Abstract
Let h(z) = z + a~2~z^2^ + ⋅⋅⋅ be analytic in the unit disc \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\cal U}$\end{document} on the complex plane \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {C}$\end{document}. For given κ < 1, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\zeta \in \mathbf {C}$\end{document}, ℜζ > κ, we prove that if ζ__h__′(z) ∈ Ω(κ, ζ) for all \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$z\in {\cal U}$\end{document}, then ℜ[ζ__h__(z)/z] > κ for all \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$z\in {\cal U}$\end{document}, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega (\kappa ,\zeta )=\lbrace w\in {\mathbf {C}}:|w-2\kappa +\overline{\zeta }|>|\Re w -\kappa |\rbrace$\end{document} is a concave domain with boundary being a parabola. Next we consider the classes of analytic functions \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${{\cal P}(\alpha )}=\lbrace p:p(0)=1,\Re [p(z)\alpha \rbrace$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\cal {P^{\prime }}}(\alpha )=\left\lbrace p:p(0)=1,\ [zp(z)]^{\prime }\in \Omega (\alpha ,1)\right\rbrace$\end{document}. For \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$q\in \cal {P^{\prime }}(\alpha )$\end{document} we find a sufficient condition on δ that implies the existence of a δ ‐neighborhood \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{N}_\delta (q)$\end{document} being contained in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\cal P}(\beta )$\end{document}, where β ⩽ α < 1. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim