Integral transforms of functions with restricted derivatives

Johnny E. Brown¹

¹Department of Mathematics Purdue University West Lafayette, IN 47907, U.S.A.

Annales Polonici Mathematici, Tome 91 (2007) no. 1, pp. 85-97

Résumé

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.

Détail

DOI : 10.4064/ap91-1-7

Keywords: functions whose derivatives lie half plane preserved under pommerenke chandra singh libera alexander bernardi integral transforms determine precisely these transforms act functions prove derivative function lies convex region derivative its pommerenke chandra singh libera alexander bernardi transforms lie strictly smaller convex region which determined consider iterates these transforms best possible results obtained