Geometric Funtion Theory, Completely Monotone Sequences and Applications in Speciale Functions

Specialeforsvar Dimitrios Askitis

Titel: Geometric Function Theory, Completely Monotone Sequences and Applications in Special Functions

Abstract: In the present thesis we elaborate the theory of universally prestarlike functions. We study the generating functions of normalised completely monotone sequences and we obtain a suitable characterization of them. Then, using Pick functions and the Hadamard convolution as our main tools, we are studying the geometric properties (convexity and starlikeness of the image sets of discs or half planes) of functions holomorphic in the slit plane $\mathbb{C}\setminus [1,\infty)$ and their connections with completely monotone sequences and their generating functions. We characterize those functions that map discs and half planes which contain $0$ in their interior to convex or starlike (with respect to the origin) sets as the universally prestarlike functions of specific orders. After we find suitable integral representations for these functions, we find necessary or sufficient conditions on the measures in them to define functions with specific geometric properties.In the end, we apply the results in Gaussian hypergeometric functions and polylogarithms.

Vejleder:  Henrik Laurberg Pedersen
Censor:    Jacob Stordal Christensen, Lunds Universitet