Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

Abstract

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform $Lf$ has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families.

1. Introduction

Since the several complex variables geometric analysis depends on the type of domains in ${\mathbb{C}}^n$(see for instance References [1,2,3]), we consider a special, but wide class of domains in ${\mathbb{C}}^n.$ We say that a domain $\mathcal{G}\subset {\mathbb{C}}^n,n\ge 1,$ is complete n-circular if $z\lambda =(z_1{\lambda }_1,\dots ,z_n{\lambda }_n)\in \mathcal{G}$ for each $z=(z_1,\dots ,z_n)\in \mathcal{G}$ and every $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in \overline{U^n}$, where $U^n$ is the open unit polydisc in ${\mathbb{C}}^n$, that is, the product of n copies of the open unit disc $U=\left\{\zeta \in \mathbb{C}:\left|\zeta \right|<1\right\}.$ From now on by $\mathcal{G}$ will be denoted a bounded complete n-circular domain in ${\mathbb{C}}^n,n\ge 1.$ Such bounded domain $\mathcal{G}$ and its boundary $\partial \mathcal{G}$ can be redefined as follows

$$\mathcal{G}=\{z\in {\mathbb{C}}^n:{\mu }_{\mathcal{G}}\left(z\right)<1\},\partial \mathcal{G}=\{z\in {\mathbb{C}}^n:{\mu }_{\mathcal{G}}\left(z\right)=1\},$$

using the Minkowski function ${\mu }_{\mathcal{G}}:{\mathbb{C}}^n\to [0,\infty )$

$${\mu }_{\mathcal{G}}\left(z\right)=inf\{t>0:\frac{1}{t}z\in \mathcal{G}\},z\in {\mathbb{C}}^n.$$

It is well-known (see e.g, Reference [4]) that ${\mu }_{\mathcal{G}}$ is a norm in ${\mathbb{C}}^n$ if $\mathcal{G}$ is a convex bounded complete n-circular domain.

The function ${\mu }_{\mathcal{G}}$ is very useful in research the space ${\mathcal{H}}_{\mathcal{G}}$ of holomorphic functions $f:\mathcal{G}\to \mathbb{C}$. By ${\mathcal{H}}_{\mathcal{G}}\left(1\right)$ will be denoted the collection of all $f\in {\mathcal{H}}_{\mathcal{G}}$, normalized by the condition $f\left(0\right)=1.$ In the paper we consider the following subfamilies of ${\mathcal{H}}_{\mathcal{G}}$

$$\begin{array}{cc}\hfill {\mathcal{B}}_{\mathcal{G}}& =\left\{f\in {\mathcal{H}}_{\mathcal{G}}:\left|f\left(z\right)\right|<1,z\in \mathcal{G}\right\},\hfill \\ \hfill {\mathcal{C}}_{\mathcal{G}}& =\{f\in {\mathcal{H}}_{\mathcal{G}}\left(1\right):Ref\left(z\right)>0,z\in \mathcal{G}\},\hfill \\ \hfill {\mathcal{V}}_{\mathcal{G}}& =\left\{f\in {\mathcal{H}}_{\mathcal{G}}\left(1\right):Re\mathcal{L}f\left(z\right)>0,z\in \mathcal{G}\right\},\hfill \end{array}$$

where $\mathcal{L}:{\mathcal{H}}_{\mathcal{G}}⟶{\mathcal{H}}_{\mathcal{G}}$ means the Temljakov [5] linear operator

$$\mathcal{L}f\left(z\right)=f\left(z\right)+Df\left(z\right)\left(z\right),z\in \mathcal{G},$$

defined by the Frechet differential $Df\left(z\right)$ of f at the point $z.$ Note that the operator $\mathcal{L}$ is invertible and its inverse has the form

$${\mathcal{L}}^{-1}f\left(z\right)=\underset{0}{\overset{1}{\int }}f\left(zt\right)dt,z\in \mathcal{G}.$$

