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

Sima Xiaoying, Tu Zhenhan, Xiong Liangpeng

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220242

Let S*γ,A,B(D) be the usual class of g -starlike functions of complex order γ in the unit disk D={ζ∈C:∣ζ∣<1} , where g(ζ)=(1+Aζ)∕(1+Bζ) , with γ∈C\{0},−1 ≤A < B ≤1,ζ∈D. First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f∈S*γ,A,B(D) when ζ=0 is a zero of order k+1 of f(ζ)−ζ . 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.