HADAMARD PRODUCT OF CERTAIN CLASSES OF FUNCTIONS

Abstract. In this paper we consider the Hadamard product? of regular functions using the concept of subordination. Let P (A,B) denote the class of regular functions subordinated to the linear fractional transfor-mation (1 +Az)/(1−Bz), where A+B 6 = 0 and |B | ≤ 1. By P (A,B)? P (C,D) we denote the set {f? g: f ∈ P (A,B), g ∈ P (C,D)}. It is known ([3], [7]), that for some complex numbers A,B,C,D there exist X and Y such that P (A,B) ? P (C,D) ⊂ P (X,Y). The purpose of this note is to find the necessary and sufficient conditions for the equality of the classes P (A,B) ? P (C,D) and P (X,Y). 1