Let us recall that every function $f\in {\mathcal{H}}_{\mathcal{G}}$ has a unique power series expansion

$$f\left(z\right)=\sum _{m=0}^{\infty }{Q}_{f,m}\left(z\right),z\in \mathcal{G},$$

(1)

where $Q_{f,m}:{\mathbb{C}}^n\to \mathbb{C},m\in \mathbb{N}\cup \left\{0\right\},$ are m-homogeneous polynomials. Usually the notion of m-homogeneous polynomial $Q_m:{\mathbb{C}}^n⟶\mathbb{C}$ is defined by the formula

$$Q_m\left(z\right)=L_m\left(z^m\right)=L_m(z,\dots ,z),z\in {\mathbb{C}}^n,$$

where $L_m:{\left({\mathbb{C}}^n\right)}^m⟶\mathbb{C}$ is an m-linear mapping $(0$-homogeneous polynomial means a constant function $Q_0:{\mathbb{C}}^n\to \mathbb{C})$. Note that the homogeneous polynomials occured in the expansion (1) have the form

$$Q_{f,m}\left(z\right)=\frac{1}{m!}{D}^{m}f\left(0\right)\left(z^m\right).$$

A simple kind of 1-homogeneous polynomial is the following linear functional $J\in {\left({\mathbb{C}}^n\right)}^{*}$

$$J\left(z\right)=\sum _{j=1}^n{z}_j,z=(z_1,\dots ,z_n)\in {\mathbb{C}}^n.$$

We will use the following generalization of the notion of the norm of m-homogeneous polynomial $Q_m:{\mathbb{C}}^n\to \mathbb{C},$ that is, the ${\mu }_{\mathcal{G}}$-balance of $Q_m$ [6,7,8]

$${\mu }_{\mathcal{G}}\left(Q_m\right)=\underset{w\in {\mathbb{C}}^n\setminus \left\{0\right\}}{sup}\frac{\left|Q_m\left(w\right)\right|}{{\left({\mu }_G\left(w\right)\right)}^m}=\underset{v\in \partial \mathcal{G}}{sup}\left|Q_m\left(v\right)\right|=\underset{u\in \mathcal{G}}{sup}\left|Q_m\left(u\right)\right|,$$

which is identical with the norm $∥Q_m∥$ if $\mathcal{G}$ is convex. The notion ${\mu }_{\mathcal{G}}$-balance of m-homogeneous polynomial brings a very useful inequality

$$\left|Q_m\left(z\right)\right|\le {\mu }_{\mathcal{G}}\left(Q_m\right){\left({\mu }_G\left(z\right)\right)}^m,$$

which generalize the well-known inequality

$$\left|Q_m\left(z\right)\right|\le ∥Q_m∥{∥z∥}^m.$$

Let us denote by I the linear functional

$$I={\left({\mu }_{\mathcal{G}}\left(J\right)\right)}^{-1}J$$

and by $I^m,m\ge 1,$ the m-homogeneous polynomial $I^m:{\mathbb{C}}^n\to \mathbb{C}$

$$I^m\left(z\right)={\left(I\left(z\right)\right)}^m,z\in {\mathbb{C}}^n.$$

It is obvious that ${\mu }_{\mathcal{G}}\left(I^m\right)=1.$

In many papers (see for instance References [9,10,11,12,13]) there are presented some sharp estimations of m-homogeneous polynomials $Q_{f,m},m\ge 1,$ for functions f of the form (1) from different subfamilies of ${\mathcal{H}}_{\mathcal{G}}.$ Below we give three Bavrin’s [9] estimates, in the case ${\mathbb{C}}^n,n\ge 1,$ in term of ${\mu }_{\mathcal{G}}$-balances of m-homogeneous polynomials, $m\ge 1$

$${\mu }_{\mathcal{G}}\left(Q_{f,m}\right)\le \left\{\begin{array}{c}1,\mathrm{for}f\in {\mathcal{B}}_{\mathcal{G}}\\ 2,\mathrm{for}f\in {\mathcal{C}}_{\mathcal{G}}\\ \frac{2}{m+1},\mathrm{for}f\in {\mathcal{V}}_{\mathcal{G}}\end{array},m\ge 1.\right.$$

