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

Abstract

Let Sγ,A,B∗(D){S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) be the usual class of gg-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+Aζ)∕(1+Bζ)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∈Sγ,A,B∗(D)f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) when ζ=0\zeta =0 is a zero of order k+1k+1 of f(ζ)−ζf\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.

Keywords

• biholomorphic mappings
• coefficient problems
• reinhardt domain
• g-starlike mappings of complex order gamma
• 32h02
• 30c45
• 26b10
• 32a30