Coefficient estimates for inverses of starlike functions of positive order

In the present paper, the coefficient estimates are found for the class S∗−1(α) consisting of inverses of functions in the class of univalent starlike functions of order α in D = {z ∈ C: |z| < 1}. These estimates extend the work of Krzyz, Libera and Zlotkiewicz [J.G. Krzyz, R.J. Libera, E. Zlotkiewicz, Coefficients of inverse of regular starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 33 (10) (1979) 103– 109] who found sharp estimates on only first two coefficients for the functions in the class S∗−1(α). The coefficient estimates are also found for the class Σ∗−1(α), consisting of inverses of functions in the class Σ∗(α) of univalent starlike functions of order α in V = {z ∈ C: 1 < |z| < ∞}. The open problem of finding sharp coefficient estimates for functions in the class Σ∗(α) stands completely settled in the present work by our method developed here. © 2006 Elsevier Inc. All rights reserved.