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.

MSC:

 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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References:

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