1 Introduction

We say that a domain \({\mathcal {G}}\subset {\mathbb {C}}^{n},n\ge 1,\) is complete n-circular if \(z\lambda =(z_{1}\lambda _{1},...,z_{n}\lambda _{n} )\in {\mathcal {G}}\) for each \(z=(z_{1},.-..,z_{n})\in {\mathcal {G}}\) and every \(\lambda =(\lambda _{1},...,\lambda _{n})\in \overline{U^{n}}\), where \(U^{n}\)is the open unit polydisc in \({\mathbb {C}}^{n}\), i.e., the Cartesian 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 \({\mathbb {C}}^{n},n\ge 1.\)

Let \({\mathcal {H}}_{{\mathcal {G}}}\) be the space of holomorphic functions \(f:\mathcal {G\rightarrow {\mathbb {C}}},{\mathcal {H}}_{{\mathcal {G}}}(1)\) be the collection of all \(f\in \mathcal {H_{G}}\), normalized by \(f(0)=1,\) \({\mathcal {C}}_{{\mathcal {G}}}\) be the collection of all \(p\in \mathcal {H_{G}}(1)\) with positive real part and \({\mathcal {M}}_{{\mathcal {G}}}\) be the family of all \(g\in \mathcal {H_{G}}(1)\) with the following factorization of its transform \({\mathcal {L}}g\) (see [1])

$$\begin{aligned} {\mathcal {L}}g(z)=p(z)g(z),z\in {\mathcal {G}}, \end{aligned}$$

with \(p\in {\mathcal {C}}_{{\mathcal {G}}}.\) Here \({\mathcal {L}}:{\mathcal {H}} _{{\mathcal {G}}}\longrightarrow {\mathcal {H}}_{{\mathcal {G}}}\) means the Temljakov [32] linear operator

$$\begin{aligned} {\mathcal {L}}g(z)=g(z)+Dg(z)(z),z\in {\mathcal {G}}, \end{aligned}$$

defined by the Fréchet differential Dg(z) of the function g at the point z (here and throughout the paper, z is treated as a column vector and Dg(z) is a one-row matrix \(\left[ g_{z_{1}}^{\prime }\left( z\right) ,...,g_{z_{n}}^{\prime }\left( z\right) \right] ).\)

Let us now specify the family \({\mathcal {R}}_{{\mathcal {G}}}\) [1], for which we solve an n-dimensional generalization of the well-known Fekete–Szegö type coefficients problem. By \(\mathcal {R_{G}}\) let us denote the family of all functions \(f\in \mathcal {H_{G}}(1)\) such that its transform \({\mathcal {L}}f\) has the following factorization

$$\begin{aligned} {\mathcal {L}}f(z)=p(z)g(z),z\in {\mathcal {G}}, \end{aligned}$$

with some functions \(p\in {\mathcal {C}}_{{\mathcal {G}}},g\in {\mathcal {M}} _{{\mathcal {G}}}.\)

The above families \({\mathcal {M}}_{{\mathcal {G}}},{\mathcal {R}}_{{\mathcal {G}}}\) correspond with the well-known families of univalent starlike and close-to-convex functions of a complex variable. This relationship is evident by the following Bavrin’s [1] geometric interpretation of \({\mathcal {M}}_{{\mathcal {G}}}\) and \({\mathcal {R}}_{{\mathcal {G}}}\) in the case \(n=2.\) A function \(f\in {\mathcal {H}}_{{\mathcal {G}}}(1)\) belongs to \(\mathcal {M_{G}}\left( \mathcal {R_{G}}\right) ,\) if and only if:

  1. (i)

    for every \(\alpha \in {\mathbb {C}}\) the function \(z_{1}f\left( z_{1},\alpha z_{1}\right) ,\) of one variable \(z_{1},\) is univalent starlike (close-to-convex) in a disc be the projection of the intersection \({\mathcal {G}}\cap \left\{ \left( z_{1},z_{2}\right) \in {\mathbb {C}}^{2} :z_{2}=\alpha z_{1}\right\} \) onto the plane \(z_{2}=0,\)

  2. (ii)

    the function \(z_{2}f\left( 0,z_{2}\right) \) of one variable \(z_{2}\) is univalent starlike (close-to-convex) function in the disc \({\mathcal {G}} \cap \left\{ \left( z_{1},z_{2}\right) \in {\mathbb {C}}^{2}:z_{1}=0\right\} .\)

Let us add that in the third chapter we will give also a connection between these families \({\mathcal {M}}_{{\mathcal {G}}},{\mathcal {R}}_{{\mathcal {G}}}\) and biholomorphic mappings in \({\mathbb {C}}^{n},n\ge 1;\) starlike, [18, 25, 30] and close-to starlike [27], respectively.

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

$$\begin{aligned} f(z)=\sum _{m=0}^{\infty }Q_{f,m}(z),z\in {\mathcal {G}}, \end{aligned}$$

where \(Q_{f,m}:{\mathbb {C}}^{n}\rightarrow {\mathbb {C}}\) are m-homogeneous polynomials, \(m\in {\mathbb {N}}\cup \left\{ 0\right\} \) (here the symbol \({\mathbb {N}}\) means the set of all positive integers). Usually the notion of m-homogeneous polynomial \(Q_{m}:{\mathbb {C}}^{n}\longrightarrow {\mathbb {C}} ,m\in {\mathbb {N}},\) is defined by the formula

$$\begin{aligned} Q_{m}(z)=A_{m}(z^{m})=A_{m}(z,...,z),z\in {\mathbb {C}}^{n}, \end{aligned}$$

where \(A_{m}:\left( {\mathbb {C}}^{n}\right) ^{m}\longrightarrow {\mathbb {C}}\) is an m-linear mapping (0-homogeneous polynomial \(Q_{0}\) means a constant function on \({\mathbb {C}}^{n})\). Note that the homogeneous polynomials, occured in the above expansion, have for \(m\in {\mathbb {N}}\) the form

$$\begin{aligned} Q_{f,m}(z)=\frac{1}{m!}D^{m}f(0)(z^{m}),z\in {\mathcal {G}}, \end{aligned}$$

where \(D^{m}f(0)\) is the m-th Fréchet differential of the function f at the point zero and \(D^{m}f(0)(a^{m})\) means the value at the Cartesian product \(a^{m}=(a,...,a)\) of m copies of a vector \(a\in {\mathbb {C}}^{n}.\) A useful kind of 1-homogeneous polynomial is the following linear functional \({\widehat{J}}\in \left( {\mathbb {C}}^{n}\right) ^{*}\)

$$\begin{aligned} {\widehat{J}}(z)=\sum \limits _{j=1}^{n}z_{j},z=(z_{1},...,z_{n})\in {\mathbb {C}} ^{n}. \end{aligned}$$

Observe that a bounded complete n-circular domain \(\mathcal {G\subset }{\mathbb {C}}^{n}\) and its boundary \(\partial {\mathcal {G}}\) can be redefined as follows

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

using the Minkowski functional \(\mu _{{\mathcal {G}}}:{\mathbb {C}} ^{n}\rightarrow [0,\infty )\)

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

Moreover, \(\mu _{{\mathcal {G}}}\) is a norm in \({\mathbb {C}}^{n}\), if \({\mathcal {G}}\) is convex (see, e.g, [29]). The value \(\mu _{{\mathcal {G}}}(z)\) of the function \(\mu _{{\mathcal {G}}}\) at a fixed point \(z\in {\mathbb {C}}^{n}\) will be referred in the sequel as Minkowski balance of z.

We will use the following generalization of the notion of the norm of m-homogeneous polynomial \(Q_{m}:{\mathbb {C}}^{n}\rightarrow {\mathbb {C}},\) i.e., the Minkowski \(\mu _{{\mathcal {G}}}\)-balance of \(Q_{m}\) (see for instance [2, 3, 23]).

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

which is identical with the norm \(\left\| Q_{m}\right\| ,\) if \({\mathcal {G}}\) is convex. The notion of Minkowski \(\mu _{{\mathcal {G}}}\)-balance of m-homogeneous polynomial leads to a very useful inequality

$$\begin{aligned} \left| Q_{m}(z)\right| \le \mu _{{\mathcal {G}}}(Q_{m})(\mu _{G}(z))^{m},z \in C^{n} \end{aligned}$$

which generalizes the well-known inequality

$$\begin{aligned} \left| Q_{m}(z)\right| \le \left\| Q_{m}\right\| \left\| z\right\| ^{m}z \in C^{n} \end{aligned}$$

Let us denote by J the linear functional

$$\begin{aligned} J=\left( \mu _{{\mathcal {G}}}({\widehat{J}})\right) ^{-1}{\widehat{J}} \end{aligned}$$

and by \(J^{m},m\in {\mathbb {N}},\) the m-homogeneous polynomial \(J^{m} :{\mathbb {C}}^{n}\rightarrow {\mathbb {C}}\)

$$\begin{aligned} J^{m}(z)=\left( J(z)\right) ^{m},z\in {\mathbb {C}}^{n}. \end{aligned}$$

It is obvious that \(\mu _{{\mathcal {G}}}(J^{m})=1\) and for \(z\in {\mathcal {G}} , \) also \(\left| J^{m}(z)\right| \le (\mu _{G}(z))^{m}<1.\)

Note also that for the transform \({\mathcal {L}}f\) of the functions \(f\in \mathcal {H_{G}},\) we have

$$\begin{aligned} {\mathcal {L}}f(z)=Q_{{\mathcal {L}}f,0}+\sum _{m=1}^{\infty }Q_{{\mathcal {L}} f,m}(z)=Q_{f,0}+\sum _{m=1}^{\infty }(m+1)Q_{f,m}(z),z\in {\mathcal {G}}. \end{aligned}$$

In many papers (see for instance [12, 16, 26]) there are presented some sharp estimations of m-homogeneous polynomials \(Q_{f,m},m\ge 1,\) for functions f from different subfamilies of \({\mathcal {H}}_{{\mathcal {G}}}.\) In particular, in the paper [6] we can found for functions f of the form

$$\begin{aligned} f(z)=1+\sum _{m=1}^{\infty }Q_{f,m}(z),z\in {\mathcal {G}}, \end{aligned}$$
(2.1)

belonging to the family \({\mathcal {R}}_{{\mathcal {G}}},\) the following sharp estimate

$$\begin{aligned} \mu _{{\mathcal {G}}}(Q_{f,m})\le m+1,m\in {\mathbb {N}}\ . \end{aligned}$$

In the papers [4, 5] we gave for a few Bavrin’s families a sharp estimate for the pair of homogeneous polynomials \(Q_{f,2},Q_{f,1},\) i.e., the sharp estimate

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

It is a generalization of a solution of the well known Fekete–Szegö coefficient problem, in complex plane [11], onto the case of several complex variables. In Sect. 2 we apply the result from Section 2 to solve a kind of Fekete–Szegö problem for close-to-starlike biholomorphic mappings in \({\mathbb {C}}^{n}.\) The research of the Fekete–Szegö problem in \({\mathbb {C}}^{n}\) and in complex Banach spaces began probably in 2014 in the paper [34] and has been continued by many authors (see e.g. [4, 34]). The most recent results can be found in the papers [8, 21]. The necessary concepts and facts for the infinite dimensional case can be found in the monographs [10, 13] and in the papers [9, 14, 15, 31].

2 A Fekete-Szegö estimate for holomorphic functions in \({\mathbb {C}}^{n}\)

In this section, we present a theorem for the family \({\mathcal {R}} _{{\mathcal {G}}},\) similar to the mentioned results from [4, 5]. To this purpose we use the following notations. Let \(M\left( \lambda ,j\right) ,j=1,2,3,4\) mean the following quantities

$$\begin{aligned} \begin{array}{ll} M\left( \lambda ,1\right) =3-4\lambda for &{} \lambda \in \Lambda _{1}=\left( -\infty ,\frac{1}{3}\right] \\ M\left( \lambda ,2\right) =\frac{1}{3}+\frac{4}{9\lambda } for &{} \lambda \in \Lambda _{2}=\left[ \frac{1}{3},\frac{2}{3}\right] \\ M\left( \lambda ,3\right) =1 \ for &{} \lambda \in \Lambda _{3}=\left[ \frac{2}{3},1\right] \\ M\left( \lambda ,4\right) =4\lambda -3 for &{} \lambda \in \Lambda _{4}=[1,\infty ) \end{array} \ \ . \end{aligned}$$
(2.2)

