# Inelastic electron transfer in olfaction: multiphonons processes

Shu-Quan Zhang  Integrated Chinese and Western Medicine Hospital, Tianjin
University, Tianjin 300354 China  Yu Cui  Tianjin Key Laboratory of Low
Dimensional Materials Physics and Preparing Technology, Department of Physics,
School of Science, Tianjin University, Tianjin 300354 China  Xue-Wei Li
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing
Technology, Department of Physics, School of Science, Tianjin University,
Tianjin 300354 China  Yong Sun  Tianjin Key Laboratory of Low Dimensional
Materials Physics and Preparing Technology, Department of Physics, School of
Science, Tianjin University, Tianjin 300354 China  Zi-Wu Wang  Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, School of Science, Tianjin University, Tianjin 300354
China<EMAIL_ADDRESS>
###### Abstract

Inelastic electron transfer being regarded as one of the potential mechanisms
to explain the odorant recognition in the atomic-scale processes is still a
matter of intense debate. Here, we propose multiphonon processes of electrons
transfer using the Markvart’ s model and calculate their lifetimes with values
of key parameters widely adopted in olfactory systems. We find that these
multiphonon processes are as quickly as the single phonon process, which
suggest that contributions from different phonon modes of odorant molecule for
electrons transfer in olfaction should be included. Meanwhile, temperature
dependence of electron transfer could be analyzed effectively based on the
reorganization energy is expanded into the linewidth of multiphonon processes.
Our theoretical results not only enrich the knowledge of the mechanism of the
olfaction recognition, but also provide insights for quantum processes in the
biological system.

odorant recognition, multiphonon processes, inelastic electron transfer

Our sense of smell (olfaction) is of important to make judgement about
surrounding environment and influence our perceptions. In spit of the
processes of olfactory recognition have been extensively studied both in
experimentalR. Bentley (2006); F. M. S. Souza (2007); M. Zarzo (2007); M. I.
Franco (2011); S. F. Huelgaa (2013); S. A. Pshenichnyuk (2018); A. P.
Horsfield (2019) and theoreticalF. M. S. Souza (2007); M. Zarzo (2007); A. P.
Horsfield (2019); J. C. Brookes (2007); Jennifer C. Brookes (2005, 2011); R.
D. Hoehn (2018) aspects during the past several decades, the fundamental
mechanism of olfaction is still hot debated. Until now, two mainstream
mechanisms have been proposed: one is the docking model (lock-and-key model),
in which the odorant size, shape and functional groups determine activation of
olfactory receptors; another is the vibrational model, in which the special
vibrational frequencies of odorant compounds contribute to odorant
recognition. In particular, the latter one accounts for the essential quantum
processes of electron transfer in atomic-scale level for biology systems,
arousing more and more attention in recent years.

Figure 1:  The schematic diagrams of inelastic electron transfer in the
olfactory receptor: (a) an electron transfers from the donor state ($E_{D}$)
to acceptor state ($E_{A}$) with the absorption of a phonon
($\hbar\omega_{0}$) provided by the odorant molecule; (2) an electron
transfers between the donor state and acceptor state by the absorption of
multiphonons processes ($n\hbar\omega_{\lambda}$, $n\geq 2$), where the
relation of $E_{D}-E_{A}=n\hbar\omega_{\lambda}$ is satisfied.

The vibrational model original proposed first by Dyson in 1938G. Dyson (1938)
and then further studied by Wright in 1977R. Wright (1977). They emphasized
the contributions of these special vibrational frequencies of odorant
compounds to the odor recognition. This model was endorsed by Turin in 1996L.
Turin (1996) and proposed a quantum description that inelastic transfer of an
electron mediated by an odorant in olfactory receptor. This process is
schematically shown in Fig. 1 (a), in which an electron can transfer between
the electron donor site ($E_{D}$) to the acceptor site ($E_{A}$) of the
olfactory receptor, but only if the energy difference
($E_{DA}$=$E_{D}$-$E_{A}$) is matched by the right vibrational energy
(frequency) of the odorant molecule, and hence an odorant is recognized by its
vibrational modes. Until 2007, Brookes et alJ. C. Brookes (2007) analyzed the
physical viability of this model based on Marcus’s electron transfer theory
again. They evaluated the lifetime of electron transfer with the appropriate
parameters of the biomolecular systems and pointed out that the time scale is
sufficient to permit odorant recognition by electron transfers assisted by
single vibrational modes. In recent years, this model has been further
explored by a number of studies both in theory and experimentsL. A. Solov’yov
(2012); J. C. Brookes (2012); A. Tirandaz (2015); J. C. Brookes (2017); J.
McFadden (2018); N. Liza (2019). However, only the single vibrational mode of
the odorant molecular was taken into account in this model for most of these
studies, the contribution of multiple vibrational modes to inelastic electron
transfer are neglected.

In this paper, we theoretically propose inelastic electron transfer assisted
by multiple phonons processes in olfactory receptors based on Markvart’ s
modelT. Markvart (1981, 1981); S. Kirchartz (2018). We present the lifetimes
of multiphonon processes for different phonon modes and give the comparisons
among them with different Huang-Rhys factors that denote the strength of
electron-phonon couplingK. Huang (1950). We find that these multiphonon
processes are enhanced obviously with the increasing of Huang-Rhys factor,
which results in multiphonon processes are as quickly as the single phonon
process. Therefore, multiphonon processes related to different phonon modes of
odorant molecule should be taken into account for electrons transfer in
odorant recognition. Moreover, we show temperature dependences of these
multiphonon processes, in which the reorganization energy, arising from the
coupling of the olfactory receptor donor and accepter states to the vibrations
of the molecular environment, is expanded into the linewidth of multiphonon
processes. We hope these theoretical results can trigger more experiments for
the odorant recognition in future.

The model of Markvart’ s multiphonon processes has been widely adopted to
study the charge and energy transfer among localized states in
semiconductorsT. Markvart (1981, 1981); S. Kirchartz (2018), in particular, to
discuss the temperature dependence of multiphonon processes. We propose the
inelastic electron transfer in olfactory receptor via multiphonon processes as
schemed in Fig. 1 (b) and the transition rate ($1/\tau$) in this model can be
expressed as

