Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions

Abstract

Since the nonlinear integral transforms $J_{\alpha }[f](z)={\int }_0^z(f^{\prime }(u){)}^{\alpha }du$ and $I_{\alpha }[f](z)={\int }_0^z(f(u)/u{)}^{\alpha }du$ with a complex number $\alpha$ 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 $J_{\alpha }[f]$ or $I_{\alpha }[f]$ 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 $J_{\alpha }[f]$ and $I_{\alpha }[f]$ 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.

arXiv comment12 pages