# Moduli difference of inverse logarithmic coefficients of univalent functions

Vasudevarao Allu  Vasudevarao Allu, School of Basic Sciences, Indian Institute
of Technology Bhubaneswar, Odisha, India<EMAIL_ADDRESS>and  Amal Shaji
Amal Shaji, School of Basic Sciences, Indian Institute of Technology
Bhubaneswar, Bhubaneswar-752050, Odisha, India<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of
normalized univalent functions with $f(0)=0$, and $f^{\prime}(0)=1$. Let $F$
be the inverse function of $f$, given by
$F(w)=w+\sum_{n=2}^{\infty}A_{n}w^{n}$ defined on some disk $|w|\leq
r_{0}(f)$. The inverse logarithmic coefficients $\Gamma_{n}$,
$n\in\mathbb{N}$, of $f$ are defined by the equation
$\log(F(w)/w)=2\sum_{n=1}^{\infty}\Gamma_{n}w^{n},\,|w|<1/4.$ In this paper,
we find the sharp upper and lower bounds for moduli difference of second and
first inverse logarithmic coefficients, i.e., $|\Gamma_{2}|-|\Gamma_{1}|$ for
functions in class $\mathcal{S}$ and for functions in some important
subclasses of univalent functions.

###### Key words and phrases:

Inverse coefficients, Successive coefficients, Univalent functions,Inverse
logarithmic coefficients

###### 2010 Mathematics Subject Classification:

30D30, 30C45, 30C50, 30C55.

##  1\. Introduction

Let $\mathcal{H}$ denote the class of analytic functions in the unit disk
$\mathbb{D}:=\\{z\in\mathbb{C}:\,|z|<1\\}$. Here $\mathcal{H}$ is a locally
convex topological vector space endowed with the topology of uniform
convergence over compact subsets of $\mathbb{D}$. Let $\mathcal{A}$ denote the
class of functions $f\in\mathcal{H}$ such that $f(0)=0$ and $f^{\prime}(0)=1$.
Let $\mathcal{S}$ denote the subclass of $\mathcal{A}$ consisting of functions
which are univalent (i.e., one-to-one) in $\mathbb{D}$. If $f\in\mathcal{S}$
then it has the following series representation

(1.1) $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n},\quad z\in\mathbb{D}.$

A domain $D\subset\mathbb{C}$ is said to be starlike with respect to a point
$z_{0}\in D$ if the line segment joining $z_{0}$ to every other point $z\in D$
lies entirely in $D$. A function $f\in\mathcal{A}$ is called starlike if
$f(\mathbb{D})$ is a starlike domain with respect to the origin. The class of
univalent starlike functions is denoted by $\mathcal{S}^{*}$. A domain
$D\subset\mathbb{C}$ is said to be convex if the line segment joining any two
arbitrary points of $D$ lies entirely in $D$; i.e., if it is starlike with
respect to each points of $D$. A function $f\in\mathcal{A}$ is said to be
convex in $\mathbb{D}$ if $f(\mathbb{D})$ is a convex domain. The class of all
univalent convex functions is denoted by $\mathcal{C}$ (see [8, 37]). The
classes $\mathcal{S}^{*}$ of starlike functions and $\mathcal{C}$ of convex
functions are analytically defined respectively as

$\displaystyle\mathcal{S}^{*}$ $\displaystyle=\left\\{f\in\mathcal{A}:{\rm
Re}\left(\cfrac{zf^{\prime}(z)}{f(z)}\right)>0,z\in\mathbb{D}\right\\},$
$\displaystyle\mathcal{C}$ $\displaystyle=\left\\{f\in\mathcal{A}:{\rm
Re}\left(1+\cfrac{zf^{\prime}(z)}{f(z)}\right)>0,z\in\mathbb{D}\right\\}.$

Given $\alpha\in[0,1)$, a function $f\in\mathcal{A}$ of the form (1.1) is
called starlike functions of order $\alpha$, if

(1.2) ${\rm Re}\left(\cfrac{zf^{\prime}(z)}{f(z)}\right)>\alpha,\quad
z\in\mathbb{D}.$

The set of all such functions is denoted by $\mathcal{S}^{*}(\alpha)$. A
function $f\in\mathcal{A}$ is called convex function of order $\alpha$, if

(1.3) ${\rm
Re}\left(1+\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)>\alpha,\quad
z\in\mathbb{D}.$

The set of all such functions is denoted by $\mathcal{C}(\alpha)$. For
$\alpha:=0$, these classes reduce to the well-known classes $\mathcal{S}^{*}$
and $\mathcal{C}$ , the class of starlike functions and the class of convex
functions, respectively.
A function $f\in\mathcal{A}$ of the form (1.1) is said to be strongly starlike
of order $\alpha$, $(0<\alpha\leq 1)$, if

(1.4)
$\left|\text{arg}\left(\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)\right|<\cfrac{\pi\alpha}{2},\quad
z\in\mathbb{D}.$

The set of all such functions is denoted by $\mathcal{S}^{*}_{\alpha}$. A
function $f\in\mathcal{A}$ of the form (1.1) belongs to
$\mathcal{C}_{\alpha}$, the class strongly convex function of order $\alpha$,
$(0<\alpha\leq 1)$, if

(1.5)
$\left|\text{arg}\left(1+\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)\right|<\cfrac{\pi\alpha}{2},\quad
z\in\mathbb{D}.$

The notion of strongly starlike functions was introduced by Stankiewicz [34]
and independently by Brannan and Kirwan[7]. An external geometric
characterisation of strongly starlike functions has been proposed by
Stankiewicz [35]. Brannan and Kirwan[7] have obtained a geometrical condition
called $\delta-$ visibility which is sufficient for functions to be starlike.

In 1985, de Branges [6] solved the famous Bieberbach conjecture, by showing
that if $f\in\mathcal{S}$ of the form (1.1), then $|a_{n}|\leq n$ for $n\geq
2$ with equality holds for Koebe function $k(z):=z/(1-z)^{2}$ or its
rotations. It was therefore natural to ask if for $f\in\mathcal{S}$, the
inequality $||a_{n+1}|-|a_{n}||\leq 1$ is true when $n\geq 2$. This problem
was first studied by Goluzin [10] with an aim to solve the Bieberbach
conjecture. In 1963, Hayman [12] proved that
$\big{|}|a_{n+1}|-|a_{n}|\big{|}\leq A$ for $f\in\mathcal{S}$, where $A\geq 1$
is an absolute constant and the best known estimate as of now is $3.61$ due to
Grinspan [11]. On the other hand, for the class $\mathcal{S}$, the sharp bound
is known only for $n=2$ (see [8, Theorem 3.11]), namely

$-1\leq|a_{3}|-|a_{2}|\leq 1.029\ldots.$

Similarly, for functions $f\in\mathcal{S}^{*}$, Pommerenke [24] has
conjectured that $||a_{n+1}|-|a_{n}||\leq 1$ which was proved later in 1978 by
Leung [15]. For convex functions, Li and Sugawa [16] investigated the sharp
upper bound of $|a_{n+1}|-|a_{n}|$ for $n\geq 2$, and sharp lower bounds for
$n=2,3$.

For $f\in\mathcal{S}$, denote by $F$ the inverse of $f$ given by

(1.6) $F(w)=w+\sum_{n=2}^{\infty}A_{n}w^{n},$

valid on some disk $|w|\leq r_{0}(f)$. Since $f(f^{-1}(w))=w$, by equating the
coefficients, we can easily obtain

(1.7) $A_{2}=-a_{2}\text{ and }A_{3}=2a_{2}^{2}-a_{3}.$

The inverse functions are studied by several authors in different perspective
(see, for instance, [37, 30] and reference therein). Recently, Sim and Thomas
[30, 31] obtained sharp upper and lower bounds on the difference of the moduli
of successive inverse coefficients for the subclasses of univalent functions.
The Logarithmic coefficients $\gamma_{n}$ of $f\in\mathcal{S}$ are defined by,

(1.8)
$F_{f}(z):=\log\frac{f(z)}{z}=2\sum\limits_{n=1}^{\infty}\gamma_{n}z^{n},\quad
z\in\mathbb{D}.$

The logarithmic coefficients $\gamma_{n}$ play a central role in the theory of
univalent functions. A very few exact upper bounds for $\gamma_{n}$ seem to
have been established. The significance of this problem in the context of
Bieberbach conjecture was pointed by Milin[18] in his conjecture. Milin [18]
has conjectured that for $f\in\mathcal{S}$ and $n\geq 2$,

$\sum\limits_{m=1}^{n}\sum\limits_{k=1}^{m}\left(k|\gamma_{k}|^{2}-\frac{1}{k}\right)\leq
0,$

which led De Branges, by proving this conjecture, to the proof of Bieberbach
conjecture [6]. For the Koebe function $k(z)=z/(1-z)^{2}$, the logarithmic
coefficients are $\gamma_{n}=1/n$. Since the Koebe function $k$ plays the role
of extremal function for most of the extremal problems in the class
$\mathcal{S}$, it is expected that $|\gamma_{n}|\leq 1/n$ holds for functions
in $\mathcal{S}$. But this is not true in general, even in order of magnitude.
Recently, various authors have taken an interest in the examination of
logarithmic coefficients within the class $\mathcal{S}$ and its subclasses
(see[1, 2, 36]).

The notion of inverse logarithmic coefficients, i.e., logartithmic
coefficients of inverse of $f$, has been proposed by Ponnusamy et al. [27].
The inverse logarithmic coefficients $\Gamma_{n}$, $n\in\mathbb{N}$, of $f$
are defined by the equation

(1.9)
$F_{f^{-1}}(w):=\log\frac{f^{-1}(w)}{w}=2\sum\limits_{n=1}^{\infty}\Gamma_{n}w^{n},\quad|w|<1/4.$

By differentiating (1.9) together with (1.7), we obtain

(1.10) $\displaystyle\Gamma_{1}=-\cfrac{1}{2}\,a_{2},$
$\displaystyle\Gamma_{2}=-\cfrac{1}{2}\,a_{3}+\cfrac{3}{4}\,a_{2}^{2}.$

In 2018, Ponnusamy et al. [27] obtained the sharp upper bound for the
logarithmic inverse coefficients for the class $\mathcal{S}$. In fact
Ponnusamy et al. [27] have proved that when $f\in\mathcal{S}$,

$|\Gamma_{n}|\leq\frac{1}{2n}\left(\begin{matrix}2n\\\
n\end{matrix}\right),\quad n\in\mathbb{N}$

and equality holds only for the Koebe function or one of its rotations.
Further, Ponnusamy et al. [27] have obtained the sharp bound for the initial
logarithmic inverse coefficients for some of the important geometric
subclasses of $\mathcal{S}$.

In 2023, Lecko and Partyka [14] studied the problem of finding the upper and
lower bound for $|\gamma_{2}|-|\gamma_{1}|$ for functions in class
$\mathcal{S}$ using Loewner technique. Recently, Obradović and Tuneski [20]
provided a simple proof of the same problem. Also kumar and cho [13] obtained
sharp upper and lower bounds for $|\gamma_{2}|-|\gamma_{1}|$ for functions in
subclasses of class $\mathcal{S}$.
In this paper, we consider the problem of finding sharp upper and lower bound
of moduli difference of second and first inverse logarithmic coefficients.

Before presenting the main results of this paper, we will discuss some
important subclasses of univalent functions

######  Definition 1.1.

A locally univalent function $f\in\mathcal{A}$ is said to belong to
$\mathcal{G}(\nu)$ for some $\nu>0$, if it satisfies the condition

${\rm
Re}\bigg{(}1+\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{)}<1+\cfrac{\nu}{2},\quad
z\in\mathbb{D}.$

In 1941, Ozaki [21] introduced the class $\mathcal{G}(1)=:\mathcal{G}$ and
proved that functions in $\mathcal{G}$ are univalent in $\mathbb{D}$. Later,
Umezawa [38] studied the class $\mathcal{G}$ and showed that this class
contains the class of functions convex in one direction. Moreover, functions
in $\mathcal{G}$ are proved to be starlike in $\mathbb{D}$ (see [25],[28]).
Thus, the class $\mathcal{G}(\nu)$ is included in $\mathcal{S}^{*}$ whenever
$\nu\in(0,1]$. It can be easily seen that functions in $\mathcal{G}(\nu)$ are
not necessarily univalent in $\mathbb{D}$ if $\nu>1$.

######  Definition 1.2.

For $-1/2<\lambda\leq 1$, the class $\mathcal{F}(\lambda)$ defined by

$\mathcal{F}(\lambda)=\left\\{f\in\mathcal{A}:\,{\rm
Re}\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)>\frac{1}{2}-\lambda~{}\mbox{
for $z\in\mathbb{D}$}\right\\}.$

We note that clearly $\mathcal{F}(1/2)=:\mathcal{C}$ is the usual class of
convex functions. Moreover, for $\lambda=1$, we obtain the class
$\mathcal{F}(1)=:\mathcal{C}(-1/2)$ which considered by many researcher in the
recent years (see [AV19, 4, 19, 26]). Also, functions in $\mathcal{C}(-1/2)$
are not necessarily starlike but are convex in some direction and so are
close-to-convex. Here, we recall that a function $f\in\mathcal{A}$ is called
close-to-convex if $f(\mathbb{D})$ is close-to-convex domain,i.e. the
complement of $f(\mathbb{D})$ in $\mathcal{C}$ is the union of closed half
lines with pairwise disjoint interiors. Pfaltzgraff et al. [23] has proved
that $F(\lambda)$ contains non-starlike functions for all $1/2<\lambda\leq 0$.

Next we will discuss about the family of spirallike functions.

######  Definition 1.3.

The family $\mathcal{S}_{\gamma}(\alpha)$ of $\gamma$-spirallike functions of
order $\alpha$ is defined by

$\mathcal{S}_{\gamma}(\alpha)=\left\\{f\in\mathcal{A}:\operatorname{Re}\left(e^{-i\gamma}\frac{zf^{\prime}(z)}{f(z)}\right)>\alpha\cos\gamma\text{
for }z\in\mathbb{D}\right\\},$

where $\alpha\in[0,1)$ and $\gamma\in(-\pi/2,\pi/2)$.

