On neighborhoods of analytic functions with positive real part

摘要

Let h(z) = z + a₂z2 + be analytic in the unit disc on the complex plane . For given κ 1, , ζ κ, we prove that if ζh′(z) ∈ Ω(κ, ζ) for all , then [ζh(z)/z] κ for all , where is a concave domain with boundary being a parabola. Next we consider the classes of analytic functions and . For we find a sufficient condition on δ that implies the existence of a δ -neighborhood being contained in , where β ⩽ 1. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim