"Some Results of Differential Subordination and Superordination for Univalent Functions

"Let 1 and 2 belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions to satisfy the double subordination chain 1() ≺() ≺ 2() , then we obtain 1() is the best subordinant, 2() is the best dominant. Also we derive some sandwich –type result.

Convex Subordination Chains in Several Complex Variables

In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ℂn. We also obtain a sufficient condition for injectivity off(z/‖z‖, ‖z‖) onBn\ ﹛0﹜, wheref(z,t) is a convex subordination chain over (0, 1).

An application of a subordination chain

LetKdenote the class of functionsg(z)=z+a2z2+⋯which are regular and univalently convex in the unit discE. In the present note, we prove that iffis regular inE,f(0)=0, then forg∈K,f(z)+αzf′(z) ≺ g(z)+αzg′(z)inEimplies thatf(z)≺g(z)inE, whereα>0is a real number and the symbol “≺” stands for subordination.