Each function in $\mathcal{S}_{\gamma}(\alpha)$ is univalent in $\mathbb{D}$
(see [17]). Clearly,
$\mathcal{S}_{\gamma}(\alpha)\subset\mathcal{S}_{\gamma}(0)\subset\mathcal{S}$
whenever $0\leq\alpha<1$. Functions in $\mathcal{S}_{\gamma}(0)$ are called
$\gamma$-spirallike, but they do not necessarily belong to the starlike family
$\mathcal{S}^{*}$. The class $\mathcal{S}_{\gamma}(0)$ was introduced by
Špaček [33]. Moreover, $\mathcal{S}_{0}(\alpha)=:\mathcal{S}^{*}(\alpha)$ is
Robertson’s class of functions that are starlike functions of order $\alpha$,
and $\mathcal{S}^{*}(0)=\mathcal{S}^{*}$ is the class of starlike functions.
The class $\mathcal{S}^{*}(\alpha)$ is meaningful even if $\alpha<0$, although
univalency will be destroyed in this situation.
We consider another family of functions that includes the class of convex
functions as a proper subfamily.

######  Definition 1.4.

The family $\mathcal{C}_{\gamma}(\alpha)$ of $\gamma$-convex functions of
order $\alpha$ is defined by

${\mathcal{C}}_{\gamma}(\alpha)=\left\\{f\in\mathcal{A}:{\rm
Re}\bigg{(}e^{-i\gamma}\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)\bigg{)}>\alpha\cos\gamma\right\\}$

where $0\leq\alpha<1$ and $-\pi/2<\gamma<\pi/2$.

We may set ${\mathcal{C}}_{0}(\alpha)=:{\mathcal{C}}(\alpha)$ which consists
of the normalized convex functions of order $\alpha$. Function in
${\mathcal{C}}_{\gamma}(0)=:{\mathcal{C}}_{\gamma}$ need not be univalent in
$\mathbb{D}$ for general values of $\gamma$ $(|\gamma|<\pi/2)$. For example,
the function $f(z)=i(1-z)^{i}-i$ is known to belong to
${\mathcal{C}}_{\pi/4}\backslash{\mathcal{S}}$. Robertson [29] has shown that
$f\in\mathcal{C}_{\gamma}$ is univalent if $0<\cos\gamma\leq 0.2315\cdots$.
Finally, Pfaltzgraff [22] has shown that $f\in\mathcal{C}_{\gamma}$ is
univalent whenever $0<\cos\gamma\leq 1/2$. This settles the improvement of the
range of $\gamma$ for which $f\in\mathcal{C}_{\gamma}$ is univalent. On the
other hand, in [32] it was also shown that functions in
${\mathcal{C}}_{\gamma}$ which satisfy $f^{\prime\prime}(0)=0$ are univalent
for all real values of $\gamma$ with $|\gamma|<\pi/2$. For the recent study of
the class for particular values of $\alpha$ and $\gamma$, we refer to [5].

##  2\. Preliminary Results

######  Theorem 2.1.

Fekete-Szegö Theorem[9]: If $f\in\mathcal{S}$ of the form (1.1), then

$|a_{3}-\mu a_{2}^{2}|\leq\left\\{\begin{array}[]{ll}4\mu-3,&{\mbox{ if
}}\mu\geq 1,\\\\[5.69054pt] 1+2e^{-2\mu/(1-\mu)},&{\mbox{ if
}}0<\mu<1,\\\\[5.69054pt] 3-4\mu,&{\mbox{ if}}\,\,\mu\leq
0,\end{array}\right.$

This bound is sharp for each $\mu$.

The functional $|a_{3}-\mu a_{2}^{2}|$ is well-known as the Fekete-Szegö
functional.

Let $\mathcal{P}$ denote the class of all analytic functions $p$ having
positive real part in $\mathbb{D}$, of the form

(2.1) $p(z)=1+c_{1}z+c_{2}z^{2}+\cdots.$

A member of $\mathcal{P}$ is called a Carathéodory function. It is known that
$|c_{n}|\leq 2$ for a function $p\in\mathcal{P}$ and for all $n\geq 1$ (see
[8]).

To prove our results, we need the following lemma.

######  Lemma 2.2.

[30] Let $B_{1}$, $B_{2}$, and $B_{3}$ be numbers such that $B_{1}>0$,
$B_{2}\in\mathbb{C},$ and $B_{3}\in\mathbb{R}$. Let $p\in\mathcal{P}$ be of
the form (2.1). Define $\Psi_{+}(c_{1},c_{2})$ and $\Psi_{-}(c_{1},c_{2})$ by

$\Psi_{+}(c_{1},c_{2})=|B_{2}c_{1}^{2}+B_{3}c_{2}|-|B_{1}c_{1}|,$

and

$\Psi_{-}(c_{1},c_{2})=-\Psi_{+}(c_{1},c_{2}).$

Then

(2.2)
$\Psi_{+}(c_{1},c_{2})\leq\left\\{\begin{array}[]{ll}|4B_{2}+2B_{3}|-2B_{1},&{\mbox{
if }}|2B_{2}+B_{3}|\geq|B_{3}|+B_{1},\\\\[14.22636pt] 2|B_{3}|,&{\mbox{
otherwise}},\end{array}\right.$

and

(2.3)
$\Psi_{-}(c_{1},c_{2})\leq\left\\{\begin{array}[]{ll}2B_{1}-B_{4},&{\mbox{ if
}}B_{1}\geq B_{4}+2|B_{3}|,\\\\[14.22636pt]
2B_{1}\sqrt{\cfrac{2|B_{3}|}{B_{4}+2|B_{3}|}},&{\mbox{ if }}B_{1}^{2}\leq
2|B_{3}|(B_{4}+2|B_{3}|),\\\\[14.22636pt]
2|B_{3}|+\cfrac{B_{1}^{2}}{B_{4}+2|B_{3}|},&{\mbox{
otherwise}},\end{array}\right.$

where $B_{4}=|4B_{2}+2B_{3}|$. All inequalities in (2.2) and (2.3) are sharp.

Our main aim of this paper is to estimate the sharp Lower and upper bounds of
$|\Gamma_{2}|-|\Gamma_{1}|$ for functions $f$ belong to $\mathcal{S}$,
$\mathcal{S}^{*}$, $\mathcal{C}$, $\mathcal{S}^{*}_{\alpha}$,
$\mathcal{C}_{\alpha}$, $\mathcal{S}^{*}(\alpha)$, $\mathcal{C}(\alpha)$,
$\mathcal{G}(\nu)$, $\mathcal{F}_{0}(\lambda)$,
$\mathcal{S}^{*}_{\gamma}(\alpha)$ and $\mathcal{C}_{\gamma}(\alpha)$.

##  3\. Main Results

We now state our first main result which provides sharp bounds for
$|\Gamma_{2}|-|\Gamma_{1}|$ when $f$ belongs to the class $\mathcal{S}$.

######  Theorem 3.1.

Let $f\in\mathcal{S}$ of the form (1.1), then

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1}{2}$

and

$|\Gamma_{2}|-|\Gamma_{1}|\geq\left\\{\begin{array}[]{ll}-\cfrac{1}{2}&{\mbox{
if }}|a_{2}|\leq 1,\\\\[8.53581pt] -0.6353...,&{\mbox{ if
}}|a_{2}|>1.\end{array}\right.$

###### Proof.

Let $f\in\mathcal{S}$ of the form (1.1). Then by (1.10), we have

(3.1)
$|\Gamma_{2}|-|\Gamma_{1}|=\cfrac{1}{2}\left(|a_{3}-\cfrac{3}{2}a_{2}^{2}|-|a_{2}|\right).$

From the Bieberbach conjecture, for all $f\in\mathcal{S}$, we have
$\cfrac{1}{2}|a_{2}|\leq 1$, and so by (3.1)

(3.2) $\displaystyle 2(|\Gamma_{2}|-|\Gamma_{1}|)$
$\displaystyle\leq\left|a_{3}-\cfrac{3}{2}a_{2}^{2}\right|-\cfrac{1}{2}\left|a_{2}^{2}\right|$
$\displaystyle\leq\left|a_{3}-\cfrac{3}{2}a_{2}^{2}+\cfrac{1}{2}a_{2}^{2}\right|$
$\displaystyle\leq\left|a_{3}-a_{2}^{2}\right|$ $\displaystyle\leq 1,$

and hence we have

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1}{2}$

Now we obtain the lower bound for $|\Gamma_{2}|-|\Gamma_{1}|$. We have

$|\Gamma_{1}|-|\Gamma_{2}|=\left(\cfrac{1}{2}\left|a_{2}\right|-\left|a_{3}-\cfrac{3}{2}a_{2}^{2}\right|\right).$

If $|a_{2}|\leq 1$, then clearly $|\Gamma_{1}|-|\Gamma_{2}|\leq 1/2$. If
$|a_{2}|>1$, then

(3.3) $\displaystyle 2(|\Gamma_{1}|-|\Gamma_{2}|)$
$\displaystyle=\left|a_{2}\right|-\left|a_{3}-\cfrac{3}{2}a_{2}^{2}\right|$
$\displaystyle\leq\left|a_{2}\right|^{2}-\left|a_{3}-\cfrac{3}{2}a_{2}^{2}\right|$
$\displaystyle\leq\left|a_{3}-\cfrac{3}{2}a_{2}^{2}+a_{2}^{2}\right|$
$\displaystyle=\left|a_{3}-\frac{1}{2}a_{2}^{2}\right|$

By taking $\mu=1/2$ in Fekete-Szegö Theorem and using (3.3), we obtain

$|\Gamma_{1}|-|\Gamma_{2}|\leq\cfrac{1+2e^{-2}}{2}=0.6353...$

For the upper bound, equality holds for the Koebe function $k(z)=z/(1-z)^{2}$.
For the lower bound, when $|a_{2}|\leq 1$, equality holds for the function

$f(z)=\cfrac{z}{1+z+z^{2}}.$

For $|a_{2}|>1$, whether the lower bound $-0.6353...$ is the best possible is
an open problem.

∎

Next, we obtain the sharp lower and upper bounds for
$|\Gamma_{2}|-|\Gamma_{1}|$ when $f$ belongs to the class
$\mathcal{S}^{*}_{\alpha}$.

######  Theorem 3.2.

Let $\alpha\in(0,1]$. If $f\in\mathcal{S}^{*}_{\alpha}$ given by (1.1), then
the following sharp inequalities holds.

(3.4)
$-\cfrac{\alpha}{\sqrt{1+3\alpha}}\leq|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{\alpha}{2}.$

###### Proof.

Fix $\alpha\in(0,1]$ and let $f\in\mathcal{S}^{*}_{\alpha}$ be of the form
$(1.1)$. Then by (1.4), we have

(3.5) $\cfrac{zf^{\prime}(z)}{f(z)}=(p(z))^{\alpha}$

for some $p\in\mathcal{P}$ of the form (2.1). By comparing the coefficient of
powers of $z$ on both sides of (3.5), we obtain

(3.6) $a_{2}=\alpha
c_{1}\,\,\,\text{and}\,\,\,a_{3}=\frac{\alpha}{4}(2c_{2}+(3\alpha-1)c_{1}^{2}).$

Now using (1.10) together with (3.6), we have

(3.7)
$|\Gamma_{2}|-|\Gamma_{1}|=\cfrac{\alpha}{4}\left(|B_{3}c_{2}+B_{2}c_{1}^{2}|-|B_{1}c_{1}|\right)=\cfrac{\alpha}{4}\,\Psi_{+}(c_{1},c_{2}),$

where

$B_{1}:=2,B_{2}:=-\frac{1}{2}(1+3\alpha),\mbox{ and }B_{3}:=1.$

For the upper bound, we see that the condition
$|2B_{2}+B_{3}|\leq|B_{3}|+B_{1}$ is equivalent to $3\alpha\leq 3$, which is
true only if $\alpha=1$. Therefore, by Lemma 2.2, we have

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{\alpha}{2}.$

Equality holds for the function $f\in\mathcal{A}$ given by (3.5), where
$p(z)=(1+z^{2})/(1-z^{2})$. Then $c_{1}=0$ and $c_{2}=2$, so by (3.6),
$a_{2}=0$ and $a_{3}=\alpha$, and therefore by (1.10),

$|\Gamma_{2}|-|\Gamma_{1}|=\cfrac{\alpha}{2}.$

For the lower bound, since $B_{4}=|4B_{2}+2B_{3}|=6\alpha$, it is easy to see
that the inequality $B_{1}\geq B_{4}+2|B_{3}|$ doesnot holds and the
inequality $B_{1}^{2}\leq 2|B_{3}|(B_{4}+2|B_{3}|)$ holds for $0<\alpha\leq
1$. Hence by Lemma 2.2, we obtain

$|\Gamma_{2}|-|\Gamma_{1}|\geq-\cfrac{\alpha}{\sqrt{1+3\alpha}}\,.$

We finally show that the left hand side inequality in (3.4) is sharp. Equality
holds for the function $f\in\mathcal{A}$ given by (3.5) with

$p(z)=\cfrac{1+2Az+z^{2}}{1-z^{2}},\quad z\in\mathbb{D},$

where $A=1/\sqrt{1+3\alpha}$, which completes the proof of the theorem. ∎

If we put $\alpha=0$ in Theorem 3.2, then we obtain the following result for
the class of starlike functions.

######  Corollary 3.3.

For every $f\in\mathcal{S}^{*}$ of the form (1.1), we have

(3.8) $-\cfrac{1}{2}\leq|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1}{2}.$

Both inequalities are sharp.

Next, we obtain the sharp lower and upper bounds for
$|\Gamma_{2}|-|\Gamma_{1}|$ when $f$ belongs to the class
$\mathcal{C}_{\alpha}$.

######  Theorem 3.4.

Let $\alpha\in(0,1]$. If $f\in\mathcal{C}_{\alpha}$ given by (1.1), then the
following sharp inequalities holds;

(3.9) $|\Gamma_{2}|-|\Gamma_{1}|\leq\frac{\alpha}{12}$

and

(3.10)
$|\Gamma_{2}|-|\Gamma_{1}|\geq\left\\{\begin{array}[]{lll}-\cfrac{1}{4}\alpha(2-\alpha)&{\mbox{
if }}\,\,0<\alpha\leq\cfrac{1}{3},\\\\[14.22636pt]
-\cfrac{\alpha}{6}\left(\cfrac{6\alpha+3}{6\alpha+14}\right)&{\mbox{ if
}}\,\,\cfrac{1}{3}<\alpha<\cfrac{5}{6},\\\\[14.22636pt]
-\cfrac{\alpha}{\sqrt{4+6\alpha}}&{\mbox{ if }}\,\,\cfrac{5}{6}\leq\alpha\leq
1.\end{array}\right.$

###### Proof.

Fix $\alpha\in(0,1]$ and let $f\in\mathcal{K}_{\alpha}$ be of the form (1.1).
Then by (1.5),

(3.11) $1+\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}=(p(z))^{\alpha}$

for some $p\in\mathcal{P}$ of the form (2.1). By comparing the coefficients of
powers of $z$ on both the sides of (3.11), we get

(3.12) $a_{2}=\cfrac{1}{2}\,\alpha
c_{1}\,\,\,\text{and}\,\,\,a_{3}=\frac{\alpha}{12}(2c_{2}+(3\alpha-1)c_{1}^{2}).$

Now using (1.10) together with (3.12), we obtain

$\displaystyle|\Gamma_{2}|-|\Gamma_{1}|$
$\displaystyle=\cfrac{1}{48}\alpha|4c_{2}-(2+3\alpha)c_{1}^{2}|-\cfrac{1}{4}\alpha|c_{1}|$
$\displaystyle=\cfrac{\alpha}{12}\left(|B_{3}c_{2}+B_{2}c_{1}^{2}|-|B_{1}c_{1}|\right)=\cfrac{\alpha}{12}\,\Psi_{+}(c_{1},c_{2})$

where

$B_{1}:=3,B_{2}:=-\frac{1}{4}(2+3\alpha),\mbox{ and }B_{3}:=1.$

It is easy to see that the condition $|2B_{2}+B_{3}|>|B_{3}|+B_{1}$ holds for
$0<\alpha\leq 1$, Therefore by Lemma 2.2, we get

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{\alpha}{6}.$

Equality holds for the function $f$ is of the form (3.11), with

$p(z)=\cfrac{1+z^{2}}{1-z^{2}},\quad z\in\mathbb{D}.$

We now consider the lower bound. Then

(3.13) $|\Gamma_{1}|-|\Gamma_{2}|=\cfrac{\alpha}{12}\,\Psi_{-}(c_{1},c_{2}),$

where $\Psi_{-}(c_{1},c_{2})=-\Psi_{+}(c_{1},c_{2})$. Since $B_{4}=3\alpha$,
the inequality $B_{1}\geq B_{4}+2|B_{3}|$ holds for $0<\alpha\leq 1/3$.
Observe that $2|B_{3}|(B_{4}+2|B_{3}|)-B_{1}^{2}=6\alpha-5\geq 0$ is true for
$5/6\leq\alpha\leq 1$. Hence, Lemma 2.2 gives

(3.14)
$\Psi_{-}(c_{1},c_{2})\leq\left\\{\begin{array}[]{lll}\cfrac{1}{4}\alpha(2-\alpha)&{\mbox{
if }}\,\,0<\alpha\leq\cfrac{1}{3},\\\\[11.38109pt]
\cfrac{\alpha}{6}\left(\cfrac{6\alpha+3}{6\alpha+14}\right)&{\mbox{ if
}}\,\,\cfrac{1}{3}<\alpha<\cfrac{5}{6},\\\\[11.38109pt]
\cfrac{\alpha}{\sqrt{4+6\alpha}}&{\mbox{ if }}\,\,\cfrac{5}{6}\leq\alpha\leq
1.\end{array}\right.$

From (3.13) and (3.14), we obtain the inequality in (3.10) as required.

We finally show that the inequalities in (3.10) are sharp. When
$\alpha\in(0,1/3]$, equality holds for the function $f\in\mathcal{A}$ given by
(3.11) with $p(z)=1+z/1-z$. In this case $a_{2}=\alpha$ and $a_{3}=\alpha^{2}$
and so by (1.10), $\Gamma_{1}=-\alpha/2$ and $\Gamma_{2}=-\alpha^{2}/4$. When
$\alpha\in(1/3,5/6)$, equality holds for the function $f\in\mathcal{A}$ given
by (3.11) with

$p(z)=\cfrac{1+2Bz+z^{2}}{1-z^{2}},$

where $B=3/(3+2\alpha)$. When $\alpha\in(5/6,1]$, equality holds for the
function $f\in\mathcal{A}$ given by (3.11) with

$p(z)=\cfrac{1+2Cz+z^{2}}{1-z^{2}},$

where $C=3/\sqrt{3+2\alpha}$. This completes the proof of the theorem. ∎

If we put $\alpha=0$ in Theorem 3.9, then we obtain the following result for
the class of convex functions.

######  Corollary 3.5.

For every $f\in\mathcal{C}$ of the form (1.1), we have

(3.15) $-\cfrac{1}{\sqrt{10}}\leq|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1}{6}.$

Both inequalities are sharp.

######  Theorem 3.6.

Let $f\in\mathcal{G}(\nu)$ of the form $(1.1)$. Then the following sharp
estimate holds.

(3.16) $|\Gamma_{2}|-|\Gamma_{1}|\leq\frac{\nu}{12}$

(3.17)
$|\Gamma_{2}|-|\Gamma_{1}|\geq\left\\{\begin{array}[]{ll}\cfrac{\nu}{12}\left(\cfrac{10\nu+34}{5\nu+8}\right)&{\mbox{
if }}\,\,\cfrac{1}{5}\leq\nu\leq 1,\\\\[11.38109pt]
\cfrac{\nu}{\sqrt{5\nu+8}},&{\mbox{ if }}\,\,\cfrac{1}{5}\leq\nu\leq
1.\end{array}\right.$

###### Proof.

Let $f\in\mathcal{G}(\nu)$. Then there exists a function $p(z)$ of the form
(2.1) satisfying the condition,

(3.18)
$p(z)=\cfrac{1}{\nu}\bigg{(}\nu-\cfrac{2zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{)}.$

Then, on equating coefficients of powers of $z$ in (3.18), we have

(3.19) $a_{2}=-\cfrac{\nu
c_{1}}{4}\,\,\,\text{and}\,\,\,a_{3}=\cfrac{\nu^{2}c_{1}^{2}-2\nu c_{2}}{24}.$

Using (1.10) and (3.19), we obtain

(3.20)
$|\Gamma_{2}|-|\Gamma_{1}|=\cfrac{\nu}{24}\left(|B_{3}c_{2}+B_{2}c_{1}^{2}|-|B_{1}C_{1}|\right)=\cfrac{\nu}{24}\,\,\Psi_{+}(c_{1},c_{2})$

where

$B_{1}:=3,B_{2}:=\frac{5}{8}\nu,\mbox{ and }B_{3}:=1.$

Also note that $B_{4}=|4B_{2}+2B_{3}|=(5\nu+4)/4$.
For the upper bound, we see that the condition
$|2B_{2}+B_{3}|\leq|B_{3}|+B_{1}$ is equivalent to $5\nu\leq 12$, which is not
true for $0<\nu\leq 1$. Therefore

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{\nu}{12}\,.$

It is easy to see that equality holds when $f\in\mathcal{G}(\nu)$ defined by
(3.18) with $p(z)=(1+z^{2})/(1-z^{2})$.
We now proceed to prove the lower bound for $|\Gamma_{2}|-|\Gamma_{1}|$. It is
easy to show that the condition $B_{1}\geq B_{4}+2|B_{3}|$ doesnot hold for
$0\leq\nu\leq 1$, but the condition $B_{1}^{2}\leq 2|B_{3}|(B_{4}+2|B_{3}|)$
holds for $\nu\geq 1/5$. Thus by Lemma 2.2, we obtain

$\Psi_{-}(c_{1},c_{2})\leq\left\\{\begin{array}[]{ll}\cfrac{10\nu+34}{5\nu+8},&{\mbox{
if }}0<\nu<1/5,\\\\[11.38109pt] \cfrac{12}{\sqrt{5\nu+8}},&{\mbox{ if
}}1/5\leq\nu\leq 1.\end{array}\right.$

By substituting the above inequality in (3.20), we get the inequality in
Theorem 3.6.
For $0<\nu<1/5$, equality holds for the function $f_{2}$ is of the form (3.18)
with

$p(z)=\cfrac{1-z^{2}}{1-2tz+z^{2}},$

where $t=6/(5\nu+8)$. For $1/5\leq\nu\leq 1$, equality holds for the function
$f_{3}$ is of the form (3.18) with

$p(z)=\cfrac{1-z^{2}}{1-2sz+z^{2}},$

where $s=2/(\sqrt{5\nu+8})$. This completes the proof of the theorem. ∎

######  Theorem 3.7.

Let $1/2\leq\lambda\leq 1$. For every $f\in\mathcal{F}_{0}(\lambda)$ be of the
form (1.1), then the following sharp inequalities holds.

