Integral representations of bounded starlike functions

Frode Rønning
1995 Annales Polonici Mathematici  
For α ≥ 0 let Fα denote the class of functions defined for |z| < 1 by integrating 1/(1 − xz) 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 Fα for α > 0. 1991
more » ... > 0. 1991 Mathematics Subject Classification: 30C45, 30E20.
doi:10.4064/ap-60-3-289-297 fatcat:kqkdk7ykxvgl7fvl7p424g6vqe