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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 transformation (1 + Az)/(1 – Bz), where A + B ≠ 0 and |B| ≤ 1. By P(A, B) ∗ P(C, D) we denote the set {ƒ ∗ g : ƒ ∈ P(A, B), g ∈ P(C, D)}. It is known ([London, Math. Japon. 43: 23–29, 1996], [Stankiewicz, Stankiewicz, Folia Sci. Univ. Tech. Resov. Math. 7: 93-101, 1988]), 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 ).