Coefficients estimate for a subclass of holomorphic mappings on the unit polydisk in Cn
Abstract: The aim of this paper is to obtain the sharp solutions of Fekete-Szeg? problems of high dimensional version for family of holomorphic mappings that are normalized on the unit polydisk Un in Cn. The main results unify some recent works, which are closely related to the starlike mappings. Moreover, some previous results are improved.
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Let S γ , A , B ∗ ( D ) {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) be the usual class of g g -starlike functions of complex order γ \gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {\mathbb{D}}=\left\{\zeta \in {\mathbb{C}}:| \zeta | \lt 1\right\} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) g\left(\zeta )=\left(1+A\zeta )/\left(1+B\zeta ) , with γ ∈ C \ { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D \gamma \left\in {\mathbb{C}}\backslash \left\{0\right\}\right,-1\le A\lt B\le 1,\zeta \in {\mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) when ζ = 0 \zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ f\left(\zeta )-\zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.
Let S γ , A , B ∗ ( D ) {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) be the usual class of g g -starlike functions of complex order γ \gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {\mathbb{D}}=\left\{\zeta \in {\mathbb{C}}:| \zeta | \lt 1\right\} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) g\left(\zeta )=\left(1+A\zeta )/\left(1+B\zeta ) , with γ ∈ C \ { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D \gamma \left\in {\mathbb{C}}\backslash \left\{0\right\}\right,-1\le A\lt B\le 1,\zeta \in {\mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) when ζ = 0 \zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ f\left(\zeta )-\zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.
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