Stability of Admissible Functions

Rabha W Ibrahim

doi:10.1155/2011/342895

Abstract

By using the concept of the weak subordination, we examine the stability (a class of analytic functions in the unit disk is said to be stable if it is closed under weak subordination) for a class of admissible functions in complex Banach spaces. The stability of analytic functions in the following classes is discussed: Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class (Qp), and Hilbert Hardy class (H2).

Highlights

We denote by U the unit disk {z ∈ C : |z| < 1} and by H U the space of all analytic functions in U
A class C of analytic functions in X is said to be stable if it is closed under weak subordination, that is, if f ∈ C whenever f and g are analytic functions in X with g ∈ C and f≺wg
By making use of the above concept of the weak subordination, we examine the stability for a class of admissible functions in complex Banach spaces G X, Y

Summary

Introduction

We denote by U the unit disk {z ∈ C : |z| < 1} and by H U the space of all analytic functions in U. A function I, analytic in U, is said to be an inner function if and only if |I z | ≤ 1 such that |I eiθ | 1 almost everywhere. If f and g are analytic functions with f, g ∈ G X, Y , f is said to be weakly subordinate to g, written as f≺wg if there exist analytic functions φ, ω : U → X, with φ an inner function φ X ≤ 1 , so that f ◦ φ g ◦ ω. A class C of analytic functions in X is said to be stable if it is closed under weak subordination, that is, if f ∈ C whenever f and g are analytic functions in X with g ∈ C and f≺wg. By making use of the above concept of the weak subordination, we examine the stability for a class of admissible functions in complex Banach spaces G X, Y. The stability of analytic functions appears in Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class Qp , and Hilbert Hardy class H2

Stability of Bloch Classes

Stability of the Hilbert Hardy Space

Stability of Qp Class

Conclusion