty )} g_{F_1 } (x) = g_{F_1 } (1) =
\frac{1}{4}.
\end{equation}

Now (\ref{eq101}) together with (\ref{eq56}) give the inequality
(\ref{eq97}).

Again let us take $s = \frac{1}{2}$ in (\ref{eq90}), we have
\begin{equation}
\label{eq102}
g_{F_1 } (x) = \frac{\sqrt x }{(x + 1)^2}.
\end{equation}

This gives
\begin{equation}
\label{eq103}
{g}'_{F_1 } (x) = - \frac{3x - 1}{2\sqrt x (x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {x \leqslant \frac{1}{3}} \\
 { \leqslant 0,} & {x \geqslant \frac{1}{3}} \\
\end{cases}.
\end{equation}

Thus from (\ref{eq103}), we conclude that the function $g_{F_1 }
(x)$ given by (\ref{eq102}) is increasing in $x \in
(0,\frac{1}{3})$ and decreasing in $x \in (\frac{1}{3},\infty )$,
and hence
\begin{equation}
\label{eq104}
M = \mathop {\sup }\limits_{x \in (0,\infty )} g_{F_1 } (x) = g_{F_1 }
(\frac{1}{3}) = \frac{3\sqrt 3 }{16}.
\end{equation}

Now (\ref{eq104}) together with (\ref{eq56}) give the inequalities
(\ref{eq98}).
\end{proof}

\begin{theorem} \label{the44} The following bounds on \textit{adjoint
} of \textit{relative JS-divergence} hold:
\begin{equation}
\label{eq105}
\frac{r^{2 - s}}{(r + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant F(Q\vert \vert P) \leqslant
\frac{R^{2 - s}}{(R + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\leqslant 0
\end{equation}

\noindent and
\begin{equation}
\label{eq106}
\frac{R^{2 - s}}{(R + 1)^2}\Phi _s (P\vert \vert Q)
 \leqslant F(Q\vert \vert P) \leqslant
\frac{r^{2 - s}}{(r + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\geqslant 2
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq107} g_{F_2 } (x) = x^{2 - s}{f}''_{F_2 } (x) =
\frac{x^{2 - s}}{(x + 1)^2}, \,\, x \in (0,\infty )
\end{equation}

\noindent where ${f}''_{F_2 } (x)$ is as given by (\ref{eq30}).

From (\ref{eq107}) one can get
\begin{equation}
\label{eq108} {g}'_{F_2 } (x) = - \frac{x^{1 - s}\left[ {sx + (s -
2)} \right]}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {s \leqslant 0} \\
 { \leqslant 0,} & {s \geqslant 2} \\
\end{cases}.
\end{equation}

In view of (\ref{eq108}), we conclude that
\begin{equation}
\label{eq109} m = \mathop {\inf }\limits_{x \in [r,R]} g_{F_2} (x)
=
\begin{cases}
 {\frac{r^{2 - s}}{(r + 1)^2},} & {s \leqslant 0} \\
 {\frac{R^{2 - s}}{(R + 1)^2},} & {s \geqslant 2} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq110} M = \mathop {\sup }\limits_{x \in [r,R]} g_{F_2} (x)
=
\begin{cases}
 {\frac{R^{2 - s}}{(R + 1)^2},} & {s \leqslant 0} \\
 {\frac{r^{2 - s}}{(r + 1)^2},} & {s \geqslant 2} \\
\end{cases}.
\end{equation}

Now (\ref{eq109}) and (\ref{eq110}) together with (\ref{eq56})
give the required result.
\end{proof}

Some particular cases of the Theorem \ref{the44} are summarized in
the following corollary.

\begin{corollary} \label{cor44} The following bounds hold:
\begin{equation}
\label{eq111}
\frac{r^3}{2(r + 1)^2}\chi ^2(Q\vert \vert P) \leqslant F(Q\vert \vert P)
\leqslant \frac{R^3}{2(R + 1)^2}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq112}
\frac{r^2}{(r + 1)^2}K(Q\vert \vert P) \leqslant F(Q\vert \vert P) \leqslant
\frac{R^2}{(R + 1)^2}K(Q\vert \vert P)
\end{equation}

\noindent and
\begin{equation}
\label{eq113}
\frac{1}{(R + 1)^2}\chi ^2(P\vert \vert Q) \leqslant F(Q\vert \vert P)
\leqslant \frac{1}{(r + 1)^2}\chi ^2(P\vert \vert Q),
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq111}) and (\ref{eq112})
follows from (\ref{eq105}) by taking $s = - 1$ and $s = 0$
respectively. The inequalities (\ref{eq113}) follow from
(\ref{eq106}) by taking $s = 2$.
\end{proof}

The cases $s = \frac{1}{2}$ and $s = 1$ are not included in the
inequalities (\ref{eq105}) and (\ref{eq106}). This we shall do
separately in the following proposition.

\begin{proposition} \label{pro44} The following bounds hold:
\begin{equation}
\label{eq114}
F(Q\vert \vert P) \leqslant \frac{3\sqrt 3 }{4}h(P\vert \vert Q),
\end{equation}

\noindent and
\begin{equation}
\label{eq115} F(Q\vert \vert P) \leqslant \frac{1}{4}K(P\vert
\vert Q).
\end{equation}
\end{proposition}

\begin{proof} For $s = \frac{1}{2}$ in (\ref{eq107}), we
have
\begin{equation}
\label{eq116}
g_{F_2 } (x) = \frac{x\sqrt x }{(x + 1)^2}.
\end{equation}

This gives
\begin{equation}
\label{eq117}
{g}'_{F_2 } (x) = - \frac{\sqrt x (x - 3)}{2(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {x \leqslant 3} \\
 { \leqslant 0,} & {x \geqslant 3} \\
\end{cases}.
\end{equation}

Thus from (\ref{eq117}) we conclude that the function $g_{F_2 }
(x)$ given by (\ref{eq116}) is increasing in $x \in (0,3)$ and
decreasing in $x \in (3,\infty )$, and hence
\begin{equation}
\label{eq118}
M = \mathop {\sup }\limits_{x \in (0,\infty )} g_{F_2 } (x) = g_{F_2 } (3) =
\frac{3\sqrt 3 }{16}.
\end{equation}

Now (\ref{eq118}) together with (\ref{eq56}) give the inequality
(\ref{eq114}).

Again for $s = 0$ in (\ref{eq107}), we have
\begin{equation}
\label{eq119}
g_{F_2 } (x) = \frac{x}{(x + 1)^2}.
\end{equation}

This gives
\begin{equation}
\label{eq120}
{g}'_{F_2 } (x) = - \frac{x - 1}{(x + 1)^3}
\begin{cases}
 { \geqslant 0,} & {x \leqslant 1} \\
 { \leqslant 0,} & {x \geqslant 1} \\
\end{cases}.
\end{equation}

Thus from (\ref{eq120}), we conclude that the function $g_{F_2 }
(x)$ given by (\ref{eq119}) is increasing in $x \in (0,1)$ and
decreasing in $x \in (1,\infty )$, and hence
\begin{equation}
\label{eq121}
M = \mathop {\sup }\limits_{x \in (0,\infty )} g_{F_2 } (x) = g_{F_2 } (1) =
\frac{1}{4}.
\end{equation}

Now (\ref{eq121}) together with (\ref{eq56}) give the inequality
(\ref{eq115}).
\end{proof}

\begin{theorem} \label{the45} The following bounds on \textit{relative
AG-divergence} hold:
\begin{equation}
\label{eq122}
\frac{1}{2r^s(r + 1)}\Phi _s (P\vert \vert Q)
 \leqslant G(P\vert \vert Q) \leqslant
\frac{1}{2R^s(R + 1)}\Phi _s (P\vert \vert Q), \,\, s \leqslant -
1
\end{equation}

\noindent and
\begin{equation}
\label{eq123}
\frac{1}{2R^s(R + 1)}\Phi _s (P\vert \vert Q)
 \leqslant G(P\vert \vert Q) \leqslant
\frac{r^{2 - s}(r + 3)}{(r + 1)^2}\Phi _s (P\vert \vert Q), \,\, s
\geqslant 0
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq124} g_{G_1 } (x) = x^{2 - s}{f}''_{G_1 } (x) = \frac{x^{
- s}}{2(x + 1)}, \,\, x \in (0,\infty ),
\end{equation}

\noindent where ${f}''_{G_1 } (x)$ is as given by (\ref{eq33}).

From (\ref{eq124}) one gets
\begin{equation}
\label{eq125}
{g}'_{G_1 } (x) = - \frac{x^{ - 1 - s}\left[ {(s + 1)x + s} \right]}{2(x +
1)^2}
\begin{cases}
 { \geqslant 0,} & {s \leqslant - 1} \\
 { \leqslant 0,} & {s \geqslant 0} \\
\end{cases}.
\end{equation}

In view of (\ref{eq125}), we conclude that
\begin{equation}
\label{eq126} m = \mathop {\inf }\limits_{x \in [r,R]} g_{G_1} (x)
= \begin{cases}
 {\frac{1}{2r^s(r + 1)},} & {s \leqslant - 1} \\
 {\frac{1}{2R^s(R + 1)},} & {s \geqslant 0} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq127} M = \mathop {\sup }\limits_{x \in [r,R]} g_{G_1} (x)
=
\begin{cases}
 {\frac{1}{2R^s(R + 1)},} & {s \leqslant - 1} \\
 {\frac{1}{2r^s(r + 1)},} & {s \geqslant 0} \\
\end{cases}.
\end{equation}

Now (\ref{eq126}) and (\ref{eq127}) together with (\ref{eq56})
give the required result.
\end{proof}

Some particular cases of the Theorem \ref{the45} are summarized in
the following corollary.

\begin{corollary} \label{cor45} The following bounds hold:
\begin{equation}
\label{eq128}
\frac{r}{4(r + 1)}\chi ^2(Q\vert \vert P) \leqslant G(P\vert \vert Q)
\leqslant \frac{R}{4(R + 1)}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq129}
\frac{1}{2(R + 1)}K(Q\vert \vert P) \leqslant G(P\vert \vert Q) \leqslant
\frac{1}{2(r + 1)}K(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq130}
\frac{2}{\sqrt R (R + 1)}h(P\vert \vert Q) \leqslant G(P\vert \vert Q)
\leqslant \frac{2}{\sqrt r (r + 1)}h(P\vert \vert Q),
\end{equation}
\begin{equation}
\label{eq131}
\frac{1}{2R(R + 1)}K(P\vert \vert Q) \leqslant G(P\vert \vert Q) \leqslant
\frac{1}{2r(r + 1)}K(P\vert \vert Q),
\end{equation}

\noindent and
\begin{equation}
\label{eq132}
\frac{1}{4R^2(R + 1)}\chi ^2(P\vert \vert Q) \leqslant G(P\vert \vert Q)
\leqslant \frac{1}{4r^2(r + 1)}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq128}) follows from
(\ref{eq122}) by taking $s = - 1$. The inequalities (\ref{eq129}),
(\ref{eq130}), (\ref{eq131}) and (\ref{eq132}) follows from
(\ref{eq123}) by taking $s = 0$, $s = \frac{1}{2}$, $s = 1$ and $s
= 2$ respectively.
\end{proof}

\begin{theorem} \label{the46} The following bounds on \textit{adjoint
}of\textit{ relative AG-divergence} holds:
\begin{equation}
\label{eq133}
\frac{r^2}{2r^s(r + 1)}\Phi _s (P\vert \vert Q)
 \leqslant G(Q\vert \vert P) \leqslant
\frac{R^2}{2R^s(R + 1)}\Phi _s (P\vert \vert Q), \,\, s \leqslant
1
\end{equation}

\noindent and
\begin{equation}
\label{eq134}
\frac{R^2}{2R^s(R + 1)}\Phi _s (P\vert \vert Q)
 \leqslant G(Q\vert \vert P) \leqslant
\frac{r^2}{2r^s(r + 1)}\Phi _s (P\vert \vert Q), \,\, s \geqslant
2
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq135} g_{G_2 } (x) = x^{2 - s}{f}''_{G_2 } (x) =
\frac{x^{2 - s}}{2(x + 1)}, \,\, x \in (0,\infty ),
\end{equation}

\noindent where ${f}''_{G_2 } (x)$ is as given by (\ref{eq36}).

From (\ref{eq135}) one gets
\begin{equation}
\label{eq136}
{g}'_{G_2 } (x) = - \frac{x^{1 - s}\left[ {(s - 1)x + (s - 2)} \right]}{2(x
+ 1)^2}
\begin{cases}
 { \geqslant 0,} & {s \leqslant 1} \\
 { \leqslant 0,} & {s \geqslant 2} \\
\end{cases}.
\end{equation}

In view of (\ref{eq136}) we conclude that
\begin{equation}
\label{eq137} m = \mathop {\inf }\limits_{x \in [r,R]} g_{G_2} (x)
=
\begin{cases}
 {\frac{r^2}{2r^s(r + 1)},} & {s \leqslant 1} \\
 {\frac{R^2}{2R^s(R + 1)},} & {s \geqslant 2} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq138} M = \mathop {\sup }\limits_{x \in [r,R]} g_{G_2} (x)
=
\begin{cases}
 {\frac{R^2}{2R^s(R + 1)},} & {s \leqslant 1} \\
 {\frac{r^2}{2r^s(r + 1)},} & {s \geqslant 2} \\
\end{cases}.
\end{equation}

Now (\ref{eq137}) and (\ref{eq138}) together with (\ref{eq56})
give the required result.
\end{proof}

Some particular cases of the Theorem \ref{the46} are summarized in
the following corollary.

\begin{corollary} \label{cor46} The following bounds hold:
\begin{equation}
\label{eq139}
\frac{r^3}{4(r + 1)}\chi ^2(Q\vert \vert P) \leqslant G(Q\vert \vert P)
\leqslant \frac{R^3}{4(R + 1)}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq140}
\frac{r^2}{2(r + 1)}K(Q\vert \vert P) \leqslant G(Q\vert \vert P) \leqslant
\frac{R^2}{2(R + 1)}K(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq141}
\frac{2r\sqrt r }{(r + 1)}h(P\vert \vert Q) \leqslant G(Q\vert \vert P)
\leqslant \frac{2R\sqrt R }{(R + 1)}h(P\vert \vert Q),
\end{equation}
\begin{equation}
\label{eq142}
\frac{r}{2(r + 1)}K(P\vert \vert Q) \leqslant G(Q\vert \vert P) \leqslant
\frac{R}{2(R + 1)}K(P\vert \vert Q)
\end{equation}

\noindent and
\begin{equation}
\label{eq143}
\frac{1}{4(R + 1)}\chi ^2(P\vert \vert Q) \leqslant G(Q\vert \vert P)
\leqslant \frac{1}{4(r + 1)}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq139}), (\ref{eq140}),
(\ref{eq141}) and (\ref{eq142}) follows from (\ref{eq133}) by
taking $s = - 1$, $s = 0$, $s = \frac{1}{2}$ and $s = 1$
respectively. The inequalities (\ref{eq143}) follows from
(\ref{eq134}) by taking $s = 2$.
\end{proof}

\begin{remark} \label{rem41} The inequalities (\ref{eq94}),
(\ref{eq95}), (\ref{eq112}), (\ref{eq113}), (\ref{eq128}),
(\ref{eq129}), (\ref{eq142}) and (\ref{eq143}) can be re-written
as
\begin{equation}
\label{eq144} r \leqslant \xi _t (P\vert \vert Q) \leqslant R,
\,\, t = 1,2,3,4,5,6,7\mbox{ and }8,
\end{equation}

\noindent where
\[
\xi _1 (P\vert \vert Q) = \frac{\sqrt {2F(P\vert \vert Q)}
}{\sqrt{\chi ^2(Q\vert \vert P)} - \sqrt {2F(P\vert \vert Q)} },
\]
\[
\xi _2 (P\vert \vert Q) = \frac{\sqrt {K(P\vert \vert Q)} - \sqrt
{F(P\vert \vert Q)} }{\sqrt {F(P\vert \vert Q)} },
\]
\[
\xi _3 (P\vert \vert Q) = \frac{\sqrt {F(Q\vert \vert P)} }{\sqrt {K(Q\vert
\vert P)} - \sqrt {F(Q\vert \vert P)} },
\]
\[
\xi _4 (P\vert \vert Q) = \frac{\sqrt {\chi ^2(P\vert \vert Q)} -
\sqrt {2 F(Q\vert \vert P)} }{\sqrt {2 F(Q\vert \vert P)} },
\]
\[
\xi _5 (P\vert \vert Q) = \frac{4G(P\vert \vert Q)}{\chi ^2(Q\vert
\vert P) - 4G(P\vert \vert Q)},
\]
\[
\xi _6 (P\vert \vert Q) = \frac{K(Q\vert \vert P) - 2G(P\vert \vert
Q)}{2G(P\vert \vert Q)},
\]
\[
\xi _7 (P\vert \vert Q) = \frac{2G(Q\vert \vert P)}{K(P\vert \vert
Q) - 2G(Q\vert \vert P)},
\]
\noindent and
\[
\xi _8 (P\vert \vert Q) = \frac{\chi ^2(P\vert \vert Q) - 4G(Q\vert \vert
P)}{4G(Q\vert \vert P)},
\]
respectively.
\end{remark}


\section{Bounds on Symmetric Divergence Measures}

In this section we shall obtain bound on \textit{symmetric
divergence measures} given by (\ref{eq2}), (\ref{eq10}) and
(\ref{eq11}) in terms of \textit{relative information of type s}
given by (\ref{eq14}). Some particular cases are also given.

\begin{theorem} \label{the51} The following bounds on
\textit{J-divergence} hold:
\begin{equation}
\label{eq145}
\frac{1 + r}{r^s}\Phi _s (P\vert \vert Q)
 \leqslant J(P\vert \vert Q) \leqslant
\frac{1 + R}{R^s}\Phi _s (P\vert \vert Q), \,\, s \leqslant 0
\end{equation}

\noindent and
\begin{equation}
\label{eq146}
\frac{1 + R}{R^s}\Phi _s (P\vert \vert Q)
 \leqslant J(P\vert \vert Q) \leqslant
\frac{1 + r}{r^s}\Phi _s (P\vert \vert Q), \,\, s \geqslant 1
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq147} g_J (x) = x^{2 - s}{f}''_J (x) = x^{ - s} + x^{1 -
s} = \frac{1 + x}{x^s}, \,\, x \in (0,\infty ),
\end{equation}

\noindent where ${f}''_J (x)$ is as given by (\ref{eq39}).

From (\ref{eq147}) one gets
\begin{equation}
\label{eq148}
{g}'_J (x) = x^{ - s - 1}\left[ {(1 - s)x + ( - s)} \right]
\begin{cases}
 { \geqslant 0,} & {s \leqslant 0} \\
 { \leqslant 0,} & {s \geqslant 1} \\
\end{cases}.
\end{equation}

In view of (\ref{eq148}), we conclude that
\begin{equation}
\label{eq149} m = \mathop {\inf }\limits_{x \in [r,R]} g_J (x) =
\begin{cases}
 {\frac{1 + r}{r^s},} & {s \leqslant 0} \\
 {\frac{1 + R}{R^s},} & {s \geqslant 1} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq150} M = \mathop {\sup }\limits_{x \in [r,R]} g_J (x) =
\begin{cases}
 {\frac{R^{1 - s}}{2(1 + R)},} & {s \leqslant 0} \\
 {\frac{r^{1 - s}}{2(1 + r)},} & {s \geqslant 1} \\
\end{cases}
\end{equation}

Now (\ref{eq149}) and (\ref{eq150}) together with (\ref{eq56})
give the required result.
\end{proof}

Some particular cases of the Theorem \ref{the51} are summarized in
the following corollary.

\begin{corollary} \label{cor51} The following bounds hold:
\begin{equation}
\label{eq151}
\frac{r + r^2}{2}\chi ^2(Q\vert \vert P) \leqslant J(P\vert \vert Q)
\leqslant \frac{R + R^2}{2}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq152}
(1 + r)K(Q\vert \vert P) \leqslant J(P\vert \vert Q) \leqslant (1 +
R)K(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq153}
\frac{1 + R}{R}K(P\vert \vert Q) \leqslant J(P\vert \vert Q) \leqslant
\frac{1 + r}{r}K(P\vert \vert Q)
\end{equation}

\noindent and
\begin{equation}
\label{eq154}
\frac{1 + R}{2R^2}\chi ^2(P\vert \vert Q) \leqslant J(P\vert \vert Q)
\leqslant \frac{1 + r}{2r^2}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq151}) and (\ref{eq152})
follows from (\ref{eq145}) by taking $s = - 1$ and $s = 0$
respectively. The inequalities (\ref{eq153}) and (\ref{eq154})
follows from (\ref{eq146}) by taking $s = 1$ and $s = 2$
respectively.
\end{proof}

The cases $s = \frac{1}{2}$ is not included in the inequalities
(\ref{eq145}) and (\ref{eq146}). This we shall do separately in
the following proposition.

\begin{proposition} \label{pro51} The following bound holds:
\begin{equation}
\label{eq155}
h(P\vert \vert Q)
 \leqslant \frac{1}{8}J(P\vert \vert Q).
\end{equation}
\end{proposition}

\begin{proof} Take $s = \frac{1}{2}$ in (\ref{eq147}), we
have
\begin{equation}
\label{eq156}
g_J (x) = \frac{x + 1}{\sqrt x }
\end{equation}

\noindent and
\begin{equation}
\label{eq157}
{g}'_J (x) = \frac{x - 1}{2x^{3 / 2}}
\begin{cases}
 { \geqslant 0,} & {x \geqslant 1} \\
 { \leqslant 0,} & {x \leqslant 1} \\
\end{cases}.
\end{equation}

Thus from (\ref{eq157}) we conclude that the function $g_J (x)$
given by (\ref{eq156}) is decreasing in $x \in (0,1)$ and
increasing in $x \in (1,\infty )$, and hence
\begin{equation}
\label{eq158}
m = \mathop {\inf }\limits_{x \in (0,\infty )} g_J (x) = g_J (1) = 2.
\end{equation}

Now (\ref{eq158}) together with (\ref{eq56}) give the inequality
(\ref{eq155}).
\end{proof}

\begin{theorem} \label{the52} The following bounds on
\textit{JS-divergence} hold:
\begin{equation}
\label{eq159}
\frac{r^{1 - s}}{2(1 + r)}\Phi _s (P\vert \vert Q)
 \leqslant I(P\vert \vert Q) \leqslant
\frac{R^{1 - s}}{2(1 + R)}\Phi _s (P\vert \vert Q), \,\, s
\leqslant 0
\end{equation}

\noindent and
\begin{equation}
\label{eq160}
\frac{R^{1 - s}}{2(1 + R)}\Phi _s (P\vert \vert Q)
 \leqslant I(P\vert \vert Q) \leqslant
\frac{r^{1 - s}}{2(1 + r)}\Phi _s (P\vert \vert Q), \,\, s
\geqslant 1.
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq161} g_I (x) = x^{2 - s}{f}''_I (x) = \frac{x^{1 -
s}}{2(x + 1)}, \,\, x \in (0,\infty ),
\end{equation}

\noindent where ${f}''_I (x)$ is as given by (\ref{eq42}).

From (\ref{eq161}) one gets
\begin{equation}
\label{eq162}
{g}'_I (x) = \frac{x^{ - s}\left[ {( - s)x + (1 - s)} \right]}{2(x + 1)^2}
\begin{cases}
 { \geqslant 0,} & {s \leqslant 0} \\
 { \leqslant 0,} & {s \geqslant 1} \\
\end{cases}.
\end{equation}

In view of (\ref{eq162}), we conclude that
\begin{equation}
\label{eq163} m = \mathop {\inf }\limits_{x \in [r,R]} g_I (x) =
\begin{cases}
 {\frac{r^{1 - s}}{2(1 + r)},} & {s \leqslant 0} \\
 {\frac{R^{1 - s}}{2(1 + R)},} & {s \geqslant 1} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq164} M = \mathop {\sup }\limits_{x \in [r,R]} g_I (x) =
\begin{cases}
 {\frac{R^{1 - s}}{2(1 + R)},} & {s \leqslant 0} \\
 {\frac{r^{1 - s}}{2(1 + r)},} & {s \geqslant 1} \\
\end{cases}
\end{equation}

Now (\ref{eq163}) and (\ref{eq164}) together with (\ref{eq56})
give the required result.
\end{proof}

Some particular cases of the Theorem \ref{the52} are summarized in
the following corollary.

\begin{corollary} \label{cor52} The following bounds hold:
\begin{equation}
\label{eq165}
\frac{r^2}{4(1 + r)}\chi ^2(Q\vert \vert P) \leqslant I(P\vert \vert Q)
\leqslant \frac{R^2}{4(1 + R)}\chi ^2(Q\vert \vert P),
\end{equation}
\begin{equation}
\label{eq166}
\frac{r}{2(1 + r)}K(Q\vert \vert P) \leqslant I(P\vert \vert Q) \leqslant
\frac{R}{2(1 + R)}K(Q\vert \vert P).
\end{equation}
\begin{equation}
\label{eq167}
\frac{1}{2(1 + R)}K(P\vert \vert Q) \leqslant I(P\vert \vert Q) \leqslant
\frac{1}{2(1 + r)}K(P\vert \vert Q),
\end{equation}

\noindent and
\begin{equation}
\label{eq168}
\frac{1}{4(R + R^2)}\chi ^2(P\vert \vert Q) \leqslant I(P\vert \vert Q)
\leqslant \frac{1}{4(r + r^2)}\chi ^2(P\vert \vert Q),
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq165}) and (\ref{eq166})
follows from (\ref{eq159}) by taking $s = - 1$ and $s = 0$
respectively. The inequalities (\ref{eq167}) and (\ref{eq168})
follows from (\ref{eq160}) by taking $s = 1$ and $s = 2$
respectively.
\end{proof}

The case $s = \frac{1}{2}$ is not included in the inequalities
(\ref{eq159}) and (\ref{eq160}). This we shall do separately in
the following proposition.

\begin{proposition} \label{pro52} The following inequality hold:
\begin{equation}
\label{eq169}
I(P\vert \vert Q) \leqslant h(P\vert \vert Q).
\end{equation}
\end{proposition}

\begin{proof} For $s = \frac{1}{2}$ in (\ref{eq161}), we
have
\begin{equation}
\label{eq170}
g_I (x) = \frac{\sqrt x }{2(x + 1)}
\end{equation}

\noindent and
\begin{equation}
\label{eq171}
{g}'_I (x) = \frac{1 - x}{4\sqrt x (x + 1)^2}
\begin{cases}
 { \geqslant 0,} & {x \leqslant 1} \\
 { \leqslant 0,} & {x \geqslant 1} \\
\end{cases}.
\end{equation}

Thus from (\ref{eq171}) we conclude that the function $g_I (x)$
given by (\ref{eq170}) is increasing in $x \in (0,1)$ and
decreasing in $x \in (1,\infty )$, and hence
\begin{equation}
\label{eq172}
M = \mathop {\sup }\limits_{x \in (0,\infty )} g_I (x) = g_I (1) =
\frac{1}{4}.
\end{equation}

Now (\ref{eq172}) and (\ref{eq56}) together give the inequality
(\ref{eq169}).
\end{proof}

\begin{theorem} \label{the53} The following bounds on
\textit{AG-divergence} hold:
\begin{equation}
\label{eq173}
\frac{r^{ - s}(1 + r^2)}{4(1 + r)}\Phi _s (P\vert \vert Q)
 \leqslant T(P\vert \vert Q) \leqslant
\frac{R^{ - s}(1 + R^2)}{4(1 + R)}\Phi _s (P\vert \vert Q), \,\, s
\leqslant - 1
\end{equation}

\noindent and
\begin{equation}
\label{eq174}
\frac{R^{ - s}(1 + R^2)}{4(1 + R)}\Phi _s (P\vert \vert Q)
 \leqslant T(P\vert \vert Q) \leqslant
\frac{r^{ - s}(1 + r^2)}{4(1 + r)}\Phi _s (P\vert \vert Q), \,\, s
\geqslant 2.
\end{equation}
\end{theorem}

\begin{proof} Let us consider
\begin{equation}
\label{eq175} g_T (x) = x^{2 - s}{f}''_T (x) = \frac{x^{2 - s}(1 +
x^2)}{4(x^2 + x^3)} = \frac{x^{ - s} + x^{2 - s}}{4(1 + x)}, \,\,
x \in (0,\infty )
\end{equation}

\noindent where ${f}''_T (x)$ is as given by (\ref{eq45}).

From (\ref{eq175}) one gets
\begin{equation}
\label{eq176} {g}'_T (x) = - \frac{x^{ - s - 1}\left[ {(s - 1)x^3
+ (s - 2)x^2 + (s + 1)x + s} \right]}{4(x + 1)^2}
\begin{cases}
 { \geqslant 0,} & {s \leqslant - 1} \\
 { \leqslant 0,} & {s \geqslant 2} \\
\end{cases}
\end{equation}

In view of (\ref{eq176}), we conclude that
\begin{equation}
\label{eq177} m = \mathop {\inf }\limits_{x \in [r,R]} g_T (x) =
\begin{cases}
 {\frac{r^{ - s}(1 + r^2)}{4(1 + r)},} & {s \leqslant - 1} \\
 {\frac{R^{ - s}(1 + R^2)}{4(1 + R)},} & {s \geqslant 2} \\
\end{cases}
\end{equation}

\noindent and
\begin{equation}
\label{eq178} M = \mathop {\sup }\limits_{x \in [r,R]} g_T (x) =
\begin{cases}
 {\frac{R^{ - s}(1 + R^2)}{4(1 + R)},} & {s \leqslant - 1} \\
 {\frac{r^{ - s}(1 + r^2)}{4(1 + r)},} & {s \geqslant 2} \\
\end{cases}
\end{equation}

Now (\ref{eq177}) and (\ref{eq178}) together with (\ref{eq56})
give the required result.
\end{proof}

Some particular cases of the Theorem \ref{the53} are summarized in
the following corollary.

\begin{corollary} \label{cor53} The following bounds hold:
\begin{equation}
\label{eq179}
\frac{r + r^3}{8(1 + r)}\chi ^2(Q\vert \vert P) \leqslant T(P\vert \vert Q)
\leqslant \frac{R + R^3}{8(1 + R)}\chi ^2(Q\vert \vert P)
\end{equation}

\noindent and
\begin{equation}
\label{eq180}
\frac{1 + R^2}{8R^2(1 + R)}\chi ^2(P\vert \vert Q) \leqslant T(P\vert \vert
Q) \leqslant \frac{1 + r^2}{8r^2(1 + r)}\chi ^2(P\vert \vert Q).
\end{equation}
\end{corollary}

\begin{proof} Inequalities (\ref{eq179}) follows from
(\ref{eq173}) by taking $s = - 1$. The inequalities (\ref{eq180})
follows from (\ref{eq174}) by taking $s = 2$.
\end{proof}

The cases $s = 0$, $s = \frac{1}{2}$ and $s = 1$ are not included
in the inequalities (\ref{eq173}) and (\ref{eq174}). This we shall
do separately in the following proposition.

\begin{proposition} \label{pro53} The following inequalities hold:
\begin{equation}
\label{eq181} \frac{\sqrt 2 - 1}{2} K(Q\vert \vert P) \leqslant
T(P\vert \vert Q),
\end{equation}
\begin{equation}
\label{eq182} \frac{\sqrt 2 - 1}{2} K(P\vert \vert Q) \leqslant
T(P\vert \vert Q)
\end{equation}

\noindent and
\begin{equation}
\label{eq183}
h(P\vert \vert Q) \leqslant T(P\vert \vert Q).
\end{equation}
\end{proposition}

\begin{proof} Take $s = 0$ in (\ref{eq175}), we have
\begin{equation}
\label{eq184}
g_T (x) = \frac{x^2 + 1}{4(x + 1)}
\end{equation}

\noindent and
\begin{equation}
\label{eq185} {g}'_T (x) = \frac{(x + 1 - \sqrt 2 )(x + 1 + \sqrt
2 )}{4(x + 1)^2}\begin{cases}
 { \geqslant 0,} & {x \geqslant \sqrt 2 - 1} \\
 { \leqslant 0,} & {x \leqslant \sqrt 2 - 1} \\
\end{cases}.
\end{equation}

In view of (\ref{eq185}) we conclude that the function $g_T (x)$
given by (\ref{eq184}) is decreasing in $x \in (0,\sqrt 2 - 1)$
and increasing in $x \in (\sqrt 2 - 1,\infty )$, and hence
\begin{equation}
\label{eq186} m = \mathop {\inf }\limits_{x \in (0,\infty )} g_T
(x) = g_T (\sqrt 2 - 1) = \frac{1 + (\sqrt 2 - 1)^2}{4\sqrt 2 } =
\frac{\sqrt 2 - 1}{2}.
\end{equation}

Now (\ref{eq186}) together with (\ref{eq56}) give the inequality
(\ref{eq181}).

Again take $s = 1$ in (\ref{eq175}), we have
\begin{equation}
\label{eq187}
g_T (x) = \frac{x^2 + 1}{4x(x + 1)},
\end{equation}

\noindent and
\begin{equation}
\label{eq188} {g}'_T (x) = \frac{(x - 1 - \sqrt 2 )(x - 1 + \sqrt
2 )}{4(x + 1)^2}\begin{cases}
 { \geqslant 0,} & {x \geqslant \sqrt 2 + 1} \\
 { \leqslant 0,} & {x \leqslant \sqrt 2 + 1} \\
\end{cases}.
\end{equation}

In view of (\ref{eq188}), we conclude that the function $g_T (x)$
given by (\ref{eq187}) is decreasing in $x \in (0,\sqrt 2 + 1)$
and increasing in $x \in (\sqrt 2 + 1,\infty )$, and hence
\begin{equation}
\label{eq189} m = \mathop {\inf }\limits_{x \in (0,\infty )} g_T
(x) = g_T (\sqrt 2 + 1) = \frac{\sqrt 2 - 1}{2}.
\end{equation}

Now (\ref{eq189}) together with (\ref{eq56}) give the inequality
(\ref{eq182}).

Finally, take $s = \frac{1}{2}$ in (\ref{eq175}), we have
\begin{equation}
\label{eq190}
g_T (x) = \frac{x^2 + 1}{4\sqrt x (x + 1)},
\end{equation}

\noindent and
\begin{equation}
\label{eq191}
{g}'_T (x) = \frac{(x - 1)\left[ {(x - 1)^2 + 6x} \right]}{8x^{3 / 2}(x +
1)^2}
\begin{cases}
 { \geqslant 0,} & {x \geqslant 1} \\
 { \leqslant 0,} & {x \leqslant 1} \\
\end{cases}.
\end{equation}

In view of (\ref{eq191}) we conclude that the function $g_T (x)$
given by (\ref{eq190}) is decreasing in $x \in (0,1)$ and
increasing in $x \in (1,\infty )$, and hence
\begin{equation}
\label{eq192}
m = \mathop {\inf }\limits_{x \in (0,\infty )} g_T (x) = g_T (1) =
\frac{1}{4}.
\end{equation}

Now (\ref{eq192}) together with (\ref{eq56}) give the inequalities
(\ref{eq183}).
\end{proof}

\begin{remark} \label{rem51}
\begin{itemize}
\item[(i)] The inequalities appearing in (\ref{eq152}),
(\ref{eq153}), (\ref{eq166}) and (\ref{eq167}) can be re-written
as
\begin{equation}
\label{eq193} r \leqslant \zeta _t (P\vert \vert Q) \leqslant R,
\,\, t = 1,2,3\mbox{ and 4},
\end{equation}

\noindent where
\[
\zeta _1 (P\vert \vert Q) = \frac{J(P\vert \vert Q) - K(Q\vert \vert
P)}{K(Q\vert \vert P)},
\]
\[
\zeta _2 (P\vert \vert Q) = \frac{K(P\vert \vert Q)}{J(P\vert \vert Q) -
K(P\vert \vert Q)},
\]
\[
\zeta _3 (P\vert \vert Q) = \frac{2I(P\vert \vert Q)}{K(Q\vert \vert P) -
2I(P\vert \vert Q)},
\]

\noindent and
\[
\zeta _4 (P\vert \vert Q) = \frac{K(P\vert \vert Q) - 2I(P\vert \vert
Q)}{2I(P\vert \vert Q)}
\]

\noindent respectively.

\item[(ii)] In view of (\ref{eq155}), (\ref{eq169}) and
(\ref{eq183}), we have the following interesting relation
\begin{equation}
\label{eq194} I(P\vert \vert Q) \leqslant h(P\vert \vert Q)
\leqslant \left( {T(P\vert \vert Q)\mbox{ or }\frac{1}{8}J(P\vert
\vert Q)} \right)
\end{equation}

A general form of the inequalities (\ref{eq194}) can be seen in
Taneja \cite{tan6} , where more kind of symmetric measures are
also studied.
\end{itemize}
\end{remark}

\begin{thebibliography}{99}
\setlength{\itemsep}{5pt}

\bibitem{bcd} N.S. BARNETT, P. CERENE and S. S. DRAGOMIR, Some
New Inequalities for Hermite-Hadamard Divergence in Information
Theory, \textit{http://rgmia.vu.edu.au, RGMIA Research Report
Collection}, \textbf{(5)}(4)(2002), Article 8.

\bibitem{bha} A. BHATTACHARYYA, Some Analogues to the Amount of
Information and Their uses in Statistical Estimation,
\textit{Sankhya}, \textbf{8}(1946), 1-14.

\bibitem{bur1} J. BURBEA, J. and C.R. RAO, Entropy Differential Metric, Distance
and Divergence Measures in Probability Spaces: A Unified Approach,
\textit{J. Multi. Analysis}, \textbf{12}(1982), 575-596.

\bibitem{bur2} J. BURBEA, J. and C.R. RAO, On the Convexity of Some Divergence
Measures Based on Entropy Functions, \textit{IEEE Trans. on
Inform. Theory}, \textbf{IT-28}(1982), 489-495.

\bibitem{cdo} P. CERONE, S.S. DRAGOMIR and F. \"{O}STERREICHER, Bounds on
Extended $f-$Divergence for a Variety of Classes,
\textit{http://rgmia.vu.edu.au, RGMIA Research Report Collection},
\textbf{6}(1)(2003), Article 7.

\bibitem{crr} P. CRESSIE and T.R.C. READ, Multinomial
goodness-of-fit tests, \textit{J. Roy. Statist. Soc. Ser. B},
\textbf{46}(1984), 440-464.

\bibitem{csi1} I. CSISZ\'{A}R,
Information Type Measures of Differences of Probability
Distribution and Indirect Observations, \textit{Studia Math.
Hungarica}, \textbf{2}(1967), 299-318.

\bibitem{csi2} I. CSISZ\'{A}R,
On Topological Properties of $f-$Divergences, \textit{Studia Math.
Hungarica}, \textbf{2}(1967), 329-339.

\bibitem{dra1} S. S. DRAGOMIR, Some Inequalities for the Csisz\'{a}r $\Phi
$-Divergence - \textit{Inequalities for Csisz\'{a}r f-Divergence
in Information Theory -} Monograph -- Chapter I -- Article 1 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra2} S. S. DRAGOMIR, Other Inequalities for Csisz\'{a}r Divergence and
Applications - \textit{Inequalities for Csisz\'{a}r f-Divergence
in Information Theory -} Monograph -- Chapter I -- Article 4 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra3} S. S. DRAGOMIR, Some Inequalities for (m, M)-Convex Mappings and
Applications for the Csisz\'{a}r $\Phi $-Divergence in Information
Theory- \textit{Inequalities for Csisz\'{a}r f-Divergence in
Information Theory -} Monograph -- Chapter I -- Article 3 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra4} S. S. DRAGOMIR, Upper and Lower Bounds for Csisz\'{a}r's
f-divergence in terms of the Kullback-Leibler Distance and
Applications - \textit{Inequalities for Csisz\'{a}r f-Divergence
in Information Theory -} Monograph -- Chapter II -- Article 1
--\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra5} S. S. DRAGOMIR, Upper and Lower Bounds for Csisz\'{a}r's
f-Divergence in terms of Hellinger Discrimination and Applications
- \textit{Inequalities for Csisz\'{a}r f-Divergence in Information
Theory -} Monograph -- Chapter II -- Article 2 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dgp} S. S. DRAGOMIR, V. GLUSCEVIC and C.E.M. PEARCE\textbf{,
}Approximation for Csisz\'{a}r $f$-Divergence via Midpoint
Inequalities- \textit{Inequalities for Csisz\'{a}r f-Divergence in
Information Theory -} Monograph -- Chapter IV -- Article 3 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{hel} E. HELLINGER, Neue Begr\"{u}ndung der Theorie der Quadratischen
Formen von Unendlichen vielen Ver\"{a}nderlichen, \textit{J. Reine
Aug. Math.}, \textbf{136}(1909), 210-271.

\bibitem{jef} H. JEFFREYS, An Invariant Form for the Prior Probability in
Estimation Problems, \textit{Proc. Roy. Soc. Lon., Ser. A},
\textbf{186}(1946), 453-461.

\bibitem{kul} S. KULLBACK and R.A. LEIBLER, On Information and Sufficiency,
\textit{Ann. Math. Statist.}, \textbf{22}(1951), 79-86.

\bibitem{liv} F. LIESE and I. VAJDA, \textit{Convex Statistical Decision Rule},
Teubner-Texte zur Mathematick, Band 95, Leipzig, 1987.

\bibitem{lin} J. LIN, Divergence Measures Based on the Shannon
Entropy, \textit{IEEE Trans. on Inform. Theory},
\textbf{IT-37}(1991), 145-151.

\bibitem{liw} J. LIN and S.K.M WONG, A New Directed Divergence
Measure and its Characterizations, \textit{Int. J. General
Systems}, \textbf{17}(1990), 73-81.

\bibitem{pea} K. PEARSON, On the Criterion that a given System of Deviations
from the Probable in the case of Correlated System of Variables is
such that it can be Reasonable supposed to have arisen from Random
Sampling, \textit{Phil. Mag.}, \textbf{50}(1900), 157-172.

\bibitem{shd} H. SHIOYA and T. DA-TE, A Generalization of Lin
Divergence and the Derivation of a New Information Divergence,
\textit{Eletronics and Communications in Japan}, Part 3
7\textbf{(78)}(1995), 34-40.

\bibitem{sib} R. SIBSON, Information Radius, \textit{Z. Wahrs. und verw Geb.},
\textbf{(14)}(1969), 149-160.

\bibitem{tan1} I.J. TANEJA, On Generalized Information Measures and Their
Applications, Chapter in: \textit{Advances in Electronics and
Electron Physics}, Ed. P.W. Hawkes, Academic Press,
\textbf{76}(1989), 327-413.

\bibitem{tan2} I.J. TANEJA, New Developments in Generalized Information Measures,
Chapter in: \textit{Advances in Imaging and Electron Physics}, Ed.
P.W. Hawkes, \textbf{91}(1995), 37-135.

\bibitem{tan3} I.J. TANEJA, \textit{Generalized Information Measures
and their Applications}, on line book:
\textit{http://www.mtm.ufsc.br/$\sim $taneja/book/book.html},
2001.

\bibitem{tan4} I.J. TANEJA, Generalized Relative Information
and Information Inequalities, \textit{Journal of Inequalities in
Pure and Applied Mathematics}, \textbf{5}(1)(2004), Art.21, pp.
1-19.

\bibitem{tan5} I.J. TANEJA, On a Difference of Jensen Inequality
and its Applications to Mean Divergence Measures -- \textit{RGMIA
Research Report Collection}, \textit{http://rgmia.vu.edu.au},
\textbf{7}(4)(2004), Art. 16. Also in:arXiv:math.PR/0501302 v1 19
Jan 2005.

\bibitem{tan6} I.J. TANEJA, Refinement Inequalities Among Symmetric
Divergence Measures � \textit{The Australian Journal of
Mathematical Analysis and Applications}, \textbf{2}(1)(2005), Art.
8, pp. 1-23, Also in: arXiv:math.ST/0501303 v2 4 Apr 2005.

\bibitem{tan7} I.J. TANEJA, Bounds on Non-Symmetric Divergence Measures in
terms of Symmetric Digergence Measures - communicated.

\bibitem{tak} I.J. TANEJA and P. KUMAR, Relative Information
of Type s, Csisz\'{a}r $f-$Divergence, and Information
Inequalities, \textit{Information Sciences},
\textbf{166}(1-4)(2004), 105-125. Also in:
\textit{http://rgmia.vu.edu.au}, \textit{RGMIA Research Report
Collection}, \textbf{6}(3)(2003), Article 12.

\end{thebibliography}
\end{document}

math0505204.extracted_bib
\begin{thebibliography}{99}
\setlength{\itemsep}{5pt}

\bibitem{bcd} N.S. BARNETT, P. CERENE and S. S. DRAGOMIR, Some
New Inequalities for Hermite-Hadamard Divergence in Information
Theory, \textit{http://rgmia.vu.edu.au, RGMIA Research Report
Collection}, \textbf{(5)}(4)(2002), Article 8.

\bibitem{bha} A. BHATTACHARYYA, Some Analogues to the Amount of
Information and Their uses in Statistical Estimation,
\textit{Sankhya}, \textbf{8}(1946), 1-14.

\bibitem{bur1} J. BURBEA, J. and C.R. RAO, Entropy Differential Metric, Distance
and Divergence Measures in Probability Spaces: A Unified Approach,
\textit{J. Multi. Analysis}, \textbf{12}(1982), 575-596.

\bibitem{bur2} J. BURBEA, J. and C.R. RAO, On the Convexity of Some Divergence
Measures Based on Entropy Functions, \textit{IEEE Trans. on
Inform. Theory}, \textbf{IT-28}(1982), 489-495.

\bibitem{cdo} P. CERONE, S.S. DRAGOMIR and F. \"{O}STERREICHER, Bounds on
Extended $f-$Divergence for a Variety of Classes,
\textit{http://rgmia.vu.edu.au, RGMIA Research Report Collection},
\textbf{6}(1)(2003), Article 7.

\bibitem{crr} P. CRESSIE and T.R.C. READ, Multinomial
goodness-of-fit tests, \textit{J. Roy. Statist. Soc. Ser. B},
\textbf{46}(1984), 440-464.

\bibitem{csi1} I. CSISZ\'{A}R,
Information Type Measures of Differences of Probability
Distribution and Indirect Observations, \textit{Studia Math.
Hungarica}, \textbf{2}(1967), 299-318.

\bibitem{csi2} I. CSISZ\'{A}R,
On Topological Properties of $f-$Divergences, \textit{Studia Math.
Hungarica}, \textbf{2}(1967), 329-339.

\bibitem{dra1} S. S. DRAGOMIR, Some Inequalities for the Csisz\'{a}r $\Phi
$-Divergence - \textit{Inequalities for Csisz\'{a}r f-Divergence
in Information Theory -} Monograph -- Chapter I -- Article 1 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra2} S. S. DRAGOMIR, Other Inequalities for Csisz\'{a}r Divergence and
Applications - \textit{Inequalities for Csisz\'{a}r f-Divergence
in Information Theory -} Monograph -- Chapter I -- Article 4 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra3} S. S. DRAGOMIR, Some Inequalities for (m, M)-Convex Mappings and
Applications for the Csisz\'{a}r $\Phi $-Divergence in Information
Theory- \textit{Inequalities for Csisz\'{a}r f-Divergence in
Information Theory -} Monograph -- Chapter I -- Article 3 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra4} S. S. DRAGOMIR, Upper and Lower Bounds for Csisz\'{a}r's
f-divergence in terms of the Kullback-Leibler Distance and
Applications - \textit{Inequalities for Csisz\'{a}r f-Divergence
in Information Theory -} Monograph -- Chapter II -- Article 1
--\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dra5} S. S. DRAGOMIR, Upper and Lower Bounds for Csisz\'{a}r's
f-Divergence in terms of Hellinger Discrimination and Applications
- \textit{Inequalities for Csisz\'{a}r f-Divergence in Information
Theory -} Monograph -- Chapter II -- Article 2 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{dgp} S. S. DRAGOMIR, V. GLUSCEVIC and C.E.M. PEARCE\textbf{,
}Approximation for Csisz\'{a}r $f$-Divergence via Midpoint
Inequalities- \textit{Inequalities for Csisz\'{a}r f-Divergence in
Information Theory -} Monograph -- Chapter IV -- Article 3 --
\textit{http://rgmia.vu.edu.au/monographs/csiszar.htm.}

\bibitem{hel} E. HELLINGER, Neue Begr\"{u}ndung der Theorie der Quadratischen
Formen von Unendlichen vielen Ver\"{a}nderlichen, \textit{J. Reine
Aug. Math.}, \textbf{136}(1909), 210-271.

\bibitem{jef} H. JEFFREYS, An Invariant Form for the Prior Probability in
Estimation Problems, \textit{Proc. Roy. Soc. Lon., Ser. A},
\textbf{186}(1946), 453-461.

\bibitem{kul} S. KULLBACK and R.A. LEIBLER, On Information and Sufficiency,
\textit{Ann. Math. Statist.}, \textbf{22}(1951), 79-86.

\bibitem{liv} F. LIESE and I. VAJDA, \textit{Convex Statistical Decision Rule},
Teubner-Texte zur Mathematick, Band 95, Leipzig, 1987.

\bibitem{lin} J. LIN, Divergence Measures Based on the Shannon
Entropy, \textit{IEEE Trans. on Inform. Theory},
\textbf{IT-37}(1991), 145-151.

\bibitem{liw} J. LIN and S.K.M WONG, A New Directed Divergence
Measure and its Characterizations, \textit{Int. J. General
Systems}, \textbf{17}(1990), 73-81.

\bibitem{pea} K. PEARSON, On the Criterion that a given System of Deviations
from the Probable in the case of Correlated System of Variables is
such that it can be Reasonable supposed to have arisen from Random
Sampling, \textit{Phil. Mag.}, \textbf{50}(1900), 157-172.

\bibitem{shd} H. SHIOYA and T. DA-TE, A Generalization of Lin
Divergence and the Derivation of a New Information Divergence,
\textit{Eletronics and Communications in Japan}, Part 3
7\textbf{(78)}(1995), 34-40.

\bibitem{sib} R. SIBSON, Information Radius, \textit{Z. Wahrs. und verw Geb.},
\textbf{(14)}(1969), 149-160.

\bibitem{tan1} I.J. TANEJA, On Generalized Information Measures and Their
Applications, Chapter in: \textit{Advances in Electronics and
Electron Physics}, Ed. P.W. Hawkes, Academic Press,
\textbf{76}(1989), 327-413.

\bibitem{tan2} I.J. TANEJA, New Developments in Generalized Information Measures,
Chapter in: \textit{Advances in Imaging and Electron Physics}, Ed.
P.W. Hawkes, \textbf{91}(1995), 37-135.

\bibitem{tan3} I.J. TANEJA, \textit{Generalized Information Measures
and their Applications}, on line book:
\textit{http://www.mtm.ufsc.br/$\sim $taneja/book/book.html},
2001.

\bibitem{tan4} I.J. TANEJA, Generalized Relative Information
and Information Inequalities, \textit{Journal of Inequalities in
Pure and Applied Mathematics}, \textbf{5}(1)(2004), Art.21, pp.
1-19.

\bibitem{tan5} I.J. TANEJA, On a Difference of Jensen Inequality
and its Applications to Mean Divergence Measures -- \textit{RGMIA
Research Report Collection}, \textit{http://rgmia.vu.edu.au},
\textbf{7}(4)(2004), Art. 16. Also in:arXiv:math.PR/0501302 v1 19
Jan 2005.

\bibitem{tan6} I.J. TANEJA, Refinement Inequalities Among Symmetric
Divergence Measures � \textit{The Australian Journal of
Mathematical Analysis and Applications}, \textbf{2}(1)(2005), Art.
8, pp. 1-23, Also in: arXiv:math.ST/0501303 v2 4 Apr 2005.

\bibitem{tan7} I.J. TANEJA, Bounds on Non-Symmetric Divergence Measures in
terms of Symmetric Digergence Measures - communicated.

\bibitem{tak} I.J. TANEJA and P. KUMAR, Relative Information
of Type s, Csisz\'{a}r $f-$Divergence, and Information
Inequalities, \textit{Information Sciences},
\textbf{166}(1-4)(2004), 105-125. Also in:
\textit{http://rgmia.vu.edu.au}, \textit{RGMIA Research Report
Collection}, \textbf{6}(3)(2003), Article 12.

\end{thebibliography}