Quasiconformal extendability of integral transforms of Noshiro-Warschawski functions

Abstract

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.