Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions

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 $\alpha$ 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.