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Hadamard product of certain classes of functions

100%

Piejko K.

tom Vol. 11, nr 1

145-151

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, {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).

Hadamard product of functions subordinated to homography

75%

Piejko K. , Sokół J. , Stankiewicz J.

tom z. 27 [212]

49-59

We consider convolution properties of regular functions using the concept of subordination. Let P(X, Y) denote the class of regular functions subordinated to the homography 1+Xz/1-Yz. It is known [10] that for some complex numbers A,B,C,D if is an element of P(A,B) and g is an element of P(C,D), then there exist X and Y such that f *p is an element of P(X,Y). In this paper we verify the reverse question: if for each h is an element of P(X, Y) it is possible to find suitable f is an element of P(A, B) and g is an element of P(C7, D) such that h=f*g.