The following theorem is a generalization of a result of Keogh and Merkes [19]:

Theorem 2.1

Let \(f\in {\mathcal {R}}_{{\mathcal {G}}}\) be a function of the form (2.1). Then, for \(\lambda \in {\mathbb {R}}\) and homogeneous polynomials \(Q_{f,2},Q_{f,1},\) there holds the sharp estimate

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2}-\lambda \left( Q_{f,1}\right) ^{2}\right) \le \begin{array}{ll} M\left( \lambda ,j\right) for&\lambda \in \Lambda _{j},j=1,2,3,4. \end{array} \ \ \end{aligned}$$
(2.3)

Proof

Let \(f\in \mathcal {R_{G}}\). Then from the definition of the family \(\mathcal {R_{G}}\) we have for \(z\in {\mathcal {G}},\)

$$\begin{aligned} {\mathcal {L}}f\left( z\right) =p(z)g(z), \end{aligned}$$

where p belongs to the family \(\mathcal {C_{G}}\) and g belongs to the family \(\mathcal {M_{G}}\). Between the functions \(p\in \) \(\mathcal {C_{G}}\) and

$$\begin{aligned} \varphi \in {\mathcal {B}}_{{\mathcal {G}}}(0)=\left\{ \varphi \in {\mathcal {H}}_{{\mathcal {G}}}(0):|\varphi (z)|<1,z\in {\mathcal {G}}\right\} \end{aligned}$$

there holds the following relationship [1]:

$$\begin{aligned} p\in \mathcal {C_{G}}\Longleftrightarrow \frac{p-1}{p+1}=\varphi \in {\mathcal {B}}_{{\mathcal {G}}}(0). \end{aligned}$$

From the above it follows that f belongs to \(\mathcal {R_{G}},\) relative to \(g\in \mathcal {M_{G}},\) iff there exists a function \(\varphi \in {\mathcal {B}}_{{\mathcal {G}}}(0)\) such that

$$\begin{aligned} \frac{{\mathcal {L}}f-g}{{\mathcal {L}}f+g}=\varphi . \end{aligned}$$

Inserting into above the expansion ( 2.1) and the below expansions

$$\begin{aligned} \varphi \left( z\right)&=\sum _{m=1}^{\infty }Q_{\varphi ,m}(z),\\ g\left( z\right)&=1+\sum _{m=1}^{\infty }Q_{g,m}(z), \end{aligned}$$

we get for \(z\in {\mathcal {G}},\)

$$\begin{aligned}&(m+1)\sum _{m=1}^{\infty }Q_{f,m}(z)-\sum _{m=1}^{\infty }Q_{g,m}(z)\\&\qquad =\left( \sum _{m=1}^{\infty }Q_{\varphi ,m}(z)\right) \left( 2+\left( m+1\right) \sum _{m=1}^{\infty }Q_{f,m}(z)+\sum _{m=1}^{\infty }Q_{g,m} (z)\right) . \end{aligned}$$

Then, comparing the m-homogeneous polynomials on both sides of the above equality, we compute the homogeneous polynomials \(Q_{f,1},Q_{f,2},\) as follows

$$\begin{aligned} Q_{f,1}&=Q_{\varphi ,1}+\frac{1}{2}Q_{g,1},\\ Q_{f,2}&=\frac{1}{3}Q_{g,2}+\frac{2}{3}\left( Q_{\varphi ,2}+Q_{\varphi ,1}^{2}\right) +\frac{2}{3}Q_{\varphi ,1}Q_{g,1}. \end{aligned}$$
(2.4)

Since \(g\in \mathcal {M_{G}},\) we get equivalently \({\mathcal {L}}g=gh,\) where the function h belongs to \(\mathcal {C_{G}}\). Likewise using the relationship between the functions \(h\in \mathcal {C_{G}}\) and \(\psi \in {\mathcal {B}} _{{\mathcal {G}}}(0)\) we obtain that

$$\begin{aligned} \frac{{\mathcal {L}}g-g}{{\mathcal {L}}g+g}=\psi \in {\mathcal {B}}_{{\mathcal {G}}}(0). \end{aligned}$$
(2.5)

Similarly, inserting expansions of functions g and \({\mathcal {L}}g\) into (2.5), we obtain

$$\begin{aligned} \sum _{m=1}^{\infty }mQ_{g,m}(z)=\left( \sum _{m=1}^{\infty }Q_{\psi ,m}(z)\right) \left( 2+\left( m+2\right) \sum _{m=1}^{\infty } Q_{g,m}(z)\right) ,z\in {\mathcal {G}}. \end{aligned}$$

Thus comparing the m-homogeneous polynomials on both sides of the above equality, we determine the homogeneous polynomials \(Q_{g,1},Q_{g,2},\) as follows

$$\begin{aligned} Q_{g,1}&=2Q_{\psi ,1},\\ Q_{g,2}&=Q_{\psi ,2}+3Q_{\psi ,1}^{2}. \end{aligned}$$
(2.6)

Hence, inserting (2.6) into (2.4), we obtain the following form of polynomials \(Q_{f,1},Q_{f,2}:\)

$$\begin{aligned} Q_{f,1}&=Q_{\varphi ,1}+Q_{\psi ,1}\\ Q_{f,2}&=\frac{1}{3}Q_{\psi ,2}+Q_{\psi ,1}^{2}+\frac{2}{3}\left( Q_{\varphi ,2}+Q_{\varphi ,1}^{2}\right) +\frac{2}{3}Q_{\varphi ,1}+2Q_{\psi ,1}. \end{aligned}$$
(2.7)

Now, for \(\lambda \in {\mathbb {R}},\) we estimate the expression \(\left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| ,z\in {\mathcal {G}}.\) Let us fix arbitrary \(z\in {\mathcal {G}}\) and \(\lambda \in {\mathbb {R}}\). Then, by (2.7),

$$\begin{aligned}&\left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| \le \frac{1}{3}\left[ \left| Q_{\psi ,2}\left( z\right) \right| +3\left| 1-\lambda \right| \left| Q_{\psi ,1}\left( z\right) \right| ^{2}\right] \\&\quad +\frac{2}{3}\left[ \left| Q_{\varphi ,2}\left( z\right) \right| +\left| 1-\frac{3}{2}\lambda \right| \left| Q_{\varphi ,1} ^{2}\left( z\right) \right| \right] +2\left[ \left| \frac{2}{3}-\lambda \right| \left| Q_{\varphi ,1}\left( z\right) \right| \left| Q_{\psi ,1}\left( z\right) \right| \right] . \end{aligned}$$