(2)

2. Main Results

In the present paper we give for $f\in {\mathcal{B}}_{\mathcal{G}}\left(0\right)=\left\{f\in {\mathcal{B}}_{\mathcal{G}}:f\left(0\right)=0\right\}$ (also for $f\in {\mathcal{C}}_{\mathcal{G}}$ and $f\in {\mathcal{V}}_{\mathcal{G}})$ a kind sharp estimate for the pair of homogeneous polynomials $Q_{f,2},Q_{f,1},$ that is, sharp estimate

$${\mu }_{\mathcal{G}}(Q_{f,2}-\lambda {\left(Q_{f,1}\right)}^2)\le M\left(\lambda \right),\lambda \in \mathbb{C}.$$

It is a generalization of a solution of the well known Fekete-Szegö coefficient problem in complex plane [14] onto the case of several complex variables. The first result we demonstrate in the following theorem, which is a generalization of a result of Keogh and Merkes [15]:

Theorem 1.

Let $\phi \in {\mathcal{B}}_{\mathcal{G}}\left(0\right)$ be a function of the form

$$\phi \left(z\right)=\sum _{m=1}^{\infty }{Q}_{\phi ,m}\left(z\right),z\in \mathcal{G}.$$

(3)

Then, for every $\gamma \in \mathbb{C}$ there holds the sharp estimate

$${\mu }_{\mathcal{G}}\left(Q_{\phi ,2}-\gamma {\left(Q_{\phi ,1}\right)}^2\right)\le max\{1,\left|\gamma \right|\}.$$

(4)

Proof. 

Let us fix arbitrarily $z\in \mathcal{G}\setminus \left\{0\right\}.$ Then using the classic Schwarz Lemma to the function $U\ni \zeta \to \phi \left(\zeta \frac{z}{{\mu }_{\mathcal{G}}\left(z\right)}\right)\in U$(at the point $\zeta ={\mu }_{\mathcal{G}}\left(z\right)\in U),$ we obtain the inequality

$$\left|\phi \left(z\right)\right|\le {\mu }_{\mathcal{G}}\left(z\right),z\in \mathcal{G}\setminus \left\{0\right\}$$

(it is also true for $z=0$).

Now, by this result we see that for every $z\in \mathcal{G}$, the function

$$\Phi \left(\zeta \right)=\left\{\begin{array}{c}\frac{\phi \left(\zeta z\right)}{\zeta },\zeta \in U\setminus \left\{0\right\}\\ \underset{\zeta \to 0}{lim}\frac{\phi \left(\zeta z\right)}{\zeta },\zeta =0\end{array}\right.$$

transforms holomorphically the disc U into itself, fixes the point $\zeta =0$ and has the expression

$$\Phi \left(\zeta \right)=\sum _{m=0}^{\infty }{\beta }_m{\zeta }^m,\zeta \in U,$$

where ${\beta }_m=Q_{\phi ,m+1}\left(z\right),$ for nonegative integers $m.$

Thus, in view of the well known [16,17] sharp coefficient estimates

$$\begin{array}{cc}\hfill \left|{\beta }_m\right|& \le 1,m=0,1,\dots ,\hfill \\ \hfill \left|{\beta }_1\right|& \le 1-{\left|{\beta }_0\right|}^2,\hfill \end{array}$$

we obtain for every $z\in \mathcal{G}$

$$\left|Q_{\phi ,m}\left(z\right)\right|\le 1,m=1,2,\dots$$

$$\left|Q_{\phi ,2}\left(z\right)\right|\le 1-{\left|Q_{\phi ,1}\left(z\right)\right|}^2.$$

Therefore, for $z\in \mathcal{G}$ and every $\gamma \in \mathbb{C}$

$$\left|Q_{\phi ,2}\left(z\right)-\gamma {\left(Q_{\phi ,1}\left(z\right)\right)}^2\right|\le \left|Q_{\phi ,2}\left(z\right)\right|+\left|\gamma \right|{\left|Q_{\phi ,1}\left(z\right)\right|}^2\le 1-{\left|Q_{\phi ,1}\left(z\right)\right|}^2+\left|\gamma \right|{\left|Q_{\phi ,1}\left(z\right)\right|}^2$$

because $\left(\left|\gamma \right|-1\right){\left|Q_{\phi ,1}\left(z\right)\right|}^2\le 0$ if $\left|\gamma \right|<1$ and $0\le \left(\left|\gamma \right|-1\right){\left|Q_{\phi ,1}\left(z\right)\right|}^2\le \left|\gamma \right|-1$ if $\left|\gamma \right|\ge 1$.

Consequently,

$$\underset{z\in \mathcal{G}}{sup}\left|Q_{\phi ,2}\left(z\right)-\gamma {\left(Q_{\phi ,1}\left(z\right)\right)}^2\right|\le max\{1,\left|\gamma \right|\}.$$

The above inequality gives the estimate (4) from the thesis by the definition of ${\mu }_{\mathcal{G}}$-balance of homogeneous polynomials and the fact that $Q_{\phi ,2}-\gamma {\left(Q_{\phi ,1}\right)}^2$ is a 2-homogeneous polynomial.

It remains the problem of the sharpness of the estimation (4). First, we prove that in the case $\left|\gamma \right|\ge 1,$ the equality in (4) is attained by the function $\tilde{\phi }\in {\mathcal{B}}_{\mathcal{G}}\left(0\right)$

$$\tilde{\phi }\left(z\right)=I\left(z\right),z\in \mathcal{G}.$$

Indeed, since $Q_{\tilde{\phi },1}=I,Q_{\tilde{\phi },2}=0$ and ${\mu }_{\mathcal{G}}\left(I^2\right)=1,$ we have

$${\mu }_{\mathcal{G}}\left(Q_{\tilde{\phi },2}-\gamma {\left(Q_{\tilde{\phi },1}\right)}^2\right)={\mu }_{\mathcal{G}}\left(-\gamma {\left(Q_{\tilde{\phi },1}\right)}^2\right)=\left|\gamma \right|{\mu }_{\mathcal{G}}\left({\left(Q_{\tilde{\phi },1}\right)}^2\right)=\left|\gamma \right|=max\{1,\left|\gamma \right|\}.$$

Now, we show that in the case $\left|\gamma \right|<1$ the equality in (4) realizes the function $\widehat{\phi }\in {\mathcal{B}}_{\mathcal{G}}\left(0\right)$

$$\widehat{\phi }\left(z\right)=I^2\left(z\right),z\in \mathcal{G}.$$

Indeed, since $Q_{\widehat{\phi },1}=0,Q_{\widehat{\phi },2}=I^2,$ we get

$${\mu }_{\mathcal{G}}\left(Q_{\widehat{\phi },2}-\gamma {\left(Q_{\widehat{\phi },1}\right)}^2\right)={\mu }_{\mathcal{G}}\left(Q_{\widehat{\phi },2}\right)=1=max\{1,\left|\gamma \right|\}.$$

This completes the proof. □

A next theorem includes a solution of the Fekete-Szegö type problem in the family ${\mathcal{C}}_{\mathcal{G}}.$

Theorem 2.

Let $\mathcal{G}\subset$${\mathbb{C}}^n$ be a bounded complete n-circular domain and let $p\in {\mathcal{C}}_{\mathcal{G}}.$If the expansion of the function p into a series of m-homogenous polynomials $Q_{p,m}$ has the form

$$p\left(z\right)=1+\sum _{m=1}^{\infty }{Q}_{p,m}\left(z\right),z\in \mathcal{G},$$

(5)

then for the homogeneous polynomials $Q_{p,2},Q_{p,1}$ and every $\lambda \in \mathbb{C}$ there holds the following sharp estimate:

$${\mu }_{\mathcal{G}}\left(Q_{p,2}-\lambda {\left(Q_{p,1}\right)}^2\right)\le 2max\left\{1,\left|2\lambda -1\right|\right\}.$$

(6)

Proof. 

