Generalization of a class of nonlinear averaging integral operators

✍ Scribed by Teodor Bulboacă

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 145 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310224

✦ Synopsis

Abstract

Let H(U) be the space of all analytic functions in the unit disk U, and let co__E__ denote the convex hull of the set E ⊂ ℂ. If K ⊂ H(U) then the operator I : K → H(U) is said to be an averaging operator if

For a function h ∈ A ⊂ H(U) we will determine simple sufficient conditions on h such that

for all f ∈ ℳ︁^′^~1/β~, where

and ℳ︁^′^~1/β~ represents the class of 1/β‐convex functions (not necessarily normalized).

As an application, we will give sufficient conditions on h to insure that the operators I~h;β,γ~ are averaging operators on certain subsets of H(U), in order to generalize the result of [5]. In addition, some particular cases of this result obtained for appropriate choices of the function h will also be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)