Evidently we should consider three cases: \(\lambda \in \left( -\infty ,\frac{2}{3}\right] ,\lambda \in \left[ \frac{2}{3},1\right] \) and \(\lambda \in \left[ 1,\infty \right) .\) We start with the case \(\lambda \in \left( -\infty ,\frac{2}{3}\right] \). Then by bound \(\left| Q_{\psi ,1}\left( z\right) \right| \le 1\) [4], we get for such \(\lambda \)

$$\begin{aligned} \left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| \le -\lambda \left| Q_{\varphi ,1}\left( z\right) \right| ^{2}+2\left( \frac{2}{3}-\lambda \right) \left| Q_{\varphi ,1}\left( z\right) \right| +\frac{5}{3}-\lambda ,z\in {\mathcal {G}}. \end{aligned}$$

Since \(z\in {\mathcal {G}}\) is fixed, the right-hand side of the above inequality can be treated as a square function S of the variable \(\left| Q_{\varphi ,1}\left( z\right) \right| \in [0,1],\) with the coefficients dependend on \(\lambda .\) Determining the maximum of the function

$$\begin{aligned} S(\left| Q_{\varphi ,1}\left( z\right) \right| )=-\lambda \left| Q_{\varphi ,1}\left( z\right) \right| ^{2}+2\left( \frac{2}{3} -\lambda \right) \left| Q_{\varphi ,1}\left( z\right) \right| +\frac{5}{3}-\lambda , \end{aligned}$$

in two subcases \(\lambda \in \Lambda _{1}=\left( -\infty ,\frac{1}{3}\right] ,\lambda \in \Lambda _{2}=\left[ \frac{1}{3},\frac{2}{3}\right] ,\) we obtain

$$\begin{aligned} \left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| \le \left\{ \begin{array}{ll} 3-4\lambda , &{} \lambda \in \left( -\infty ,\frac{1}{3}\right] \\ \frac{1}{3}+\frac{4}{9\lambda }, &{} \lambda \in \left[ \frac{1}{3},\frac{2}{3}\right] \end{array} \ \ \ .\right. \end{aligned}$$

Consequently, by the arbitrariness of \(z\in {\mathcal {G}}\), also

$$\begin{aligned} \underset{z\in {\mathcal {G}}}{\sup }\left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| \le \left\{ \begin{array}{ll} 3-4\lambda , &{} \lambda \in \left( -\infty ,\frac{1}{3}\right] \\ \frac{1}{3}+\frac{4}{9\lambda }, &{} \lambda \in \left[ \frac{1}{3},\frac{2}{3}\right] \end{array} \ \ \ .\right. \end{aligned}$$

Finally, by the fact that the mapping \(Q_{f,1}^{2}\) is a 2-homogenous polynomial, we obtain

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2}-\lambda \left( Q_{f,1}\right) ^{2}\right) \le \left\{ \begin{array}{ll} M\left( \lambda ,1\right) , &{} \lambda \in \Lambda _{1} \\ M\left( \lambda ,2\right) , &{} \lambda \in \Lambda _{2} \end{array} \ \ \ .\right. \end{aligned}$$

Now we consider the case \(\lambda \in \Lambda _{3}=\left[ \frac{2}{3},1\right] .\) Since \(\left| Q_{\psi ,1}\left( z\right) \right| \le 1,\left| Q_{\varphi ,1}\left( z\right) \right| \le 1,\) at \(z\in {\mathcal {G}},\) (see [4]), we get for \(\lambda =1,\)

$$\begin{aligned} \left| Q_{f,2}\left( z\right) -Q_{f,1}^{2}\left( z\right) \right|&\le 1-\frac{1}{3}\left( \left| Q_{\psi ,1}\left( z\right) \right| ^{2}+\left| Q_{\varphi ,1}\left( z\right) \right| ^{2}-2\left| Q_{\varphi ,1}\left( z\right) \right| \left| Q_{\psi ,1}\left( z\right) \right| \right) \\&=1-\frac{1}{3}\left( \left| Q_{\psi ,1}\left( z\right) \right| -\left| Q_{\varphi ,1}\left( z\right) \right| \right) ^{2}\le 1. \end{aligned}$$

On the other hand, by the fact that \(\lambda =\frac{2}{3}\in \Lambda _{2},\) we get also

$$\begin{aligned} \left| Q_{f,2}\left( z\right) -\frac{2}{3}Q_{f,1}^{2}\left( z\right) \right| \le 1. \end{aligned}$$

Thus, for \(\lambda \in \left[ \frac{2}{3},1\right] ,\)

$$\begin{aligned}&\left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| \\&\quad = \left| \left[ \left( 3\lambda -2\right) Q_{f,2}\left( z\right) +3\left( 1-\lambda \right) Q_{f,2}\left( z\right) \right] -\left[ \left( 3\lambda -2\right) Q_{f,1}^{2}\left( z\right) +3\left( 1-\lambda \right) \frac{2}{3}Q_{f,1}^{2}\left( z\right) \right] \right| \\&\quad =\left| \left( 3\lambda -2\right) \left[ Q_{f,2}\left( z\right) -Q_{f,1}^{2}\left( z\right) \right] +3\left( 1-\lambda \right) \left[ Q_{f,2}\left( z\right) -\frac{2}{3}Q_{f,1}^{2}\left( z\right) \right] \right| \\&\quad \le \left| 3\lambda -2\right| \left| Q_{f,2}\left( z\right) -Q_{f,1}^{2}\left( z\right) \right| +3\left| 1-\lambda \right| \left| Q_{f,2}\left( z\right) -\frac{2}{3}Q_{f,1}^{2}\left( z\right) \right| \\&\quad \le \left| 3\lambda -2\right| +3\left| 1-\lambda \right| =1. \end{aligned}$$