It is known, that between the functions $p\in$${\mathcal{C}}_{\mathcal{G}}$ and $\phi \in {\mathcal{B}}_{\mathcal{G}}\left(0\right),$ there holds the following relationship [9]:

$$p\in {\mathcal{C}}_{\mathcal{G}}⟺\frac{p-1}{p+1}=\phi \in {\mathcal{B}}_{\mathcal{G}}\left(0\right).$$

(7)

Inserting the expansions (3) and (5) of functions into (7), we receive

$$\sum _{m=1}^{\infty }{Q}_{p,m}\left(z\right)=\left(\sum _{m=1}^{\infty }{Q}_{\phi ,m}\left(z\right)\right)\left(2+\sum _{m=1}^{\infty }{Q}_{p,m}\left(z\right)\right),z\in \mathcal{G}.$$

Then, comparing the m-homogeneous polynomials on both sides of the above equality, we determine the homogeneous polynomials $Q_{\phi ,1},Q_{\phi ,2},$ as follows

$$\begin{array}{cc}\hfill Q_{\phi ,1}& =\frac{1}{2}{Q}_{p,1},\hfill \\ \hfill Q_{\phi ,2}& =\frac{1}{2}{Q}_{p,2}-\frac{1}{4}{\left(Q_{p,1}\right)}^2.\hfill \end{array}$$

Putting the above equalities into Theorem 2.1 and using the fact that the mapping ${\left(Q_{f,1}\right)}^2$ is a 2-homogenous polynomial, we obtain

$$\frac{1}{2}{\mu }_{\mathcal{G}}\left[Q_{p,2}-\frac{1}{2}\left(1+\gamma \right){\left(Q_{p,1}\right)}^2\right]\le max\{1,\left|\gamma \right|\}.$$

Denoting

$$\lambda =\frac{1}{2}\left(1+\gamma \right),$$

we get

$${\mu }_{\mathcal{G}}\left(Q_{p,2}-\lambda {\left(Q_{p,1}\right)}^2\right)\le 2max\left\{1,\left|2\lambda -1\right|\right\}.$$

Now, we show the sharpness of the estimate. To do it, let us consider two cases.

At the beginning, we prove that, in the case

$$\left|2\lambda -1\right|\ge 1$$

the equality in (6) is attained by the function $p=\tilde{p}$ with

$$\tilde{p}\left(z\right)=\frac{1+I\left(z\right)}{1-I\left(z\right)},z\in \mathcal{G}.$$

Indeed. The function $\tilde{p}$ belongs to ${\mathcal{C}}_{\mathcal{G}}$ and $Q_{\tilde{p},1}=2I,Q_{\tilde{p},2}=2I^2.$

From this, by the case condition for $\lambda ,$ we have step by step:

$$\begin{array}{cc}\hfill {\mu }_{{}_{\mathcal{G}}}\left(Q_{\tilde{p},2}-\lambda {\left(Q_{\tilde{p},1}\right)}^2\right)& ={\mu }_{{}_{\mathcal{G}}}\left(2I^2-\lambda 4I^2\right)=2\left|1-2\lambda \right|{\mu }_{{}_{\mathcal{G}}}\left(I^2\right)=2\left|2\lambda -1\right|\hfill \\ \hfill & =2max\left\{1,\left|2\lambda -1\right|\right\}.\hfill \end{array}$$

Now, we show that, in the case

$$\left|2\lambda -1\right|<1$$

the equality in (6) realizes the function $p=\widehat{p},$ with

$$\widehat{p}\left(z\right)=\frac{1+I^2\left(z\right)}{1-I^2\left(z\right)},z\in \mathcal{G}.$$

