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A note on the quasicontinuity

Grande Z.

Demonstratio Mathematica

2006

Vol. 39, nr 3

515-518

Let (X,px) and (V, pv) be metric spaces. A function f: X - Y is said graph quasicontinuous if there is a quasicontinuous function g : X -> Y with the graph Gr(g) contained in the closure cl(Gr(f)) of Gr(f). If the space (V, pv) is compact and if there is a dense subset A C X such that the restricted function f/A is continuous then f is graph quasicontinuous. Moreover each locally bounded function f: R -> R is graph quasicontinuous.

Hadamard product of certain classes of functions

Piejko K.

Journal of Applied Analysis

2005

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

Piejko Krzysztof, Sokół Janusz, Stankiewicz Jan

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2004

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.

Convolution properties of a class of analytic functions

Piejko K., Sokół J.

Demonstratio Mathematica

2004

Vol. 37, nr 1

63--69

In this paper we consider convolution properties of a class of bounded analytic functions investigated by J. Stankiewicz and Z. Stankiewicz in [6]. We give some examples which verify a conjecture connected with this paper.