(3.21)
$-\cfrac{1+2\lambda}{2\sqrt{5+10\lambda}}\leq|\Gamma_{2}|-|\Gamma_{1}|\leq\frac{\nu}{12}.$

###### Proof.

Let $f\in\mathcal{F}_{0}(\lambda)$ be of the form (1.1). Then we have

(3.22)
$1+\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}=\bigg{(}\cfrac{1}{2}+\lambda\bigg{)}p(z)+\cfrac{1}{2}-\lambda,\quad
z\in\mathbb{D}.$

By comparing the coefficients of powers of $z$ on both the sides of (3.22), we
obtain

(3.23)
$a_{2}=\cfrac{(1+2\lambda)c_{1}}{4}\,\,\,\text{and}\,\,\,a_{3}=\cfrac{(1+2\lambda)(2c_{2}+(1+2\lambda)c_{1}^{2})}{24}.$

Now using (1.10) together with (3.23), we have

(3.24)
$|\Gamma_{2}|-|\Gamma_{1}|=\cfrac{(1+2\lambda)}{24}\left(|B_{3}c_{2}+B_{2}c_{1}^{2}|-|B_{1}c_{1}|\right)=\cfrac{(1+2\lambda)}{24}\psi_{+}(c_{1},c_{2})$

where

$B_{1}:=3,B_{2}:=-\frac{5}{8}(1+2\lambda),\mbox{ and }B_{3}:=1.$

Note that the inequality $|2B_{2}+B_{3}|\geq|B_{3}|+B_{1}$ is equivalent to
$1+10\lambda\geq 16$, which is not true for $1/2\leq\lambda\leq 1$. So using
Lemma together with (3.24), we can conclude that

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1+2\lambda}{12},$

and equality holds for the function $f_{4}$ given by

(3.25) $f_{2}(z)=z+\cfrac{(2\lambda+1)}{6}\,z^{3}+\cdots$

We now consider the lower bound. Then

(3.26)
$|\Gamma_{1}|-|\Gamma_{2}|=\cfrac{(1+2\lambda)}{24}\psi_{-}(c_{1},c_{2})$

where $\psi_{-}(c_{1},c_{2})=-\psi_{+}(c_{1},c_{2})$. Since
$B_{4}=(1+10\lambda)/2$, it is easy to see that the inequality $B_{1}\geq
B_{4}+2|B_{3}|$ is not true. Further we have,

$2|B_{3}|(B_{4}+2|B_{3}|)-B_{1}^{2}=10\lambda-4\geq 0.$

Hence by (3.26) and Lemma 2.2 we get

$|\Gamma_{1}|-|\Gamma_{2}|\leq\cfrac{1}{2}\left(\cfrac{1+2\lambda}{\sqrt{5+10\lambda}}\right),$

and equality holds for the function $f\in\mathcal{F}_{0}(\lambda)$ satisfying
(3.22) with

$p(z)=\cfrac{1+2tz+z^{2}}{1-z^{2}},$

where $t=2/\sqrt{5+10\lambda}$. This completes the proof of the theorem. ∎

Next, we obtain the sharp lower and upper bounds for
$|\Gamma_{2}|-|\Gamma_{1}|$ when $f$ belongs to the class
$\mathcal{S}^{*}_{\gamma}(\alpha)$.

######  Theorem 3.8.

Let $-\pi/2<\gamma<\pi/2$ and $0\leq\alpha<1$. For every
$f\in\mathcal{S}^{*}_{\gamma}(\alpha)$ of the form (1.1), the following sharp
inequalities holds.

$-\cfrac{(1-\alpha)\cos\gamma}{\sqrt{|\eta|+1}}\leq|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{(1-\alpha)\cos\gamma}{2}$

where $\eta=4(1-\alpha)\mu-1$ with $\mu=e^{i\gamma}\cos\gamma$.

###### Proof.

Let $f\in\mathcal{S}^{*}_{\gamma}(\alpha)$, then there exists a function
$p\in\mathcal{P}$ such that

(3.27)
$p(z)=\cfrac{1}{1-\alpha}\bigg{(}\cfrac{1}{\cos\gamma}\bigg{(}e^{-i\gamma}\bigg{(}\cfrac{zf^{\prime}(z)}{f(z)}\bigg{)}+i\sin\gamma\bigg{)}-\alpha\bigg{)}.$

Equating the coefficients of $z^{n}$ on both the sides of (3.27) for $n=1,2$,
we obtain

(3.28) $a_{2}=(1-\alpha)\mu
c_{1}\,\,\,\text{and}\,\,\,a_{3}=\cfrac{(1-\alpha)\mu}{2}\left((1-\alpha)\mu
c_{1}^{2}+c_{2}\right),$

where $\mu=e^{i\gamma}\cos\gamma$. Now using (1.10) together with (3.28), we
get

(3.29) $\displaystyle|\Gamma_{2}|-|\Gamma_{1}|$
$\displaystyle=\cfrac{(1-\alpha)\cos\gamma}{4}\left(|B_{3}c_{2}+B_{2}c_{1}^{2}|-|B_{1}c_{1}|\right),$

where

$B_{1}:=2,\,B_{2}:=-2(1-\alpha)\mu,\mbox{ and }B_{3}=1.$

It is easy to see that the inequality $|2B_{2}+B_{3}|\leq|B_{3}|+B_{1}$ is
true for $-\pi/2<\gamma<\pi/2$ and $0\leq\alpha<1$. Therefore by Lemma 2.2, we
get

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{(1-\alpha)\cos\gamma}{2}$

Note that $B_{4}=|8(1-\alpha)\mu-2|$. For the lower bound, it is easy to see
that the inequality $B_{1}\geq B_{4}+2|B_{3}|$ doesnot holds and the
inequality $B_{1}^{2}\leq 2|B_{3}|(B_{4}+2|B_{3}|)$ holds for
$-\pi/2<\gamma<\pi/2$ and $0\leq\alpha<1$. Hence by Lemma 2.2, we get

(3.30)
$|\Gamma_{2}|-|\Gamma_{1}|\geq-\cfrac{(1-\alpha)\cos\gamma}{\sqrt{|\eta|+1}}$

where $\eta=4(1-\alpha)\mu-1$. Equality holds for the function $f$ defined by
(3.27) with

(3.31)
$p(z)=\cfrac{1+q_{1}(q_{2}+1)z+q_{2}z^{2}}{1+q_{1}(q_{2}-1)z-q_{2}z^{2}},$

where

$q_{1}=\cfrac{1}{\sqrt{|\tau|+1}}\mbox{ and }q_{2}=e^{i\arg\tau}.$

This completes the proof. ∎

If we put $\gamma=0$ in Theorem 3.8, then we obtain the following result for
the class of starlike function of order $\alpha$.

######  Corollary 3.9.

Let $0\leq\alpha<1$ and $f\in\mathcal{S}^{*}(\alpha)$ of the form (1.1). Then

(3.32) $|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1-\alpha}{2}$

and

(3.33)
$|\Gamma_{2}|-|\Gamma_{1}|\geq\left\\{\begin{array}[]{ll}-\cfrac{\sqrt{1-\alpha}}{2}&{\mbox{
if }}\,\,0\leq\alpha\leq\cfrac{3}{4},\\\\[11.38109pt]
-\cfrac{1-\alpha}{\sqrt{4\alpha-2}}&{\mbox{ if
}}\,\,\cfrac{3}{4}\leq\alpha<1,\end{array}\right.$

Next, we obtain the sharp lower and upper bounds for
$|\Gamma_{2}|-|\Gamma_{1}|$ when $f$ belongs to the class
$\mathcal{C}_{\gamma}(\alpha)$.

######  Theorem 3.10.

Let $-\pi/2<\gamma<\pi/2$ and $0\leq\alpha<1$. For every
$f\in\mathcal{C}_{\gamma}(\alpha)$ of the form (1.1), the following sharp
inequality holds.

(3.34) $|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{(1-\alpha)\cos\gamma}{6}$

and

(3.35)
$|\Gamma_{2}|-|\Gamma_{1}|\geq\left\\{\begin{array}[]{lll}\cfrac{(\alpha-1)\cos\gamma\,}{12}(6-|\beta|),&{\mbox{
if }}|\beta|\leq 1,\\\\[14.22636pt]
\left(\cfrac{(\alpha-1)\cos\gamma}{12}\right)\left(2+\cfrac{9}{|\beta|+2}\right),&{\mbox{
if }}1\leq|\beta|\leq 5/2,\\\\[14.22636pt]
\left(\cfrac{(\alpha-1)\cos\gamma}{2}\right)\left(\sqrt{\cfrac{2}{|\beta|+2}}\right),&{\mbox{
if }}|\beta|\geq 5/2.\end{array}\right.$

where $\beta=5(1-\alpha)\mu-2$ with $\mu=e^{i\gamma}\cos\gamma$.

###### Proof.

Let $f\in\mathcal{C}_{\gamma}(\alpha)$, then there exists a function
$p\in\mathcal{P}$ such that

(3.36)
$p(z)=\cfrac{1}{1-\alpha}\bigg{\\{}\cfrac{1}{\cos\gamma}\bigg{(}e^{-i\gamma}\bigg{(}1+\cfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{)}+i\sin\gamma\bigg{)}-\alpha\bigg{\\}}.$

Equating the coefficients of $z^{n}$ on both the sides of (3.36) for $n=1,2$,
we obtain

(3.37)
$a_{2}=\cfrac{(1-\alpha)\mu}{2}c_{1}\,\,\,\text{and}\,\,\,a_{3}=\cfrac{(1-\alpha)\mu}{6}\left((1-\alpha)\mu
c_{1}^{2}+c_{2}\right),$

where $\mu=e^{i\gamma}\cos\gamma$. Now using (1.10), we obtain

$\displaystyle|\Gamma_{2}|-|\Gamma_{1}|$
$\displaystyle=\cfrac{(1-\alpha)\cos\gamma}{12}\bigg{(}|c_{2}-\cfrac{5}{4}(1-\alpha)\mu
c_{1}^{2}|-|3c_{1}|\bigg{)}$ (3.38)
$\displaystyle=\cfrac{(1-\alpha)\cos\gamma}{12}\bigg{(}|B_{3}c_{2}+B_{2}c_{1}^{2}|-|B_{1}c_{1}|\bigg{)},$

where

$B_{1}:=3,\,B_{2}:=-\cfrac{5}{4}(1-\alpha)\mu,\mbox{ and }B_{3}=1.$

For the upper bound, it is easy to see that the first condition
$|2B_{2}+B_{3}|\geq|B_{3}|+B_{1}$ is not satisfied since
$|5(1-\alpha)\mu-2|\leq 3$. By using Lemma 2.2 and the equation (3), we obtain
that

$|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{(1-\alpha)\cos\gamma}{6}.$

This proves the inequality (3.34).

We next prove the lower bound in (3.35) by checking the condition of Lemma 2.2
for $\Psi_{-}(c_{1},c_{2})$. Note that the inequality $B_{1}\geq
B_{4}+2|B_{3}|$ is true for $|5(1-\alpha)\mu-2|\leq 1$ and the inequality
$B_{1}^{2}\leq 2|B_{3}|(B_{4}+2|B_{3}|)$ is true for $|5(1-\alpha)\mu-2|\geq
5/2$. Thus, by applying Lemma 2.2, we obtain

$\Psi_{-}(c_{1},c_{2})\leq\left\\{\begin{array}[]{lll}6-|\beta|,&{\mbox{ if
}}|\beta|\leq 1,\\\\[8.53581pt] 2+\cfrac{9}{|\beta|+2},&{\mbox{ if
}}1\leq|\beta|\leq 5/2,\\\\[14.22636pt]
6\left(\sqrt{\cfrac{2}{|\beta|+2}}\right),&{\mbox{ if }}|\beta|\geq
5/2.\end{array}\right.$

where $\beta=5(1-\alpha)\mu-2$. By Substituting the above inequality in (3) we
obtain the required inequality (3.35).

Equality holds in (3.34) when $f$ is defined by (3.36) with
$p(z)=(1+z^{2})/(1-z^{2})$. For $|\beta|\geq 5/2$, equality holds for the
function $f$ defined by (3.36) with

(3.39)
$p(z)=\cfrac{1+q_{1}(q_{2}+1)z+q_{2}z^{2}}{1+q_{1}(q_{2}-1)-q_{2}z^{2}},$

where $q_{1}=1/\sqrt{1+|\beta|}\mbox{ and }q_{2}=e^{i\arg\beta}$. For
$|\beta|\leq 5/4$, equality holds for the function $f$ given by (3.36) where
$p(z)$ is of the form (3.39) with $q_{1}=3/2(1+|\beta|)\mbox{ and
}q_{2}=e^{i\arg\beta}$. This completes the proof of this theorem. ∎

If we put $\gamma=0$ in Theorem 3.10, then we obtain the following result for
the class of convex function of order $\alpha$.

