Coefficient estimates for inverses of starlike functions of positive order. (English) Zbl 1153.30301

Summary: 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 J. G. Krzyz, R. J. Libera and E. Zlotkiewicz [“Coefficients of inverse of regular starlike functions”, Ann. Univ. Mariae Curie-Sklodowska Sect. A 33, No. 10, 103–109 (1979)] 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 \(\Sigma ^{* - 1}(\alpha )\), consisting of inverses of functions in the class \(\Sigma ^{*}(\alpha )\) 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 \(\Sigma ^{*}(\alpha )\) stands completely settled in the present work by our method developed here.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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