$\displaystyle\tau^{-1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2\pi}|\gamma|^{2}\Omega}{\hbar[\sqrt{S_{LA}\overline{(\hbar\omega_{LA})^{2}_{T}}}+p\hbar\omega_{\alpha}]}$
(1)
$\displaystyle\exp[-\frac{(E_{DA}-p\hbar\omega_{\alpha}-S_{LA}\overline{\hbar\omega_{LA}}}{2S_{LA}\overline{(\hbar\omega_{LA})^{2}_{T}}}],$

with

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p(1+\chi_{0}^{2})^{1/2}}}\exp[p(\frac{\hbar\omega_{\alpha}}{2K_{B}T}+\sqrt{1+\chi_{0}^{2}}$
$\displaystyle-\chi_{0}\cosh(\frac{\hbar\omega_{\alpha}}{2K_{B}T})-\ln(\frac{1+\sqrt{1+\chi_{0}^{2}}}{\chi_{0}}))],$
$S_{LA}\overline{(\hbar\omega_{LA})^{2}_{T}}=S_{LA}\overline{(\hbar\omega_{LA})^{2}}\coth[\overline{\hbar\omega_{LA}}/(2K_{B}T)],$
(3) $\chi_{0}=\frac{S_{0}}{p\sinh[\hbar\omega_{\alpha}/(2K_{B}T)]},$ (4)

where $\gamma$ is the hopping integral accounting for coupling strength
between the donor state and acceptor state. The reorganization energy
$S_{LA}\overline{\hbar\omega_{LA}}$, arising from the the coupling of the
olfactory receptor donor and accepter states to the vibrations of the
molecular environment, is attributed from the acoustic phonon modes with an
average phonon energy $\overline{\hbar\omega_{LA}}$ and an effective Huang-
Rhys factor $S_{LA}$. In the present model, this reorganization energy is
expanded into the linewidth of these multiphonon processes as given in Eq.
(1), so as to the temperature dependence of multiphonon processes could be
analyzed qualitatively. $p=1,2,3\cdots$ is the phonon numbers of
$\hbar\omega_{\alpha}$ with the proper branches of modes $\alpha$
(frequencies) for the inelastic electron transfer. In fact, probabilities of
these multiphonon processes depend on another Huang-Rhys factor $S_{0}$, which
describes the coupling strength between the olfactory receptor and odorant
molecule. $E_{DA}$ is the energy separation between the donor and acceptor
states in the olfactory receptor, KB is the Bolzmann constant and $T$ is the
temperature.

The characteristic values of these physical qualities above mentioned in Eq.
(1) have been discussed very widely for bimolecular systems, such as the
hopping integral $\gamma=1$ meV, the reorganization energy
$S_{LA}\overline{\hbar\omega_{LA}}$ is expected to lies in the range of 30-50
meVJ. C. Brookes (2007); Jennifer C. Brookes (2011); J. C. Brookes (2012) and
Huang-Rhys factor could be varied in the range of 0-0.35 due to the
contribution from different phonon modesL. A. Solov’yov (2012). The adopted
values of these quantities for the simulations in this paper have been listed
in table I.

Table 1: These adopted parameters in the numerical simulations for multiphonon processes of inelastic electron transfer in olfactory receptor. Parameter | Values
---|---
Hopping integral constant ($\gamma$) | 1.0 (meV)8,10
The reorganization energy ($S_{LA}\overline{\hbar\omega_{LA}}$) | 30 (meV)8,10,15
Average phonon energy ($\overline{\hbar\omega_{LA}}$) | 6.0 (meV)
Donor-acceptor energy separation ($E_{DA}$) | 200 (meV) 8,10,15
Huang-Rhys factor ($S_{0}$) | 0$\sim$0.358,15,16
The energy of phonon mode ($\hbar\omega_{\alpha}$) | Variable
Figure 2: Lifetimes of inelastic electron transfer assisted by one
($\hbar\omega_{0}$=200 meV) phonon, two ($\hbar\omega_{1}$=100 meV), three
($\hbar\omega_{2}$=70 meV) and four ($\hbar\omega_{3}$=50 meV) phonons
processes at $E_{DA}$=200 meV and $T$=309 K (body temperature). (a), (b) and
(c) for Huang-Rhys factor $S_{0}$=0.1, 0.3, 0.5, respectively. Figure 3:
Temperature dependence of inelastic electron transfer assisted by different
phonon modes at $E_{DA}$=200 meV and $S_{0}$=0.3. (a), (b) and (c) for one
phonon ($\hbar\omega_{0}$=200 meV), two ($\hbar\omega_{1}$=100 meV) and three
($\hbar\omega_{2}$=70 meV) phonons processes, respectively.

The value of energy difference $E_{DA}$=200 meV between the donor and acceptor
states in the olfactory receptor has been widely adopted by Brookes and
othersJ. C. Brookes (2007); Jennifer C. Brookes (2011); L. A. Solov’yov
(2012); J. C. Brookes (2012). They studied the process of an electron
transfers when $E_{DA}$ is matching the energy of the single phonon based on
the well-known Marcus’ model. They pointed out that the corresponding lifetime
of electron transfer should be much faster than the process of elastic
electron tunneling, making the inelastic signal sufficiently strong relative
to the elastic signal in olfactory receptor. But energies of multiple phonons
matching the energy difference $E_{DA}$ have been neglected in these works
even though the odorant molecule provides different phonon modes. We present
multiphonon processes for four types of phonon modes $\hbar\omega_{0}=200$
meV, $\hbar\omega_{1}=100$ meV, $\hbar\omega_{2}=70$ meV and
$\hbar\omega_{3}=50$ meV, respectively, and give the comparisons for them in
Fig. 2 at different values of Huang-Rhys factor ($S_{0}$) . From Fig. 2 (a),
one can see that the lifetime of single phonon process is in the range of tens
of picosecond given by this model, which is much faster than the predicted
time scale (nanosecond) by Brookes et alJ. C. Brookes (2007); Jennifer C.
Brookes (2011); L. A. Solov’yov (2012). Moreover, two- and three-phonons
processes are also in the range of nanoseconds, which are also sufficient to
permit detection on the observed time scaleJ. C. Brookes (2007). These
multiphonon processes are enhanced obviously with increasing Huang-Rhys factor
$S_{0}$ as shown from Fig. 2 (a) to (c). Therefore, these multiphonon
processes should give the significant contributions to inelastic electron
transfer, in particular, for the energy difference $E_{DA}$ is much larger
than the single phonon energy that the odorant molecules provided. Here, we
must emphasize that Huang-Rhys factor $S_{0}$ is a very important parameter
for these multiphonon processes and its value can varies in the range of 0.0
$\sim$ 0.35 for the typical odorants in some previous studiesL. A. Solov’yov
(2012); J. C. Brookes (2012). However, this factor is related to many
microscopic electronic structures of the olfactory receptor, which results in
the accurate evaluation of Huang-Rhys factor is still a challenge work both in
the theoretical and experimental aspects. In addition, the combined
multiphonon processes consisting of different phonon modes, such as
$\hbar\omega_{1}+2\hbar\omega_{3}$ and $2\hbar\omega_{2}+\hbar\omega_{3}$,
should also be taken into account if the larger value of Huang-Rhys factor
could be obtained.

