Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂ ⁿ

Abstract

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.

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

“…Therefore, the study of the Reinhardt domain allows us to discover a deep interplay between them. An increasing number of papers on various types of Reinhardt domains [6], on the Schwarz lemma [7], on the rigidity theorem [8], on Bohr radii [9,10], on Bergman kernels [11], and on the bounds of all the coefficients of homogeneous expansions [12] for the domain show the importance of this topic. A few recent papers [13,14] initiated an intensive study of functions that are analytical in a complete Reinhardt domain by methods of the Wiman-Valiron theory.…”