Subordination by univalent functions

Abstract

Let K K be the class of functions f ( z ) = z + a 2 z 2 + ⋯ f(z) = z + {a_2}{z^2} + \cdots , which are regular and univalently convex in | z | > 1 \left | z \right | > 1 . In this paper we establish certain subordination relations between an arbitrary member f f of K K , its partial sums and the functions ( λ / z ) ∫ 0 z f ( t ) d t (\lambda /z)\int _0^z {f(t)dt} and μ ∫ 0 z t − 1 f ( t ) d t \mu \int _0^z {{t^{ - 1}}f(t)dt} . The well-known result that z / 2 z/2 is subordinate to f ( z ) f(z) in | z | > 1 \left | z \right | > 1 for every f f belonging to K K follows as a particular case from our results. We also improve certain results of Robinson regarding subordination by univalent functions. A sufficient condition for a univalent function to be convex of order α \alpha is also given.