The temperature dependences of inelastic electron transfer via three types of
phonon modes are shown in Fig. 3, where an effective phonon modes
$\overline{\hbar\omega_{LA}}$ =6 meV is assumed for the reorganization energy.
One can see that the more broader energy window is opened for electron
transfer with increasing the temperature, so the influence of the thermal
fluctuation of the molecular environment on the odorant recognition could be
analyzed qualitatively based on the present model, which is also an advantage
of this model to study the quantum mechanism of the olfaction. In fact, the
similar results could be obtained when the adopted effective phonon mode
$\overline{\hbar\omega_{LA}}$ is in the range of several meV for the
reorganization energy. Lastly, the olfactory discrimination of enantiomers
with chiral properties based on the inelastic electron transfer mechanism has
been proposedA. Tirandaz (2015). These multiphonon processes for the chiral
odorants are not considered, which will also be very important for the
olfaction study and need to further investigate in the following work.

In conclusion, we theoretically study electrons transfer via multiphonon
processes based on the Markvart’ s model and stimulate these processes with
values of key parameters usually adopted in olfactory receptors. We find that
(1) these multiphonon processes are as quickly as the single phonon process,
which suggest that inelastic electrons transfer in odorant recognition should
include contributions from different phonon modes of odorant molecule; (2)
more broader energy window is opened for electron transfer with increasing the
temperature, since the influence of the thermal fluctuation of the molecular
environment on the odorant recognition could be analyzed effectively with an
effective phonon modes $\overline{\hbar\omega_{LA}}$ is adopted for the
reorganization energy.

##  I Acknowledgement

This work was supported by National Natural Science Foundation of China (Grant
No 11674241).

##  II Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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]Essential norm of composition operators harmonic Bloch spaces
Y. Estaremi, S. Esmaeili and A. Ebadian
Y. estaremi, S. Esmaeili and A. Ebadian estaremi<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of mathematics, Payame Noor university , P. O. Box: 19395-3697, Tehran,

In this paper we characterize essential norm of composition operators on the spaces of Harmonic Bloch functions. These results extends the similar results that were proven for composition operators on Bloch spaces.

§ INTRODUCTION

The essential norm $\|T\|_{e}$ of a continuous linear operator $T$ between Banach spaces $X$ and $Y$ is defined as the distance from $T$ to the space of compact operators from $X$ to $Y$.
The essential norm of composition operators have been studied on analytic function spaces in [17, 16, 7, 14, 15, 4, 12, 6].

Let $D$ be the open unit disk in the complex plane. Let $\varphi$ be an analytic self-map of $D$, i. e., an analytic function $\varphi$ in $D$ such that $\varphi(D)\subset D$. The composition operator $C_{\varphi}$ induced by such $\varphi$ is the linear map on the spaces of all harmonic functions on the unit disk defined by

The main goal of this paper is to compute the essential norm of $C_{\varphi}$ in terms of an asymptotic bound involving the quantity

Further, we obtain an other essential norm formula for composition operators on $HB(\alpha)$ for any $0<\alpha<\infty$ in terms of $\varphi^{n}$, where, $\varphi^{n}$ means the $n$-th power of $\varphi$.

Let $D$ be the open unit disk in the complex plane. For a continuously differentiable complex-valued $f(z)=u(z)+i\upsilon(z),$ $z=x+iy,$ we use the common notation for its formal derivatives:

A twice continuously differentiable complex-valued function $f=u+i\upsilon$ on $D$ is called a harmonic function if and only if the real-valued function $u$ and $\upsilon$ satisfy Laplace's equation $\Delta u=\Delta \upsilon=0$.

A direct calculation shows that the Laplacian of $f$ is

$$\Delta f=4f_{z\bar{z}}.$$

Thus for functions $f$ with continuous second partial derivatives, it is clear that $f$ is harmonic if ana only if $\Delta f=0.$
We consider complex-valued harmonic function $f$ defined in a simply connected domain $D\subset C.$ The function $f$ has a canonical decomposition $f=h+\bar{g},$ where $h$ and $g$ are analytic in $D$ [8].
A planar complex-valued harmonic function $f$ in $D$ is called a harmonic Bloch function if and only if

$$\beta_{f}=\sup_{z,w\in D,z\neq w}\frac{|f(z)-f(w)|}{\varrho(z,w)}<\infty.$$

Here $\beta_{f}$ is the Lipschitz number of $f$ and

\begin{align*}
\varrho(z,w)=\arctan h|\frac{z-w}{1-\bar{z}w}|,
\end{align*}

denotes the hyperbolic distance between $z$ and $w$ in $D$, where here $\rho(z,w)$ is the pseudo-hyperbolic distance on $D$.

In [3] Colonna proved that

\begin{align*}
\beta_{f}=\sup_{z\in D}(1-|z|^2)[|f_{z}(z)|+|f_{\bar{z}}(z)|].
\end{align*}

Moreover, the set of all harmonic Bloch mappings, denoted by the symbol $HB(1)$ or $HB$, forms a complex Banach space with the norm $\|.\|$ given by

$$\|f\|_{HB(1)}=|f(0)|+\sup_{z\in D}(1-|z|^2)[|f_{z}(z)|+|f_{\bar{z}}(z)|].$$

Now we define the Harmonic $\alpha$-Bloch space $HB(\alpha)$.
For $\alpha\in(0,\infty)$, the Harmonic $\alpha$-Bloch space $HB(\alpha)$ consists of complex-valued harmonic function $f$ defined on $D$ such that

$$|||f|||_{HB(\alpha)}=\sup_{z\in D}(1-|z|^2)^\alpha[|f_{z}(z)|+|f_{\bar{z}}(z)|]<\infty,$$

and the harmonic little $\alpha$-Bloch space $HB_{0}(\alpha)$ consists of all function in $HB(\alpha)$ such that

Obviously, when $\alpha=1$, we have $|||f|||_{HB(\alpha)}=\beta_{f}$. Each $HB(\alpha)$ is a Banach space with the norm given by

\begin{align*}
\|f\|_{HB(\alpha)}&=|f(0)|+\sup_{z\in D}(1-|z|^2)^\alpha[|f_{z}(z)|+|f_{\bar{z}}(z)|],
\end{align*}

and $HB_{0}(\alpha)$ is a closed subspace of $HB(\alpha)$.

§ MAIN RESULTS

In this section we characterize essential norm of composition operator $C_{\varphi}$ on $HB(\alpha)$.

Here we recall the next lemma from [19] that we need it in the sequel.
If $\alpha>0$,$n\in N$, $0<x<1$ and

then $H_{n,\alpha}$ has the following properties:

$$\max_{0\leq x\leq1}H_{n,\alpha}(x)=$$

\begin{array}{ll}
1 , & \hbox{n=1;} \\$$

$$(\frac{2\alpha}{n-1+2\alpha})^{\alpha} $$
$$& n\geq2.$$

\right.$$

In which

\begin{array}{ll}
0 , & \hbox{n=1;} \\$$

$$& n\geq2.$$

\right.$$

b) For $n\geq 1,H_{n,\alpha}$ is increasing on $[0,r_{n}]$ and decreasing on $[r_{n},1]$.

c) For $n\geq 1,H_{n,\alpha}$ is decreasing on $[r_{n},r_{n+1}],$ and so

By using the Lemma <ref> we can find a lower bound for the essential norm of $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$.