To do it observe that $\widehat{p}$ belongs to ${\mathcal{C}}_{\mathcal{G}}$ and $Q_{\widehat{p},1}=0,Q_{\widehat{p},2}=2I^2.$ From this, by the case condition for $\lambda ,$ we have:

$${\mu }_{{}_{\mathcal{G}}}\left(Q_{\widehat{p},2}-\lambda {\left(Q_{\widehat{p},1}\right)}^2\right)={\mu }_{{}_{\mathcal{G}}}\left(2I^2\right)=2=2max\left\{1,\left|2\lambda -1\right|\right\}.$$

This completes the proof. □

In the sequel we apply the Fekete-Szegö type result in ${\mathcal{C}}_{\mathcal{G}}$ to study the family ${\mathcal{V}}_{\mathcal{G}}.$

We start with the observation that for the transform $\mathcal{L}f$ of the functions $f\in {\mathcal{H}}_{\mathcal{G}}\left(1\right),$ we have

$$\mathcal{L}f\left(z\right)=1+\sum _{m=1}^{\infty }{Q}_{\mathcal{L}f,m}\left(z\right)=1+\sum _{m=1}^{\infty }(m+1)Q_{f,m}\left(z\right),z\in \mathcal{G}.$$

(8)

We present the Fekete-Szegö type result in the family ${\mathcal{V}}_{\mathcal{G}}$ in the following theorem:

Theorem 3.

Let $\mathcal{G}\subset$${\mathbb{C}}^n$ be a bounded complete n-circular domain and the expansion of the function $f\in {\mathcal{V}}_{\mathcal{G}}$ into a series of m-homogenous polynomials $Q_{f,m}$ has the form (1) with $Q_{f,0}=1.$ Then for the homogeneous polynomials $Q_{f,2},Q_{f,1}$ and $\eta \in \mathbb{C}$ there holds the following sharp estimate:

$${\mu }_{\mathcal{G}}\left(Q_{f,2}-\eta {\left(Q_{f,1}\right)}^2\right)\le \frac{2}{3}max\left\{1,\left|\frac{3}{2}\eta -1\right|\right\}.$$

(9)

Proof. 

Let $f\in {\mathcal{V}}_{\mathcal{G}}$. Then $p=\mathcal{L}f$ belongs to the family ${\mathcal{C}}_{\mathcal{G}}$. Inserting into this equality the expansions (5) of functions $p\in {\mathcal{C}}_{\mathcal{G}}$ and the expansions (8) of $\mathcal{L}f$ of functions $f\in {\mathcal{V}}_{\mathcal{G}},$ we obtain

$$1+\sum _{m=1}^{\infty }{Q}_{p,m}\left(z\right)=1+\sum _{m=1}^{\infty }\left(m+1\right)Q_{f,m}\left(z\right),z\in \mathcal{G}.$$

Then, comparing the m-homogeneous polynomials on both sides of the above equality, we can determine the homogeneous polynomials $Q_{p,1},Q_{p,2},$ as follows

$$\begin{array}{cc}\hfill Q_{p,1}& =2Q_{f,1},\hfill \\ \hfill Q_{p,2}& =3Q_{f,2}.\hfill \end{array}$$

Putting the above equalities into Theorem 2.2 and using the fact that the mapping ${\left(Q_{f,1}\right)}^2$ is a 2-homogenous polynomial, we obtain

$${\mu }_{\mathcal{G}}\left[3Q_{f,2}-4\lambda {\left(Q_{f,1}\right)}^2\right]\le 2max\{1,\left|2\lambda -1\right|\}$$

and consequently

$${\mu }_{\mathcal{G}}\left[Q_{f,2}-\frac{4}{3}\lambda {\left(Q_{f,1}\right)}^2\right]\le \frac{2}{3}max\{1,\left|2\lambda -1\right|\}.$$

Denoting

$$\eta =\frac{4}{3}\lambda ,$$

we get

$${\mu }_{\mathcal{G}}\left(Q_{f,2}-\eta {\left(Q_{f,1}\right)}^2\right)\le \frac{2}{3}max\left\{1,\left|\frac{3}{2}\eta -1\right|\right\}.$$

Now, we will show the sharpnes of the estimates (9). To this aim, we consider two cases.