Hence, by the arbitrariness of \(z\in {\mathcal {G}},\) for \(\lambda \in \left[ \frac{2}{3},1\right] ,\)

$$\begin{aligned} \underset{z\in {\mathcal {G}}}{\sup }\left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| \le 1. \end{aligned}$$

Consequently,

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2}-\lambda \left( Q_{f,1}\right) ^{2}\right) \le M\left( \lambda ,3\right) ,\lambda \in \Lambda _{3}. \end{aligned}$$

In the closing case \(\lambda \in \Lambda _{4}=\left[ 1,\infty \right) \) observe that \(\left| Q_{f,2}\left( z\right) -Q_{f,1}^{2}\left( z\right) \right| \le 1\) for \(z\in {\mathcal {G}},\) because \(\lambda =1\in \Lambda _{3}.\) This and the well-known bound \(\mu _{{\mathcal {G}}}\left( Q_{f,2}\right) \le 2\) for \(f\in \mathcal {R_{G}}\) (see: [6]), give for \(\lambda \in \left[ 1,\infty \right) \),

$$\begin{aligned} \underset{z\in {\mathcal {G}}}{\sup }\left| Q_{f,2}\left( z\right) -\lambda Q_{f,1}^{2}\left( z\right) \right| =\underset{z\in {\mathcal {G}}}{\sup }\left| Q_{f,2}\left( z\right) -Q_{f,1}^{2}\left( z\right) +(1-\lambda )Q_{f,1}^{2}\left( z\right) \right| \\ \le \underset{z\in {\mathcal {G}}}{\sup }\left| Q_{f,2}\left( z\right) -Q_{f,1}^{2}\left( z\right) \right| +\left| 1-\lambda \right| \underset{z\in {\mathcal {G}}}{\sup }\left| Q_{f,1}^{2}\left( z\right) \right| =4\lambda -3. \end{aligned}$$

Hence,

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2}-\lambda Q_{f,1}^{2}\right) \le M\left( \lambda ,4\right) ,\lambda \in \Lambda _{4}. \end{aligned}$$

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

In the case when \(\lambda \in \Lambda _{1}\) the equality in (2.3) is realized by the function

$$\begin{aligned} f_{1}(z)=\frac{1}{\left( 1-J(z)\right) ^{2}},z\in {\mathcal {G}}. \end{aligned}$$
(2.8)

Of course, it is holomorphic, because by the properties of Minkowski balances \(\mu _{{\mathcal {G}}}(z),\mu _{{\mathcal {G}}}(J),\) we have for \(z\in {\mathcal {G}},\)

$$\begin{aligned} \left| J(z)\right| \le \mu _{{\mathcal {G}}}(J)\mu _{{\mathcal {G}}}(z)\le \mu _{{\mathcal {G}}}(z)<1. \end{aligned}$$

Moreover, the function (2.8) belongs to the family \(\mathcal {R_{G}},\) because \(f_{1}\in \mathcal {M_{G}}\) and \(\mathcal {M_{G}}\subset \mathcal {R_{G}}\) (see [1, 2]).

From the expansion of the function \(f_{1}\) into m-homogeneous polynomial series, we get that

$$\begin{aligned} Q_{f_{1},1}=2J,Q_{f_{1},2}=3J^{2}. \end{aligned}$$

From this, by the case condition for \(\lambda ,\) we have:

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}\left( Q_{f_{1},2}-\lambda \left( Q_{f_{2},1}\right) ^{2}\right)&=\mu _{_{{\mathcal {G}}}}\left( \left( 3-4\lambda \right) J^{2}\right) =\left| 3-4\lambda \right| \mu _{_{{\mathcal {G}}}}\left( J^{2}\right) \\&=3-4\lambda =M\left( \lambda ,1\right) . \end{aligned}$$

Now, we go to the next case when \(\lambda \in \Lambda _{2}.\) Then, denoting by \({\mathcal {Z}}\) the analytic set \(\{z\in {\mathcal {G}}:J(z)=0\},\) which is nowhere dense and closed in \({\mathcal {G}},\) we define the extremal function as follows

$$\begin{aligned} f_{2}\left( z\right) =\left\{ \begin{array}{ll} t\frac{1}{\left( 1-J(z)\right) ^{2}}+\left( 1-t\right) \left[ \frac{1}{4J(z)}\log \frac{1+J(z)}{1-J(z)}+\frac{1}{2}\frac{1}{\left( 1-J(z)\right) ^{2}}\right] &{} \text { for }z\in \mathcal {G\diagdown Z}\\ 1&{}\text { for }z\in {\mathcal {Z}} \end{array} \right. , \end{aligned}$$
(2.9)

where \(t=\frac{2}{3\lambda }-1\) and the branch of the function \(\log (1-\zeta ),\zeta \in U,\) takes the value 0 at the point \(\zeta =0.\) Of course, \(f_{2}\) is holomorphic in \(\mathcal {G\diagdown Z}\) (compare holomorphicity of \(f_{1})\) and extends holomorphically onto \({\mathcal {G}}\).

Moreover, we observe that \(f_{2}\in {\mathcal {R}}_{{\mathcal {G}}},\) because \( g=f_{1}\) belongs to \({\mathcal {M}}_{{\mathcal {G}}},\)

$$\begin{aligned} h(z)=t\frac{1+J\left( z\right) }{1-J\left( z\right) }+\left( 1-t\right) \frac{1+J^{2}\left( z\right) }{1-J^{2}\left( z\right) },z\in {\mathcal {G}} \end{aligned}$$

belongs to the family \({\mathcal {C}}_{{\mathcal {G}}}\) and (for\(\ t=\frac{2}{3\lambda }-1\)) there holds the equality

$$\begin{aligned} g(z)h(z)={\mathcal {L}}f_{2}\left( z\right) ,z\in {\mathcal {G}}. \end{aligned}$$

Observe that the power series expansions of the functions \(\frac{1}{\left( 1-\zeta \right) ^{2}},\) \(\log (1+\zeta ),\log (1-\zeta )\) in the disc U,  imply the expression

$$\begin{aligned} f_{2}\left( z\right) =1+J\left( z\right) \frac{2}{3\lambda }+J^{2}\left( z\right) \left( \frac{4}{3}\left( \frac{2}{3\lambda }-1\right) +\frac{5}{3}\right) +...,z\in {\mathcal {G}}. \end{aligned}$$

Thus

$$\begin{aligned} Q_{f_{2},1}=\frac{2}{3\lambda }J,Q_{f_{2},2}=\left( \frac{8}{9\lambda }+\frac{1}{3}\right) J^{2}. \end{aligned}$$

Therefore, we have step by step

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}\left( Q_{f_{2},2}-\lambda \left( Q_{f_{2},1}\right) ^{2}\right)&=\mu _{_{{\mathcal {G}}}}\left( \left( \frac{8}{9\lambda } +\frac{1}{3}\right) J^{2}-\lambda \frac{4}{9\lambda }J^{2}\right) =\left| \frac{4}{9\lambda }+\frac{1}{3}\right| \mu _{_{{\mathcal {G}}}}\left( J^{2}\right) \\&=\frac{1}{3}+\frac{4}{9\lambda }=M\left( \lambda ,2\right) . \end{aligned}$$

Now, we show that in the case \(\lambda \in \Lambda _{3}\) the equality in (2.3) is realized by the function

$$\begin{aligned} f_{3}(z)=\frac{1}{1-J^{2}(z)},z\in {\mathcal {G}}. \end{aligned}$$
(2.10)

Of course, \(f_{3}\) is holomorphic (compare holomorphicity of \(f_{1}).\) Moreover, \(f_{3}\in {\mathcal {M}}_{{\mathcal {G}}}\) (see [3]) and consequently, from [2] the function (2.10) belongs to \({\mathcal {R}}_{{\mathcal {G}}}.\) By expanding the function \(f_{3}\) into a series of m-homogeneous polynomials, we obtain that \(Q_{f_{3},1}=0,\) and \(Q_{f_{3},2}=J^{2}.\) From this, by the case condition for \(\lambda ,\) we have:

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}\left( Q_{f_{3},2}-\lambda \left( Q_{f_{3},1}\right) ^{2}\right) =\mu _{_{{\mathcal {G}}}}\left( J^{2}-\lambda 0\right) =\mu _{_{{\mathcal {G}}}}\left( J^{2}\right) =1=M\left( \lambda ,3\right) . \end{aligned}$$

We finally assume that \(\lambda \in \Lambda _{4}.\) Then the equality in inequality (2.3) is realized by the function

$$\begin{aligned} f_{4}(z)=\frac{1}{\left( 1-iJ(z)\right) ^{2}},z\in {\mathcal {G}}. \end{aligned}$$
(2.11)

Of course, \(f_{4}\) is holomorphic (compare holomorphicity of \(f_{1})\). Now, we confirm that \(f_{4}\in \mathcal {M_{G}}.\) To this aim let us observe that by the linearlity of the functional J,  by equality (2.8), and by the differentiation properties, we have for \(z\in {\mathcal {G}},\)

$$\begin{aligned} {\mathcal {L}}f_{4}(z)=f_{4}(z)+Df_{4}(z)\left( z\right) =\frac{1+iJ(z)}{\left( 1-iJ(z)\right) ^{3}}. \end{aligned}$$

Hence,

$$\begin{aligned} {\mathcal {L}}f_{4}(z)=f_{4}(z)\cdot p\left( z\right) ,z\in {\mathcal {G}}, \end{aligned}$$

with the function

$$\begin{aligned} p\left( z\right) =\frac{1+J(iz)}{1-J(iz)},z\in {\mathcal {G}}. \end{aligned}$$

Since this function belongs to the family \(\mathcal {C_{G}},\) we have that \(f_{4}\in \mathcal {M_{G}\subset R_{G}}\). Moreover,

$$\begin{aligned} Q_{f_{4},2}=-3J^{2},Q_{f_{4},1}=2iJ. \end{aligned}$$

Therefore,

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}\left( Q_{f_{4},2}-\lambda \left( Q_{f_{4},1}\right) ^{2}\right)&=\mu _{_{{\mathcal {G}}}}\left( -3J^{2}-\lambda \left( 2iJ\right) ^{2}\right) \\&=\left| -3+4\lambda \right| \mu _{_{{\mathcal {G}}}}\left( J^{2}\right) =M\left( \lambda ,4\right) . \end{aligned}$$

This completes the proof. \(\square \)

3 A Fekete–Szegö estimate for biholomorphic mappings in \({\mathbb {C}}^{n}\)

Let \({\mathbb {B}}^{n}\) be the open unit Euclidean ball and I be the identity matrix in \({\mathbb {C}}^{n}\). A mapping \(G:{\mathbb {B}}^{n}\rightarrow {\mathbb {C}}^{n},G(0)=0,DG(0)=I,\) is called starlike if G is biholomorphic and \(G\left( {\mathbb {B}}^{n}\right) \) is a starlike domain in \({\mathbb {C}}^{n}.\) The family of such mappings we denote \(S^{*}({\mathbb {B}}^{n})\). In the well known characterization of starlikeness [30] (see also [18, 25]) the collection \(P\left( {\mathbb {B}}^{n}\right) \) of all holomorphic mappings \(H:{\mathbb {B}}^{n}\rightarrow {\mathbb {C}}^{n},H(0)=0,DH(0)=I,\) with \({\text {*}}{Re}\left\langle H\left( z\right) ,z\right\rangle >0,z\in {\mathbb {B}}^{n}\backslash \left\{ 0\right\} ,\) plays the main role (here \(\left\langle \cdot ,\cdot \right\rangle \) means the Euclidean inner product). This mentioned characterization is included in the following theorem:

Theorem A

A locally biholomorphic mapping \(G:{\mathbb {B}}^{n}\rightarrow {\mathbb {C}}^{n}\) normalized by the conditions \(G(0)=0,DG(0)=I,\) belongs to \(S^{*}({\mathbb {B}}^{n}),\) iff the mapping

$$\begin{aligned} H\left( z\right) =\left( DG\left( z\right) \right) ^{-1}G\left( z\right) ,z\in {\mathbb {B}}^{n}, \end{aligned}$$
(3.1)

belongs to \(P\left( {\mathbb {B}}^{n}\right) .\)

According to this theorem, Pfaltzgraff and Suffridge [27] considered the family \(CS^{*}({\mathbb {B}}^{n})\) of close-to-starlike mappings. A locally biholomorphic mapping \(F:{\mathbb {B}}^{n}\rightarrow {\mathbb {C}}^{n},\) normalized by the conditions \(F(0)=0,DF(0)=I,\) will be called close-to-starlike mapping relative to a mapping \(G\in S^{*}({\mathbb {B}} ^{n}),\) if the mapping

$$\begin{aligned} H\left( z\right) =\left( DF\left( z\right) \right) ^{-1}G\left( z\right) ,z\in {\mathbb {B}}^{n}, \end{aligned}$$
(3.2)

belongs to \(P\left( {\mathbb {B}}^{n}\right) .\) From equalities (3.1) and (3.2) it follows that every mapping \(F\in S^{*}({\mathbb {B}}^{n})\) is close-to starlike relative to itself.

Pfaltzgraff and Suffridge proved that every mapping \(F\in CS^{*} ({\mathbb {B}}^{n})\) transforms biholomorphically each ball \(r{\mathbb {B}} ^{n},0<r<1,\) onto domain \(F(r{\mathbb {B}}^{n})\) which complement to \({\mathbb {C}}^{n}\) is the union of nonintersecting rays, hence having similar property as the one-dimensional Kaplan-Lewandowski [17, 22] close-to-convex univalent functions. For a wide collection of references in the area of various biholomorphic mappings in \({\mathbb {C}}^{n},\) see the monographs [13, 20]. In this part of the paper we consider the family \(\widetilde{CS^{*}}({\mathbb {B}}^{n})\) of mappings \(F\in CS^{*} ({\mathbb {B}}^{n})\)

$$\begin{aligned} F(z)=zf(z),z\in {\mathbb {B}}^{n}, \end{aligned}$$
(3.3)

relative to mappings \(G\in S^{*}({\mathbb {B}}^{n})\) of the form

$$\begin{aligned} G(z)=zg(z),z\in {\mathbb {B}}^{n}, \end{aligned}$$
(3.4)

where f,g \(\in \) H\(_{{B}^{n}}(1),f(z)g(z)\ne 0,z\in {\mathbb {B}}^{n}.\)

For a characterization of starlike mappings G of the form (3.4), see, e.g., [7, 13, 24, 28, 33]. For instance in the paper [28] we can found the following result:

Theorem B

A mapping \(G:{\mathbb {B}}^{n}\rightarrow {\mathbb {C}}^{n}\) of the form ( 3.3), with the normalization \(G(0)=0,DG(0)=I,\) is biholomorphic starlike mapping, iff the function\(\ g\) belongs to the Bavrin’s family \({\mathcal {M}}_{{\mathbb {B}}^{n}}.\)

Now, we present a similar characterization of the subfamily \(\widetilde{CS^{*}}({\mathbb {B}}^{n}).\)

Lemma 1

A mapping F of the form (3.3) is a close-to-starlike mapping relative to starlike mapping G of the form (3.4) , iff \(f\in {\mathcal {R}}_{{\mathbb {B}}^{n}}\) with \(g\in {\mathcal {M}}_{{\mathbb {B}} ^{n}}.\)

Proof

At first observe that using a generalization of the differential product rule for F of the form (3.3) we have

$$\begin{aligned} DF(z)=f(z)\left[ I+\frac{1}{f(z)}zDf(z)\right] ,z\in {\mathbb {B}}^{n}. \end{aligned}$$

Thus, by the fact that the columns of the square matrix \(\frac{zDf(z)}{f(z)}\) are proportional for \(z\in {\mathbb {B}}^{n}\), we get ( [28])

$$\begin{aligned} \det DF(z)=(f(z))^{n}\left[ 1+Trace\left( \frac{1}{f(z)}zDf(z)\right) \right] ,z\in {\mathbb {B}}^{n}. \end{aligned}$$

Hence, ( [5])

$$\begin{aligned} \det DF(z)=(f(z))^{n-1}{\mathcal {L}}f(z),z\in {\mathbb {B}}^{n}, \end{aligned}$$

and F is locally biholomorphic, iff \(f(z){\mathcal {L}}f(z)\ne 0\), \(z\in {\mathbb {B}}^{n}\) (of course, if \(f\in {\mathcal {R}}_{{\mathbb {B}}^{n}},\) then \(f(z){\mathcal {L}}f(z)\ne 0\) at each \(z\in {\mathbb {B}}^{n},\) see [1]).

Moreover,

$$\begin{aligned} \left( DF\left( z\right) \right) ^{-1}=\frac{1}{f(z)}\left[ I-\frac{1}{{\mathcal {L}}f(z)}zDf(z)\right] ,z\in {\mathbb {B}}^{n}. \end{aligned}$$

Consequently, (3.2) is equivalent to

$$\begin{aligned} H(z)=zh(z),z\in {\mathbb {B}}^{n}, \end{aligned}$$

where

$$\begin{aligned} h(z)=\frac{g(z)}{{\mathcal {L}}F(z)},z\in {\mathbb {B}}^{n}. \end{aligned}$$

This follows from the following equalities for \(z\in {\mathbb {B}}^{n},\)

$$\begin{aligned} \left( DF\left( z\right) \right) ^{-1}G\left( z\right)&=\frac{1}{f(z)}\left[ I-\frac{zDf(z)}{{\mathcal {L}}f(z)}\right] zg(z)=\frac{1}{f(z)}\left[ z-\frac{zDf(z)z}{{\mathcal {L}}f(z)}\right] g(z)\\&=z\frac{1}{f(z)}\left[ 1-\frac{Df(z)z}{{\mathcal {L}}f(z)}\right] g(z)=z\frac{g(z)}{{\mathcal {L}}f(z)}. \end{aligned}$$

Using the above facts, the relation

$$\begin{aligned} {\text {Re}}\left\langle H(z),z\right\rangle>0,z\in {\mathbb {B}} ^{n}\backslash \left\{ 0\right\} \Longleftrightarrow {\text {Re}} \frac{g(z)}{{\mathcal {L}}F(z)}>0,z\in {\mathbb {B}}^{n} \end{aligned}$$

and the relation

$$\begin{aligned} {\text {Re}}\left( \frac{1}{w}\right)>0\Leftrightarrow {\text {Re}}w>0,w\in {\mathbb {C}}\backslash \left\{ 0\right\} , \end{aligned}$$

we get that \(H\in P\left( {\mathbb {B}}^{n}\right) ,\) iff \(h\in {\mathcal {C}}_{{\mathbb {B}}^{n}}.\) This gives the thesis of the lemma. \(\square \)

The main result of this section is presented in the following theorem.

