On P-Stable Functions

Abstract

Let h_1, h_2 be two analytic functions defined in the open unit disc Δ:={z∈C:|z|<1} which are normalized by the condition h_1 (0)=1=h_2 (0). Then h_1 is P-stable with respect to h_2, whenever 
(P_n (h_1,z))/(h_1 (z))≺1/(h_2 (z))       (z∈Δ),
holds for all n∈N. Here ’≺’ stands for subordination and P_n (h,z)=P_n (z)*h(z) where P_n (z) denote the n-degree polynomial induced by the (n+1)th row entities in an admissible lower triangular matrix. The main purpose of this article is to prove that the function ((Az+1)/(Bz+1))^δ is P-stable with respect to (Bz+1)^(-δ), for δ∈(0,1] and -1≤B<A≤0 but not P-stable with respect to itself, when -1≤B<A<0 and δ∈(0,1]. As an application, considered different admissible lower triangular matrices to derive various results related on stability.