At the begining, we prove that the equality in (9) holds in the case

$$\left|\frac{3}{2}\eta -1\right|\ge 1.$$

To do it let us denote by $\mathcal{Z}$ the analytic set $\{z\in \mathcal{G}:I(z)=0\}.$ In this case the extremal function has the form

$$\tilde{f}\left(z\right)=\left\{\begin{array}{c}-1-\frac{2}{I\left(z\right)}log\left(1-I(z\right)),\mathrm{for}z\in \mathcal{G}\backslash \mathcal{Z}\\ 1,\mathrm{for}z\in \mathcal{Z}\end{array}\right.,$$

where the branch of the function $log(1-\zeta ),\zeta \in U,$ takes the value 0 at the point $\zeta =0$.

First we observe that $\tilde{f}\in {\mathcal{V}}_{\mathcal{G}},$ because $\mathcal{L}\tilde{f}=\frac{1+I}{1-I}$$\in {\mathcal{C}}_{\mathcal{G}}.$

Now we show that $\tilde{f}$ realizes the equality in the thesis. To do it observe that the power series expansion of the function $log(1-\zeta ),\zeta \in U,$ implies the expression

$$\tilde{f}\left(z\right)=-1+\frac{2}{I\left(z\right)}\left(I\left(z\right)+\frac{1}{2}{I}^2\left(z\right)+\frac{1}{3}{I}^3\left(z\right)+\dots \right),z\in \mathcal{G}.$$

Thus

$$\begin{array}{cc}\hfill Q_{\tilde{f},1}\left(z\right)& =I\left(z\right)\hfill \\ \hfill Q_{\tilde{f},2}\left(z\right)& =\frac{2}{3}{I}^2\left(z\right).\hfill \end{array}$$

Hence, we have step by step:

$${\mu }_{{}_{\mathcal{G}}}\left(Q_{\tilde{f},2}-\eta {\left(Q_{\tilde{f},1}\right)}^2\right)={\mu }_{{}_{\mathcal{G}}}\left(\frac{2}{3}{I}^2-\eta I^2\right)=\left|\frac{2}{3}-\eta \right|{\mu }_{{}_{\mathcal{G}}}\left(I^2\right)=\left|\frac{2}{3}-\eta \right|=\frac{2}{3}max\left\{1,\left|\frac{3}{2}\eta -1\right|\right\}.$$

Now, we show that, in the case

$$\left|\frac{3}{2}\eta -1\right|<1$$

the extremal function has the form

$$\widehat{f}\left(z\right)=\left\{\begin{array}{c}-1+log\frac{1+I\left(z\right)}{1-I\left(z\right)},\mathrm{for}z\in \mathcal{G}\backslash \mathcal{Z},\\ 1,\mathrm{for}z\in \mathcal{Z}\end{array}\right.,$$

where the branch of the function $log(1-\zeta ),\zeta \in U,$ takes the value 0 at the point $\zeta =0.$

Of course, $\widehat{f}\in {\mathcal{V}}_{\mathcal{G}},$ because $\mathcal{L}\widehat{f}=\frac{1+I^2}{1-I^2}$$\in {\mathcal{C}}_{\mathcal{G}}.$

Observe that using the power series expansion of the function $log(1-\zeta ),\zeta \in U,$ we get the expression

$$\widehat{f}\left(z\right)=-1+\frac{1}{I\left(z\right)}\left[2I\left(z\right)+\frac{2}{3}{I}^3\left(z\right)+\dots \right],z\in \mathcal{G}$$

and consequently

$$Q_{\widehat{f},1}=0,Q_{\widehat{f},2}=\frac{2}{3}{I}^2.$$

Therefore, we have step by step

$${\mu }_{{}_{\mathcal{G}}}\left(Q_{\widehat{f},2}-\eta {\left(Q_{\widehat{f}\widehat{,}1}\right)}^2\right)={\mu }_{{}_{\mathcal{G}}}\left(Q_{\widehat{f},2}\right)={\mu }_{{}_{\mathcal{G}}}\left(\frac{2}{3}{I}^2\right)=\frac{2}{3}=\frac{2}{3}max\left\{1,\left|\frac{3}{2}\eta -1\right|\right\}.$$

