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Integral representations of bounded starlike functions

Frode Rønning (1995)

Annales Polonici Mathematici

For α ≥ 0 let ℱ α denote the class of functions defined for |z| < 1 by integrating 1 / ( 1 - x z ) α if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to ℱ 0 . In general, it is only known that bounded starlike functions belong to ℱ α for α > 0.

Integral transforms of functions with restricted derivatives

Johnny E. Brown (2007)

Annales Polonici Mathematici

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....

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