Coefficient Estimates for Inverses of Starlike Functions of Positive Order

Abstract

In the present paper, the coefficient estimates are found for the class $\mathcal S^{*-1}(\alpha)$ 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 $\mathcal S^{*-1}(\alpha)$. The coefficient estimates are also found for the class $\sum^{*-1}(\alpha)$, consisting of inverses of functions in the class $\sum^*(\alpha)$ 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 $\sum^*(\alpha)$ stands completely settled in the present work by our method developed here.