Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions

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Since the nonlinear integral transforms Jα[f](z) = ∫₀ᶻ(f'(u))α du and Iα[f](z) =∫₀ᶻ (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 Jα[f] or Iα[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 Jα[f] and Iα[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.