Rocky Mountain Journal of Mathematics

Quasiconformal extendability of integral transforms of Noshiro-Warschawski functions

Ikkei Hotta and Li-Mei Wang

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Since the nonlinear integral transforms $$ J_{\alpha }[f](z) = \int _{0}^{z}(f'(u))^{\alpha } du $$ and $$ \ \ I_{\alpha }[f](z) =\int _0^z (f(u)/u)^{\alpha } du $$ with a complex number $\alpha $ were introduced, a great number of studies have been dedicated to deriving sufficient conditions for univalence on the unit disk. However, little is known about the conditions where $J_{\alpha }[f]$ or $I_{\alpha }[f]$ produce a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper, we discuss quasiconformal extendability of the integral transforms $J_{\alpha }[f]$ and $I_{\alpha }[f]$ for holomorphic functions which satisfy the Noshiro-Warschawski criterion. Various approaches using pre-Schwarzian derivatives, differential subordination and Loewner theory are applied to this problem.

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Rocky Mountain J. Math., Volume 47, Number 1 (2017), 185-204.

First available in Project Euclid: 3 March 2017

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Zentralblatt MATH identifier

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C62: Quasiconformal mappings in the plane 44A15: Special transforms (Legendre, Hilbert, etc.)

Univalent function integral transform Noshiro-Warschawski theorem quasiconformal extension pre-Schwarzian derivative differential subordination


Hotta, Ikkei; Wang, Li-Mei. Quasiconformal extendability of integral transforms of Noshiro-Warschawski functions. Rocky Mountain J. Math. 47 (2017), no. 1, 185--204. doi:10.1216/RMJ-2017-47-1-185.

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