Let $0< \alpha < \infty$, $\varphi$ be an analytic self-map of the unit disk $D$ and $C_{\varphi}$ be a bounded operator from $HB(\alpha)$ into $HB(\alpha)$. Then

$$\|C_{\varphi}\|_{e}\geq \lim_{s\rightarrow 1}\sup _{|\varphi(z)|> s}\frac{(1-|z|^{2})^\alpha}{(1-|\varphi(z)|^{2})^\alpha} |\varphi^{'}(z)|.$$

Let $n\in \mathbb{N}$ and consider the function $z^{n}+\bar{z}^{n}$. By the Lemma <ref> we have

\begin{align*}
\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}&=\max_{z\in D}2n |z|^{n-1}(1-|z|^2)^{\alpha}\\
\end{align*}

in which the maximum is attained at any point on the circle with radius $r_{n}=(\frac{n-1}{n-1+2\alpha})^{\frac{(n-1)}{2}}$.

Let$f_{n}=\frac{z^{n}+\bar{z}^{n}}{\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}}$. Then $f_{n}$ converges to $0$ weakly in $HB(\alpha)$. In particular, if $K$ is any compact operator on $HB(\alpha)$, then $\lim_{n\rightarrow\infty}\|Kf_{n}\|_{HB(\alpha)}=0$. For

$$D_{n}=\{z\in D : r_{n}\leq |\varphi(z)|\leq r_{n+1}\},$$

we have

\begin{align*}
\min_{z\in D_{n}}(1-|z|^2)^{\alpha}[|(f_{n})_{z}(z)|+|(f_{n})_{\bar{z}}(z)|]&=(1-|r_{n+1}|^{2})^{\alpha}[|h_{n}'(z)|+|g_{n}'(z)|]\\
\end{align*}

It is easy to see that this minimum tends to $1$ when $n\rightarrow\infty$. Also for each $n\geq2$ the minimum  is attained at any point of the circle centered at the origin and of radius $r_{n+1}$. For any compact operator $K$ on $HB(\alpha)$, we get that

\begin{align*}
\|C_{\varphi}-K\|&\geq \limsup_{n\rightarrow\infty}\|(C_{\varphi}-K)f_{n}\|_{HB(\alpha)}\\
\end{align*}

Thus we obtain the followings:

\begin{align*}
\|C_{\varphi}\|_{e}&=\inf_{K}\|C_{\varphi}-K\|\\
%&=\limsup_{n\rightarrow\infty}\sup_{z\in D}(1-|z|^2)^\alpha [|(f_{n}o\varphi)_{z}(z)|]+[|(f_{n}o\varphi)_{\bar{z}}(z)|]\\
&=\limsup_{n\rightarrow\infty}\sup_{z\in D}\frac{(1-|z|^2)^\alpha}{(1-|\varphi(z)|^2)^\alpha}|\varphi^{'}(z)|(1-|\varphi(z)|^2)^\alpha [|h_{n}^{'}(\varphi(z))|+|g_{n}^{'}(\varphi(z))|]\\
%&\geq\lim_{n\rightarrow\infty}\sup_{\varphi(z)\in D_{n}}\frac{(1-|z|^2)^\alpha}{(1-|\varphi(z)|^2)^\alpha}|\varphi^{'}(z)|(1-|\varphi(z)|^2)^\alpha [|h_{n}^{'}(\varphi(z))|+|g_{n}^{'}(\varphi(z))|]\\
&\geq\lim_{n\rightarrow\infty}\sup_{\varphi(z)\in D_{n}}\frac{(1-|z|^2)^\alpha}{(1-|\varphi(z)|^2)^\alpha}|\varphi^{'}(z)|\min_{\varphi(z)\in D_{n}}(1-|\varphi(z)|^2)^\alpha [|h_{n}^{'}(\varphi(z))|+|g_{n}^{'}(\varphi(z))|]\\
&\geq\lim_{n\rightarrow\infty}\sup_{\varphi(z)\in D_{n}}\frac{(1-|z|^2)^\alpha}{(1-|\varphi(z)|^2)^\alpha}|\varphi'(z)|\min_{w\in D_{n}}(1-|w|^2)^\alpha [|h_{n}^{'}(w)|+|g_{n}^{'}(w)|],
\end{align*}

$$\limsup_{n\rightarrow\infty}\min_{w \in D_{n}}(1-|w|^2)^\alpha [|h_{n}^{'}(w)|+|g_{n}^{'}(w)|]=1,$$

then we conclude that

$$\|C_{\varphi}\|_{e}\geq \lim_{n\rightarrow\infty}\sup_{\varphi(z)\in D_{n}}\frac{(1-|z|^2)^\alpha}{(1-|\varphi(z)|^2)^\alpha}|\varphi^{'}(z)|.$$

We need some Lemmas to obtain some upper bounds for the essential norm of $C_{\varphi}$. For $r\in(0,1)$, let $K_{r}f(z)=f(rz)$. Then by Theorems 2. 6 and 2.7 of [9] we get that $K_{r}$ is a compact operator on the spaces $HB(\alpha)$ and $HB_{0}(\alpha)$ for any positive number $\alpha$, with $\|K_{r}\|\leq1$.

For $0<\alpha<1$, there exists a sequence $\{r_{k}\}$, $0<r_{k}<1$, tending to $1$, such that the compact operator $L_{n}=(\frac{1}{n})\sum_{k=1}^{n}K_{r_{k}}$ on $HB_{0}(\alpha)$ satisfies:

a) For every $t\in[0,1)$ we have

$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]=0.$$

$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{z\in D}|(I-L_{n})f(z)|=0.$$

In order to prove the Lemma we need to prove that, for any $t$, $0<t<1$, and any $\varepsilon>0$, there is an $n>0$ such that, for any $n>N,$

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]<\varepsilon,$$

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{z\in D}|(I-L_{n})f(z)|<\varepsilon,$$

Let $S_{HB_{0}(\alpha)}$ denote the unit ball of $HB_{0}(\alpha)$ and the positive number $s_{1}$ such that $t<s_{1}<1$. Since $S_{HB_{0}(\alpha)}$ is a relatively compact subset with respect to the topology $\tau$ of the uniform convergence on compact subsets of the unit disk and $(I-K_{r})f$ tends to $0$ with respect to $\tau$, then $((I-K_{r})f)^{'}$ tends to $0$. And so there is $0<r_{1}<1$ such that

\begin{align*}
&\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq s_{1}}(1-|z|^2)^\alpha[|((I-K_{r_{1}})f)_{z}(z)|+|((I-K_{r_{1}})f)_{\bar{z}}(z)|]\\
\end{align*}

Since $K_{r_{1}}$ is compact, then $K_{r_{1}}S_{HB_{0}(\alpha)}$ is relatively compact and we can find $s_{2}>s_{1}$ such that

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|> s_{2}}(1-|z|^2)^\alpha[|(K_{r_{1}}f)_{z}(z)|+|(K_{r_{1}}f)_{\bar{z}}(z)|]<\varepsilon.$$

And so we get that

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|> s_{2}}(1-|z|^2)^\alpha[|((I-K_{r_{1}})f)_{z}(z)|+|((I-K_{r_{1}})f)_{\bar{z}}(z)|]<1+\varepsilon.$$

By repeating this method we can find the sequences $\{s_{k}\}$ and $\{r_{k}\}$ satisfying

\begin{align*}
&\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq s_{k}}(1-|z|^2)^\alpha[|((I-K_{r_{k}})f)_{z}(z)|+|((I-K_{r_{k}})f)_{\bar{z}}(z)|]\\
\end{align*}

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|> s_{k+1}}(1-|z|^2)^\alpha[|((I-K_{r_{k}})f)_{z}(z)|+|((I-K_{r_{k}})f)_{\bar{z}}(z)|]<1+\varepsilon.$$

If we take $n>\frac{1}{\varepsilon}$ and set $L_{n}=(\frac{1}{n})\sum_{k=1}^{n}K_{r_{k}}$, the by using the fact that $(1-|s_{1}|^2)^\alpha\}\leq(1-|z|^2)^\alpha$ for $|z|\leq s_{1}$, we have

\begin{align*}
&\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]\\
&\leq\frac{1}{n}\sum_{k=1}^{n}\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq s_{1}}[|((I-K_{r_{k}})f)_{z}(z)|+|((I-K_{r_{k}})f)_{\bar{z}}(z)|]\\
\end{align*}

This gives (a). Now we prove (c). We know that for $n\geq1$ and $0<\alpha<1$,

\begin{align*}
\|I-L_{n}\|&=\sup_{\|f\|_{HB(\alpha)}\leq1}\|(I-L_{n})f\|_{HB(\alpha)}\\
&=\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}(1-|z|^2)^\alpha[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]\\
&+\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|>t}(1-|z|^2)^\alpha[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]\\
\end{align*}

In which

$$I_{1}=\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}(1-|z|^2)^\alpha[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]<\varepsilon$$

$$I_{2}=\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{s_{l}<|z|<s_{l+1},k\neq l}(1-|z|^2)^\alpha[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]+$$

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|>s_{k+1}}(1-|z|^2)^\alpha[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|].$$

For $k=l$, $s_{l}<|z|<s_{l+1}$ and $\|f\|\leq1$, we have

except possibly for $k\neq l$. However, in this case we get that

and so

\begin{align*}
&\leq\frac{1}{n}\sum_{k\neq l}(1-|z|^2)^\alpha[|((I-k_{r_{k}})f)_{z}(z)|+|((I-k_{r_{k}})f)_{\bar{z}}(z)|]\\
\end{align*}

Therefore we get (c). The proof of the property (b) is exactly as in the proof of the Lemma 1 of [15].

Similar to the Lemma <ref> for the case $\alpha=1$ we have the next Lemma.

There is a sequence $\{r_{k}\}$, $0<r_{k}<1$, tending to 1, such that the compact operator $L_{n}=(\frac{1}{n})\sum_{k=1}^{n}K_{r_{k}}$ on $HB_{0}(\alpha)$ satisfies:

a) For any $t\in[0,1)$ we have

$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]=0.$$

b) $\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|>s}|(I-L_{n})f(z)|(-\log(1-|z|^2))^{-1}\leq1$ for $s$ sufficiently close to 1

$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq s}|(I-L_{n})f(z)|=0.$$

c) $\limsup_{n\rightarrow\infty}\|I-L_{n}\|\leq1$.

The proof of (a) and (b) is exactly the same as the proof of Lemma <ref>. Also the proof of (b) is given in given in [12].

Also for the case $\alpha>1$, we have the following Lemma.

For $\alpha>1$, there is a sequence $\{r_{k}\}$ with $0<r_{k}<1$, tending to $1$. Also, the compact operator $L_{n}=(\frac{1}{n})\sum_{k=1}^{n}K_{r_{k}}$ on $HB_{0}(\alpha)$ has the followings properties:

a) For any $s\in[0,1)$,
$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq s}[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]=0.$$

b) For any $t\in[0,1)$,
$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}|(I-L_{n})f(z)|=0.$$

c) $\limsup_{n\rightarrow\infty}\|I-L_{n}\|\leq1$.

The proof of (a) and (c) is the same as the proof of the Lemma <ref>. Hence we prove (b). For harmonic function $f=h+\bar{g}$ we have

\begin{align*}
\end{align*}

For each $t\in[0,1)$ and any $\varepsilon>0$ we can find $s\in[0,1)$ sufficiently close to $1$ such that

Hence we get that

\begin{align*}
I_{1}&=\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}\int_{D\setminus D_{s}}\frac{(1-|w|^2)^{1+\alpha}}{|w||1-\bar{z}w|^{2+\alpha}}[|((I-L_{n})f)_{w}(z)|+|((I-L_{n})f)_{w}(z)|]dA(w)\\
&\leq\sup_{\|f\|_{HB(\alpha)}\leq1}\|(I-L_{n})f\|_{HB(\alpha)}\sup_{|z|\leq t}|\int_{D\setminus D_{s}}\frac{(1-|w|^2)^{1+\alpha}}{|w||1-\bar{z}w|^{2+\alpha}}dA(w)\\
%&\leq\frac{1}{s(1-t)^{2+\alpha}}\sup_{|z|\leq t}|\int_{D\setminus D_{s}}(1-|w|^2)dA(w)\\
\end{align*}

when $s$ is sufficiently close to $1$. By (a) we can find $N>0$ such that for all $n>N$,

$$\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq s}[|((I-L_{n})f)_{z}(z)|+|((I-L_{n})f)_{\bar{z}}(z)|]<\varepsilon,$$

and so we conclude that

\begin{align*}
I_{2}&=\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}\int_{D_{s}}\frac{(1-|w|^2)^{1+\alpha}}{|w||1-\bar{z}w|^{2+\alpha}}[|((I-L_{n})f)_{w}(z)|+|((I-L_{n})f)_{w}(z)|]dA(w)\\
%&\leq\varepsilon\sup_{|z|\leq t}\int_{D_{s}}\frac{(1-|w|^2)^{1+\alpha}}{|w||1-\bar{z}w|^{2+\alpha}}dA(w)\\
&\leq\frac{\varepsilon}{(1-t)^{2+\alpha}}\sup_{|z|\leq t}\int_{D_{s}}\frac{(1-|w|^2)^{1+\alpha}}{|w|}dA(w)<C\varepsilon,
\end{align*}

for some $C>0$. Therefore by combining the above inequalities, we get that for $n>N$,

\begin{align*}
&\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}|(I-L_{n})f(z)|\\
&\leq\sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}\int_{D}\frac{(1-|w|^2)^{1+\alpha}}{|w||1-\bar{z}w|^{2+\alpha}}[|((I-L_{n})f)_{w}(z)|+|((I-L_{n})f)_{w}(z)|]dA(w)\\
&\leq I_{1}+I_{2}<(1+C)\varepsilon.
\end{align*}

This implies that

$$\lim_{n\rightarrow\infty} \sup_{\|f\|_{HB(\alpha)}\leq1} \sup_{|z|\leq t}|(I-L_{n})f(z)|=0.$$

Now we can find an upper bound for the essential norm of $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$.

