Completely monotone sequences and universally prestarlike functions

Abstract

We introduce universally convex, starlike and prestarlike functions in the slit domain C \ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions.

A note on generating functions for Hausdorff moment sequences

“…Wirths [5] has shown that f ∈ T implies that the function zf (z) is univalent in the half-plane Re z < 1, and recently the theory of universally prestarlike mappings has been developed, showing a close link to T ; see [4]. Many classical functions belong to T or are closely related to it.…”

“…Fundamental for the proof of Theorem 1.5 is the following result, which is based on a general theorem in [4].…”

On Completely Monotonic and Related Functions

“…Universally starlike functions can also be defined and characterized in an analogous way by means of the class T , see [66] for details.…”

Universally prestarlike functions as convolution multipliers

“…In [3] the notion of prestarlike functions has been extended from the unit disk to other disks and half-planes containing the origin. Let be such a disk or half-plane.…”

“…For some general information concerning universally prestarlike functions, in particular about universally convex functions ( f ∈ R u 0 ) and universally starlike functions ( f ∈ R u 1/2 ), see [3].…”