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arXiv · 1401.5647

Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions

Ikkei Hotta·Li-Mei Wang
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Abstract

Since the nonlinear integral transforms $J_α[f](z) = \int_{0}^{z}(f'(u))^α du$ and $I_α[f](z) =\int_0^z (f(u)/u)^α du$ with a complex number $α$ have been introduced, a great number of studies were dedicated to deriving sufficient conditions for univalence on the unit disk. On the other hand, little is known about the conditions that $J_α[f]$ or $I_α[f]$ produces a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper we discuss quasiconformal extendibility of the integral transforms $J_α[f]$ and $I_α[f]$ for holomorphic functions which satisfy the Noshiro-Warschawski criterion. Various approaches using pre-Schwarzian derivatives, differential subordinations and Loewner theory are taken to this problem.

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Ikkei Hotta, Li-Mei Wang. 2015-05-29. Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions. https://arxiv.org/abs/1401.5647

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