Let $\alpha>0$ and $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$ be  bounded. Then we have

$$\|C_{\varphi}\|_{e}\leq \lim_{s\rightarrow 1}\sup _{|\varphi(z)|> s}\frac{(1-|z|^{2})^\alpha}{(1-|\varphi(z)|^{2})^\alpha} |\varphi^{'}(z)|.$$

Let $L_{n}$ be a sequence of operators given in Lemma <ref>. Since each $L_{n}$ is a compact operator from $HB(\alpha)$ to $HB(\alpha)$, then $C_{\varphi}L_{n}$ is compact and we have

\begin{align*}
\|C_{\varphi}\|_{e}&\leq\|C_{\varphi}-C_{\varphi}L_{n}\|\\
&+\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{z\in D}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|].
\end{align*}

By the Lemma <ref> we can suppose the term $\sup_{\|f\|_{HB(\alpha)}\leq1}|(I-L_{n})f(\varphi(0))|$ sufficiently small for some large enough $n$.
Now we need only consider the term

$$\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{z\in D}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|].$$

For every $0<s<1$, we bound this expression from above by

\begin{align*}
&\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{|\varphi(z)|\leq s}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|]\\
&+\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{|\varphi(z)|> s}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|].
\end{align*}

Since $C_{\varphi}$ is bounded from $HB(\alpha)$ into $HB(\alpha)$, then we conclude that

$$(1-|z|^2)^\alpha |\varphi^{'}(z)|\leq (1-|\varphi(z)|^{2})^{\alpha}\sup_{z\in D}(1-|z|^2)^\alpha |\varphi^{'}(z)|<\infty,$$

and so

$$\sup_{|\varphi(z)|\leq s}(1-|z|^2)^\alpha |\varphi^{'}(z)|<\infty.$$

Hence by the Lemma <ref> we see that

$$\lim_{n\rightarrow\infty}\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{|\varphi(z)|\leq s}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|]=0.$$

Easily we find that the term

$$\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{|\varphi(z)|> s}\frac{(1-|z|^2)^\alpha |\varphi^{'}(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}(1-|\varphi(z)|^{2})^{\alpha}[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|]$$

is bounded above by

\begin{align*}
&\sup_{\|f\|_{HB(\alpha)}\leq1}\|(I-L_{n})f\|_{HB(\alpha)}\sup_{|\varphi(z)|> s}\frac{(1-|z|^2)^\alpha |\varphi^{'}(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}\\
&=\|I-L_{n}\|\sup_{|\varphi(z)|> s}\frac{(1-|z|^2)^\alpha }{(1-|\varphi(z)|^{2})^{\alpha}}|\varphi^{'}(z)|.
\end{align*}

Therefore we have

\begin{align*}
&\limsup_{n\rightarrow\infty}\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{|\varphi(z)|> s}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|]\\
&\leq\sup_{|\varphi(z)|> s}\frac{(1-|z|^2)^\alpha }{(1-|\varphi(z)|^{2})^{\alpha}}|\varphi^{'}(z)|,
\end{align*}

and so we get that

$$\|C_{\varphi}\|_{e}\leq\sup_{|\varphi(z)|> s}\frac{(1-|z|^2)^\alpha }{(1-|\varphi(z)|^{2})^{\alpha}}|\varphi^{'}(z)|.$$

Consequently by taking the limit as $s\rightarrow1$, we get the result.

We remark that a similar result can be obtained for the essential norm of $C_{\varphi}$ acting boundedly on $HB_{0}(\alpha)$, for $\alpha>0$, where in the essential norm formula in the Theorem <ref> we may replace $\lim_{s\rightarrow 1}\sup _{|\varphi(z)|> s}$ by $\limsup_{|z|\rightarrow1}$. This equivalence will be shown in the following Theorem:

If $\alpha>0$ and $C_{\varphi}:HB_{0}(\alpha)\rightarrow HB_{0}(\alpha)$ is  bounded. Then

$$\|C_{\varphi}\|_{e}=\limsup_{|z|\rightarrow1}\frac{(1-|z|^{2})^\alpha}{(1-|\varphi(z)|^{2})^\alpha} |\varphi^{'}(z)|.$$

By Theorems <ref>, it would be sufficient to prove the equality

$$\limsup_{|z|\rightarrow1}\frac{(1-|z|^{2})^\alpha}{(1-|\varphi(z)|^{2})^\alpha} |\varphi^{'}(z)|=\lim_{s\rightarrow 1}\sup _{|\varphi(z)|> s}\frac{(1-|z|^{2})^\alpha}{(1-|\varphi(z)|^{2})^\alpha} |\varphi^{'}(z)|.$$

This equivalence can be shown by a similar argument of Theorem 2.2 in [14] and using the characterization of boundedness of $C_{\varphi}:HB_{0}(\alpha)\rightarrow HB_{0}(\alpha)$ for $\alpha>0$.

Recall that the composition operator $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$ is said to be bounded below, if there exists $\varepsilon>0$ such that

for any $f\in HB(\alpha)$. In the next theorem we characterize boundedness of below of $C_{\varphi}$ on $HB(\alpha)$.

The composition operator $C_{\varphi}$ is bounded below on $HB(\alpha)$ if and only if there exists $\delta>0$ such that $\|C_{\varphi}(f)\|_{HB(\alpha)}\geq \delta \|f\|_{HB(\alpha)}$ for $f\in HB(\alpha)$.