Theorem 3.1

For maps \(F\in \widetilde{CS^{*}}({\mathbb {B}}^{n}),\ \)all points \(z\in {\mathbb {B}}^{n}\backslash \left\{ 0\right\} \) and parameters \(\lambda \in \Lambda _{j},j=1,...,4,\) there hold the following estimates

$$\begin{aligned} \left| \frac{T_{z}\left( D^{3}F(0)(z^{3}\right) }{3!\left\| z\right\| ^{3}}-\lambda \left( \frac{T_{z}\left( D^{2}F(0)(z^{2}\right) }{2!\left\| z\right\| ^{2}}\right) ^{2}\right| \le M\left( \lambda ,j\right) , \end{aligned}$$
(3.5)

where \(T_{z}:{\mathbb {C}}^{n}\rightarrow {\mathbb {C}}\) is the linear functional fulfilling the conditions \(\left\| T_{z}\right\| =1,T_{z}\left( z\right) =\left\| z\right\| \) and \(M\left( \lambda ,j\right) ,j=1,...,4,\) is defined by (2.2).

Proof

First let us observe that for every mapping \(F\in \) \(\widetilde{CS^{*} }({\mathbb {B}}^{n})\) and the associated, by (3.3), function \(f\in {\mathcal {R}}_{{\mathbb {B}}^{n}},\) we obtain the following equalities

$$\begin{aligned} \frac{D^{3}F(0)(z^{3})}{3!}=z\frac{D^{2}f(0)(z^{2})}{2!}, \frac{D^{2}F(0)(z^{2})}{2!}=zDf(0)(z),z\in {\mathbb {B}}^{n}. \end{aligned}$$
(3.6)

These patterns were earlier used in a few papers of Xu et al. (see for instance [33]) for pairs (F, f) of associated elements F, f of another families.

From this and estimate (2.3) in the family \({\mathcal {R}} _{{\mathbb {B}}^{n}},\) by properties of the Minkowski balance of homogeneous polynomials, we get step by step for \(\lambda \in \Lambda _{j}, j\in \left\{ 1,2,3,4\right\} ,\)

$$\begin{aligned}&\left| \frac{T_{z}\left( D^{3}F(0)(z^{3}\right) }{3!\left\| z\right\| ^{3}}-\lambda \left( \frac{T_{z}\left( D^{2}F(0)(z^{2}\right) }{2!\left\| z\right\| ^{2}}\right) ^{2}\right| \\&\quad =\left| \frac{T_{z}\left( z\right) D^{2}f(0)(z^{2})}{2\left\| z\right\| ^{3}}-\lambda \left( \frac{T_{z}\left( z\right) Df(0)(z)}{\left\| z\right\| ^{2}}\right) ^{2}\right| \\&\quad =\left| \frac{D^{2}f(0)(z^{2})}{2\left\| z\right\| ^{2}} -\lambda \left( \frac{Df(0)(z)}{\left\| z\right\| }\right) ^{2}\right| =\left| \frac{Q_{f,2}(z)}{\left\| z\right\| ^{2} }-\lambda \left( \frac{Q_{f,1}(z)}{\left\| z\right\| }\right) ^{2}\right| \\&\quad \le \sup _{z\in {\mathbb {B}}^{n}}\frac{\left| Q_{f,2}(z)-\lambda \left( Q_{f,1}(z)\right) ^{2}\right| }{\left\| z\right\| ^{2}} =\mu _{{\mathbb {B}}^{n}}\left( Q_{f,2}-\lambda \left( Q_{f,1}\right) ^{2}\right) \le M\left( \lambda ,j\right) . \end{aligned}$$

It remains to show the sharpness of estimate (3.5). To do it, observe first that by the maximum principle for the modulus of holomorphic functions of several complex variables, there exists \(a\in {\mathbb {B}}^{n}\backslash \left\{ 0\right\} \) such that \(|J(\frac{a}{\left\| a\right\| })|=1,\) because \(\left\| J\right\| =\mu _{{\mathbb {B}}^{n}}(J)=1.\)

Now, we will show that the equality in (3.5) in such points a are attained by the mappings

$$\begin{aligned} F_{j}\left( z\right) =zf_{j}\left( z\right) ,z\in {\mathbb {B}}^{n}, \end{aligned}$$

where \(f_{j}, j\in \left\{ 1,2,3,4\right\} ,\) are defined in \({\mathcal {G}} ={\mathbb {B}}^{n}\) by formulas (2.8), (2.9), (2.10), (2.11), in the j-th case \(\lambda \in \Lambda _{j},j=1,...,4,\) respectively. To this purpose, using (3.6) and the forms of the homogeneous polynomials \(Q_{f_{j},1},Q_{f_{j},2}, j\in \left\{ 1,2,3,4\right\} \) (see the Section 2 of the paper), we get for \(j\in \left\{ 1,2,3,4\right\} \)

$$\begin{aligned}&\left| \frac{T_{a}\left( D^{3}F_{j}(0)(a^{3}\right) }{3!\left\| a\right\| ^{3}}-\lambda \left( \frac{T_{a}\left( D^{2}F_{j}(0)(a^{2} \right) }{2!\left\| a\right\| ^{2}}\right) ^{2}\right| \\&\qquad =\left| \frac{T_{a}\left( a\right) D^{2}f_{j}(0)(a^{2})}{2\left\| a\right\| ^{3}}-\lambda \left( \frac{T_{a}\left( a\right) Df_{j} (0)(a)}{\left\| a\right\| ^{2}}\right) ^{2}\right| \\&\qquad =\left| Q_{f_{j},2}\left( \frac{a}{\left\| a\right\| }\right) -\lambda \left( Q_{f_{j},1}\left( \frac{a}{\left\| a\right\| }\right) \right) ^{2}\right| =M\left( \lambda ,j\right) \left| J^{2}\left( \frac{a}{\left\| a\right\| }\right) \right| =M\left( \lambda ,j\right) . \end{aligned}$$

Therefore, the proof is complete. \(\square \)