This completes the proof. □

3. Complementary Remarks

Bavrin [9] declared that every of the estimations (2) is sharp in this sense that there exists an n-circular complete bounded domain $\mathcal{G}$ and a function f from appropriate family $(f\in {\mathcal{B}}_{\mathcal{G}},f\in {\mathcal{C}}_{\mathcal{G}}$, $f\in {\mathcal{V}}_{\mathcal{G}})$ for which the equality in an inequality of $\left(2\right)$ holds. Actually we know that the estimations $\left(2\right)$ are sharp in the sense that for every domain $\mathcal{G}$ there exists an extremal function in appropriate family which realizes equality in required inequality from (2). Another problem, connected with the above type estimates, is a characterization of the set of all extremal functions. An information in this direction follows from the main result of Reference [12]. Here we present its part connected with the family ${\mathcal{C}}_{\mathcal{G}}$(in the term of ${\mu }_{\mathcal{G}}$-balance of m-homogeneous polynomials).

If the function p of the form (5) belongs to ${\mathcal{C}}_{\mathcal{G}},$ then for every $m\ge 1$

$$2-{\mu }_{\mathcal{G}}\left(Q_{p,m}\right)\le m^2\left[2-{\mu }_{\mathcal{G}}\left(Q_{p,1}\right)\right].$$

Observe that this result implies that the equality ${\mu }_{\mathcal{G}}\left(Q_{p,1}\right)=2$ for a function $p\in {\mathcal{C}}_{\mathcal{G}}$ implies equalities ${\mu }_{\mathcal{G}}\left(Q_{p,m}\right)=2,m\ge 1.$ In others words if a function $p\in {\mathcal{C}}_{\mathcal{G}}$ is extremal in the estimation $\left(2\right)$ for $m=1,$ then it is also extremal for each $m\ge 1.$

Actually, we also have a similar result for the family ${\mathcal{V}}_{\mathcal{G}}.$ More precisely, it is true the following statement. If the function f of the form (1) with $Q_{f,0}=1,$ belongs to ${\mathcal{V}}_{\mathcal{G}},$ then for every $m\ge 1$

$$\frac{2}{m+1}-{\mu }_{\mathcal{G}}\left(Q_{f,m}\right)\le \frac{2m^2}{m+1}\left[1-{\mu }_{\mathcal{G}}\left(Q_{f,1}\right)\right].$$

To this aim it suffices to recall that, by the assumptions, the function

$$p\left(z\right)=\mathcal{L}f\left(z\right)=1+\sum _{m=1}^{\infty }\left(m+1\right)Q_{f,m}\left(z\right),z\in \mathcal{G},$$

belongs to the family ${\mathcal{C}}_{\mathcal{G}}$ and use the previous original inequality in ${\mathcal{C}}_{\mathcal{G}}$. Therefore, if a function $f\in {\mathcal{V}}_{\mathcal{G}}$ is extremal in appropriate estimate $\left(2\right)$ for $m=1,$ that is, if ${\mu }_{\mathcal{G}}\left(Q_{f,1}\right)=1,$ then it is also extremal in required estimate $\left(2\right)$ for each $m\ge 1,$ that is, ${\mu }_{\mathcal{G}}\left(Q_{f,m}\right)=\frac{2}{m+1}.$

We close the paper with a suggestion of characterization of the set of all extremal functions in different estimates of homogeneous polynomials (also of Fekete-Szegö type) in series of functions from subfamilies of the family ${\mathcal{H}}_{\mathcal{G}}.$