Integral transforms of functions with restricted derivatives
Volume 91 / 2007
Annales Polonici Mathematici 91 (2007), 85-97
MSC: Primary 30C45; Secondary 30C75.
DOI: 10.4064/ap91-1-7
Abstract
We show that functions whose derivatives lie in a half-plane are preserved under the Pommerenke, Chandra–Singh, Libera, Alexander and Bernardi integral transforms. We determine precisely how these transforms act on such functions. We prove that if the derivative of a function lies in a convex region then the derivative of its Pommerenke, Chandra–Singh, Libera, Alexander and Bernardi transforms lie in a strictly smaller convex region which can be determined. We also consider iterates of these transforms. Best possible results are obtained.
