Integral transforms of functions with restricted derivatives

Volume 91 / 2007

Johnny E. Brown 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.

Authors

  • Johnny E. BrownDepartment of Mathematics
    Purdue University
    West Lafayette, IN 47907, U.S.A.
    e-mail

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