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LetSγ,A,B∗(D){S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}})be the usual class ofgg-starlike functions of complex orderγ\gammain the unit diskD={ζ∈C:∣ζ∣&lt;1}{\mathbb{D}}=\left\{\zeta \in {\mathbb{C}}:| \zeta | \lt 1\right\}, whereg(ζ)=(1+Aζ)∕(1+Bζ)g\left(\zeta )=\left(1+A\zeta )/\left(1+B\zeta ), withγ∈C\{0},−1≤A&lt;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 functionsf∈Sγ,A,B∗(D)f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}})whenζ=0\zeta =0is a zero of orderk+1k+1off(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.

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