First we assume that $C_{\varphi}$ is bounded below on $HB(\alpha)$, so there exists a $\delta>0$ such that $|||go\varphi|||_{HB(\alpha)}\geq \delta |||g|||_{HB(\alpha)}$ for $g\in HB(\alpha)$. Let $g(z)=f(z)-f(\varphi(0))$ for $f\in HB(\alpha)$. Hence $g\in HB(\alpha)$, $g(\varphi(0))=0$, $\|g\|_{HB(\alpha)}=\|f\|_{HB(\alpha)}$ and $\|go\varphi\|_{HB(\alpha)}=\|fo\varphi\|_{HB(\alpha)}$. Therefore

\begin{align*}
\|C_{\varphi}(f)\|_{HB(\alpha)}&=\|fo\varphi\|_{HB(\alpha)}\\
%&\geq \delta |||g|||_{HB(\alpha)}\\
&\geq \delta \|g\|_{HB(\alpha)}=\delta \|f\|_{HB(\alpha)}.
\end{align*}

Conversely, suppose that there exists $\delta>0$ such that $\|C_{\varphi}(f)\|_{HB(\alpha)}\geq \delta \|f\|_{HB(\alpha}$ for all $f\in HB(\alpha)$. Then for every $f\in HB(\alpha)$, we have

\begin{align*}
|f(0)|&\leq |f(\varphi(0))|+\int_{o}^{\varphi(0)}[|(f)_{z}(z)|+|(f)_{\bar{z}}(z)|]|dz|\\
&\leq |f(\varphi(0))|+\|f\|_{HB(\alpha)}\int_{o}^{\varphi(0)}\frac{|dz|}{(1-|z|^2)^{\alpha}}.
\end{align*}

If $\alpha=1$, then

$$|f(0)|\leq |f(\varphi(0))|+\frac{\|f\|_{HB(\alpha)}}{2}\ln \frac{1+|\varphi(0)|}{1-|\varphi(0)|}$$

and so

\begin{align*}
%&\leq |f(\varphi(0))|+\frac{\|fo\varphi\|_{HB(\alpha)}}{\delta}(1+\frac{1}{2}ln\frac{1+|\varphi(0)|}{1-|\varphi(0)|})\\
&\leq\frac{|||fo\varphi|||_{HB(\alpha)}}{\delta}(1+\frac{1}{2}ln \frac{1+|\varphi(0)|}{1-|\varphi(0)|}).
\end{align*}

Since, $\|fo\varphi\|_{HB(\alpha)}\geq \delta \|f\|_{HB(\alpha)}$.
As the same way if $\alpha\neq1$,

\begin{align*}
|f(0)|&\leq |f(\varphi(0))|+\|f\|_{HB(\alpha)}\int_{o}^{\varphi(0)}\frac{|dz|}{(1-|z|^2)^{\alpha}}\\
\end{align*}

\begin{align*}
|f(0)|&\leq |f(\varphi(0))|+\frac{\|f\|_{HB(\alpha)}}{-\alpha+1}[(1-|\varphi(0)|)^{-\alpha+1}-1]|||f|||_{HB(\alpha)}\\
&\leq \frac{|||fo\varphi|||_{HB(\alpha)}}{\delta}(1+\frac{1}{-\alpha+1}[(1-|\varphi(0)|)^{-\alpha+1}-1]).
\end{align*}

By these observations we get that $C_{\varphi}$ is bounded below. [4]

Here we give an equivalent condition for boundedness of $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$.

Let $0<\alpha<\infty$ and $\varphi$ be an analytic self-map of the unit disk $D$. Then $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$ is bounded if and only if

$$\sup_{n\in N}n^{\alpha-1}\|\varphi^{n}+\bar{\varphi}^{n}\|_{HB(\alpha)}<\infty.$$

Let $n\in N$ and $z^{n}+\bar{z}^{n}$. By Theorem <ref> we have

\begin{align*}
\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}&=\max_{z\in D}2n |z|^{n-1}(1-|z|^2)^{\alpha}\\
\end{align*}

where the maximum is attained at any point on the circle with radius $r_{n}=(\frac{n-1}{n-1+2\alpha})^{\frac{(n-1)}{2}}$. Therefore

And so there exists a constant $K>0$, independent of$n$, such that $\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}\leq K n^{1-\alpha}$. For $f_{n}=\frac{z^{n}+\bar{z}^{n}}{\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}}$ we get that $|||f_{n}|||_{HB(\alpha)}=\|f_{n}\|_{HB(\alpha)}=1$. Hence we have

\begin{align*}
\infty&>\|C_{\varphi}\|\\
\end{align*}

Conversely, by the assumption obviously we have $\|\varphi+\bar{\varphi}\|_{HB(\alpha)}<\infty$.  If $\sup_{z\in D}|\varphi(z)|<1$, then we can find $0<r<1$ such that $\sup_{z\in D}|\varphi(z)|<r$. In this case easily we get that $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$ is bounded. Now we assume that $\sup_{z\in D}|\varphi(z)|=1$ and put for any $n\geq1$

$$D_{n}=\{z\in D : r_{n}\leq |\varphi(z)|\leq r_{n+1}\},$$

where $r_{n}$ is given as before. Let $m$ be the smallest positive integer such that $D_{m}\neq\varnothing$. Since $\sup_{z\in D}|\varphi(z)|=1$, $D_{n}$ is not empty for every integer $n\geq m$, and $D=\bigcup_{n=m}^{\infty}D_{n}$. Then for every $n\geq m$ we conclude that

\begin{align*}
\min_{z\in D_{n}}2 n^{\alpha}|\varphi(z)|^{n-1}(1-|\varphi(z)|^{2})^{\alpha}&\geq2 n^{\alpha}H_{n,\alpha}(r_{n+1})\\
&=2(\frac{2\alpha n}{n+2\alpha})^{\alpha}(\frac{n}{n+2\alpha})^{\frac{(n-1)}{2}},
\end{align*}

where $H_{n,\alpha}(x)=x^{n-1}(1-x^2)^{\alpha}$. Thus

$$\lim_{n\rightarrow\infty}\min_{z\in D_{n}}2 n^{\alpha}|\varphi(z)|^{n-1}(1-|\varphi(z)|^{2})^{\alpha}\geq2(\frac{2\alpha}{e}).$$

Therefore, there exists $\delta>0$ such that, for any $n\geq m,$

$$\min_{z\in D_{n}}2 n^{\alpha}|\varphi(z)|^{n-1}(1-|\varphi(z)|^{2})^{\alpha}\geq\delta.$$

Also for every $f\in HB(\alpha)$ we have

\begin{align*}
\|C_{\varphi}f\|_{HB(\alpha)}&=\sup_{z\in D}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|f_{z}(z)|+|f_{\bar{z}}(z)|]\\
&=\sup_{n\geq m}\sup_{z\in D_{n}}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|f_{z}(z)|+|f_{\bar{z}}(z)|]\\
&\leq\frac{1}{\delta}\sup_{n\geq m}\sup_{z\in D_{n}}n^{\alpha-1}(1-|\varphi(z)|^{2})^{\alpha}[|f_{z}(z)|+|f_{\bar{z}}(z)|]
\end{align*}

This implies that $C_{\varphi}$ is bounded on $HB(\alpha)$.

It is well-known that the composition operator $C_{\varphi}$ is bounded on the Bloch space $HB$ for any analytic self-map $\varphi$ of $D$. Hence, if we let $\alpha=1$ in Theorem <ref>, we get the following corollary.

For any analytic self-map $\varphi$ of the unit disk $D$,

$$\sup_{n\in N}\|\varphi^{n}+\bar{\varphi}^{n}\|_{HB}<\infty.$$

Let $0< \alpha < \infty$ and $\varphi$ be an analytic self-map of the unit disk $D$. Then the essential norm of the composition operator $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$ is

Let $n\in N$ and $z^{n}+\bar{z}^{n}$. By Theorem <ref> we have

\begin{align*}
\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}&=\max_{z\in D}2n |z|^{n-1}(1-|z|^2)^{\alpha}\\
\end{align*}

where the maximum is attained at any point on the circle with radius

Thus we have

Let $f_{n}=\frac{z^{n}+\bar{z}^{n}}{\|z^{n}+\bar{z}^{n}\|_{HB(\alpha)}}$ and so $\|f_{n}\|_{HB(\alpha)}=1$. Also easily we get that $\{f_{n}\}$ converges to $0$ weakly in $HB(\alpha)$. If $K$ is a compact operator on $HB(\alpha)$, then $\lim_{n\rightarrow\infty}\|Kf_{n}\|_{HB(\alpha)}=0$.

\begin{align*}
\|C_{\varphi}-K\|&\geq \limsup_{n\rightarrow\infty}\|(C_{\varphi}-K)f_{n}\|_{HB(\alpha)}\\
\end{align*}

By taking the $\inf$ on both sides of this inequality over all compact operator $K$, for $C_{\varphi}:HB(\alpha)\rightarrow HB(\alpha)$, we obtain the that

\begin{align*}
\|C_{\varphi}\|_{e}&=\inf_{K}\|C_{\varphi}-K\|\\
\end{align*}

then $C_{\varphi}$ is not bounded on $HB(\alpha)$. Now suppose that

Hence $C_{\varphi}$ is a bounded operator from $HB(\alpha)$ to $HB(\alpha)$ and, clearly,
Notice that if $\sup_{Z\in D}|\varphi(z)|<1$, then $C_{\varphi}$ is compact and we have the conditions of theorem. Hence, in the sequel we assume that $\sup_{Z\in D}|\varphi(z)|=1$. Let $L_{n}$ be a sequence of operators as we used in the last results. Since each $L_{n}$ is compact as an operator on $HB(\alpha)$, so is $C_{\varphi}L_{n}$. And we have

\begin{align*}
\|C_{\varphi}\|_{e}&=\inf_{C_{\varphi}L_{n} is compact}\|C_{\varphi}-C_{\varphi}L_{n}\|\\
&+\limsup_{n\rightarrow\infty}\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{z\in D}(1-|z|^2)^\alpha |\varphi^{'}(z)|
\end{align*}

Consequently, we have

Now we need only consider the term

$$J=\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{z\in D}(1-|z|^2)^\alpha |\varphi^{'}(z)|

For every $n\geq1$ put

$$D_{n}=\{z\in D : r_{n}\leq |\varphi(z)|\leq r_{n+1}\},$$

where $r_{n}$ is as given in the Lemma 2.16. Let $m$ be the smallest positive integer such that $D_{m}\neq\varnothing$. Since $\sup_{z\in D}|\varphi(z)|=1$, $D_{n}$ is not empty for every integer $n\geq m$ and $D=\bigcup_{n=m}^{\infty}D_{n}$. Now we divide $J$ into two parts:

\begin{align*}
J&=\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{m\leq k\leq N-1}\sup_{z\in D_{k}}(1-|z|^2)^\alpha |\varphi^{'}(z)|[|((I-L_{n})f)_{z}(\varphi(z))|
&+\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{k\geq N}\sup_{z\in D_{k}}(1-|z|^2)^\alpha |\varphi^{'}(z)|
\end{align*}

For $J_{1}$ we have

\begin{align*}
\limsup_{n\rightarrow\infty}J_{1}&=\limsup_{n\rightarrow\infty}\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{m\leq k\leq N-1}\sup_{z\in D_{k}}(1-|z|^2)^\alpha
&=\limsup_{n\rightarrow\infty}\sup_{\|f\|_{HB(\alpha)}\leq1}\sup_{r_{m}\leq|\varphi(z)|\leq r_{N-1}}[|((I-L_{n})f)_{z}(\varphi(z))|+[|((I-L_{n})f)_{\bar{z}}(\varphi(z))|]=0.
\end{align*}

Write the function under three supremum signs in $J_{2}$ as

$$\frac{(1-|z|^2)^\alpha |\varphi^{'}(z)|

For $z\in D_{k}$,

$$2k^{\alpha}|\varphi(z)|^{k-1}(1-|\varphi(z)|^{2})^{\alpha}\geq2(\frac{2\alpha k}{k+2\alpha})^{\alpha}(\frac{k}{k+2\alpha})^{\frac{(k-1)}{2}}.$$

It is easy to see that

$$\lim_{n\rightarrow\infty}[2(\frac{2\alpha k}{k+2\alpha})^{\alpha}(\frac{k}{k+2\alpha})^{\frac{(k-1)}{2}}]^{-1}=\frac{1}{2}(\frac{e}{2\alpha})^{\alpha}.$$

Hence for every $\varepsilon>0$, we can choose $N>m+1$ large enough such that for any $k\geq N$,

$$[2(\frac{2\alpha k}{k+2\alpha})^{\alpha}(\frac{k}{k+2\alpha})^{\frac{(k-1)}{2}}]^{-1}<\frac{1}{2}(\frac{e}{2\alpha})^{\alpha}+\varepsilon.$$

For such $N$ we have

\begin{align*}
J_{2}&\leq[\frac{1}{2}(\frac{e}{2\alpha})^{\alpha}+\varepsilon]\sup_{\|f\|_{HB(\alpha)}\leq1}\|(I-L_{n})f\|_{HB(\alpha)}\sup_{k\geq N}\sup_{z\in D_{k}}2k^{\alpha-1}|(\varphi^{k})^{'}(z)|(1-|z|^2)^\alpha\\
&\leq[\frac{1}{2}(\frac{e}{2\alpha})^{\alpha}+\varepsilon]\|I-L_{n}\|\sup_{k\geq N}k^{\alpha-1}\|\varphi^{k}+\bar{\varphi}^{k}\|_{HB(\alpha)}.
\end{align*}

Thus by the Theorem <ref> we obtain

$$\limsup_{k\rightarrow\infty}J_{2}\leq[\frac{1}{2}(\frac{e}{2\alpha})^{\alpha}+\varepsilon]\sup_{k\geq N}k^{\alpha-1}\|\varphi^{k}+\bar{\varphi}^{k}\|_{HB(\alpha)}.$$

Hence for any $N$ sufficiently large we have

\begin{align*}
\|C_{\varphi}\|_{e}\leq\limsup_{n\rightarrow\infty}J_{2}\leq[\frac{1}{2}(\frac{e}{2\alpha})^{\alpha}+\varepsilon]\sup_{k\geq N}k^{\alpha-1}\|\varphi^{k}+\bar{\varphi}^{k}\|_{HB(\alpha)}.
\end{align*}

Thus we have

Since $\varepsilon$ is an arbitrary positive number, therefore we get

This completes the proof.

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