######  Corollary 3.11.

Let $0\leq\alpha<1$ and $f\in\mathcal{C}(\alpha)$ of the form (1.1). Then

(3.40) $|\Gamma_{2}|-|\Gamma_{1}|\leq\cfrac{1}{6}(1-\alpha)$

and

(3.41)
$|\Gamma_{1}|-|\Gamma_{2}|\leq\left\\{\begin{array}[]{lllll}\sqrt{\left(\cfrac{1-\alpha}{10}\right)},&{\mbox{
if }}0\leq\alpha\leq\cfrac{1}{10},\\\\[14.22636pt]
\cfrac{1}{60}(19-10\alpha),&{\mbox{ if
}}\cfrac{1}{10}\leq\alpha\leq\cfrac{2}{5},\\\\[14.22636pt]
\cfrac{1}{12}(1-\alpha)(3+5\alpha),&{\mbox{ if
}}\cfrac{2}{5}\leq\alpha\leq\cfrac{3}{5},\\\\[14.22636pt]
\cfrac{1}{12}(1-\alpha)(9-5\alpha),&{\mbox{ if
}}\cfrac{3}{5}\leq\alpha\leq\cfrac{4}{5},\\\\[14.22636pt]
\cfrac{(1-\alpha)(10\alpha+7)}{12(5\alpha-1)},&{\mbox{ if
}}\cfrac{4}{5}\leq\alpha\leq 1.\\\\[14.22636pt] \end{array}\right.$

All the inequalities are sharp.

Acknowledgement: The research of the first named author is supported by SERB-
CRG, Govt. of India and the second named author’s research work is supported
by UGC-SRF.

## References

  * [1] M. F. Ali and A. Vasudevarao, On logarithmic coefficients of some close-to-convex functions, Proc. Amer. Math. Soc. 146 (2017), 1131–1142.
  * [2] V. Allu, A. Lecko, and D. K. Thomas, Hankel, Toeplitz and Hermitian-Toeplitz Determinants for Ozaki Close-to-convex Functions, Mediterr. J. Math. 19 (2022).
  * [3] M. F. Ali and A. Vasudevarao, On Coefficient Estimates of Negative Powers and Inverse Coefficients for Certain Starlike Functions, Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017), 449–462.
  * [4] V. Arora, Initial Successive coefficients for certain classes of univalent functions, Lobachevskii J Math 43 (2022), 2080–2091.
  * [5] V. Arora, S. Ponnusamy, and S. K. Sahoo, Successive coefficients for spirallike and related functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113(4) (2019), 2969–2979.
  * [6] L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137-152.
  * [7] D.A. Brannan and W.E. Kirwan, On some classes of bounded univalent functions, J. Lond. Math. Soc. 2(1) (1969), 431–443.
  * [8] P. L. Duren, Univalent Functions, Springer-Verlag, New York, 1983.
  * [9] M. Fekete and G. Szegö, Eine Bemerkung ber ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85–89.
  * [10] G. M. Goluzin, On distortion theorems and coefficients of univalent functions, Mat. Sb. 19 (61) (1946), 183–202 (in Russian).
  * [11] A. Z. Grinspan, Improved bounds for the difference of adjacent coefficients of univalent functions (Russian), Questions in the mordern theory of functions (Novosibirsk), Sib. Inst. Mat. 38 (1976), 41–45.
  * [12] W. K. Hayman, On successive coefficients of univalent functions, J. London. Math. Soc. 38 (1963), 228–243.
  * [13] V. Kumar and N.E. Cho, Moduli difference of successive inverse and logarithmic coefficients for a class of close-to-convex functions, Asian-Eur. J. Math.16 (2023). https://doi.org/10.1007/s11587-021- 00682-1.
  * [14] A. Lecko and D. Partyka, Successive Logarithmic Coefficients of Univalent Functions, Computational Methods and Function Theory (2023), https://doi.org/10.1007/s40315-023-00500-9.
  * [15] Y. Leung, Successive coefficients of starlike functions, Bull. Lond. Math. Soc. 10 (1978), 193–196.
  * [16] M. Li and T. Sugawa, A note on successive coefficients of convex functions, Comput. Methods Funct. Theory 17(2) (2017), 179–193.
  * [17] R. J. Libera Univalent $\alpha$-spiral functions, Canad. J. Math. 19 (1967) 249–456.
  * [18] I. M. Milin, Univalent functions and orthonormal systems, Translations of Mathematical Monographs, Volume 49 (1977).
  * [19] M. Obradović, S. Ponnusamy, and K.-J. Wirths, Disk of univalence of the ratio of two analytic functions, Complex Var. Elliptic Equ. 60(3) (2015), 392–404.
  * [20] M. Obradović and N. Tuneski Simple proofs of certain inequalities with logarithmic coefficients of univalent functions submitted. arXiv:2311.09901
  * [21] S. Ozaki, On the theory of multivalent functions. II, Sci. Rep. Tokyo Bunrika Daigaku. Sect. A. 4 (1941) 45–87.
  * [22] J. A. Pfaltzgraff, Univalence of the integral of $f^{\prime}(z)^{\lambda}$, Bull. London Math. Soc. 7 (1975), 254–256.
  * [23] J. A. Pfaltzgraff, M. O. Reade, and T. Umezawa, Sufficient conditions for univalence, Ann. Fac. Sci. Univ. Nat. Zaïm̈issingre (Kinshasa) Sect. Math.-Phys. 2(2) (1976), 211–218.
  * [24] Ch. Pommerenke, Probleme aus der Funktionentheorie, Jber. Deutsch. Math.-Verein. 73 (1971), 1–5.
  * [25] S. Ponnusamy and S. Rajasekaran, New sufficient conditions for starlike and univalent functions, Soochow J. Math. 21(2) (1995), 193–201.
  * [26] S. Ponnusamy, S. K. Sahoo, and H. Yanagihara, Radius of convexity of partial sums of functions in the close-to-convex family, Nonlinear Anal. 95 (2014), 219–228.
  * [27] S. Ponnusamy, N.L. Sharma and K.J. Wirths, Logarithmic Coefficients of the Inverse of Univalent Functions, Results Math 73 (2018), DOI: 10.1007/s00025-018-0921-7.
  * [28] S. Ponnusamy and A. Vasudevarao, Region of variability of two subclasses of univalent functions, J. Math. Anal. Appl. 332(2) (2007), 1323–1334.
  * [29] M. S. Robertson, Univalent functions $f(z)$ for which $zf^{\prime}(z)$ is spirallike, Michigan Math. J. 16 (1969), 97–101.
  * [30] Y. J. Sim and D. K. Thomas, On the difference of inverse coefficients of univalent functions, Symmetry 12(12) (2020).
  * [31] Y. J. Sim and D. K. Thomas, A note on spirallike functions, Bull. Aust. Math. Soc., https://doi.org/10.1017/S0004972721000198.
  * [32] V. Singh and P. N. Chichra, Univalent functions $f(z)$ for which $zf^{\prime}(z)$ is $\alpha$-spirallike, Indian J. Pure Appl. Math. 8 (1977), 253–259.
  * [33] L. Špaček Contributiona la theorie des fonctions univalentes, Casop Pest. Mat.-Fys. 62 (1933) 12–19.
  * [34] J. Stankiewicz, Quelques problèmes extrémaux dans les classes des fonctions $\alpha$-angulairement étoilées, Ann. Univ. Mariae Curie-Skłodowska Sect. A. 20 (1966), 59–75.
  * [35] J. Stankiewicz, On a family of starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A. 22-24 (1968-1970), 175–181.
  * [36] D. K. Thomas, On the logarithmic coefficients of close-to-convex functions, Proc. Amer. Math. Soc. 144 (2016), 1681–1687.
  * [37] D. K. Thomas, N. Tuneski and A. Vasudevarao Univalent Functions: A Primer, De Gruyter Studies in Mathematics 69, Berlin, Boston, (2018).
  * [38] T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 195–202.