Coefficient Estimates for Inverses of Starlike Functions of Positive Order

Abstract

In the present paper, the coefficient estimates are found for the class ${\mathcal{S}}^{\ast -1}(\alpha )$ consisting of inverses of functions in the class of univalent starlike functions of order $\alpha$ in $\mathcal{D}=\{z\in \mathbb{C}:|z|<1\}$. These estimates extend the work of {\it Krzyz, Libera and Zlotkiewicz [Ann. Univ. Marie Curie-Sklodowska, 33(1979), 103-109]} who found sharp estimates on only first two coefficients for the functions in the class ${\mathcal{S}}^{\ast -1}(\alpha )$. The coefficient estimates are also found for the class ${\sum }^{\ast -1}(\alpha )$, consisting of inverses of functions in the class ${\sum }^{\ast }(\alpha )$ of univalent starlike functions of order $\alpha$ in $\mathcal{V}=\{z\in \mathbb{C}:1<|z|<\infty \}$. The open problem of finding sharp coefficient estimates for functions in the class ${\sum }^{\ast }(\alpha )$ stands completely settled in the present work by our method developed here.

12 pages