Involvement of Hypergeometric Functions in The Theory of Harmonic Functions

Harmonic univalent mappings have attracted the serious attention of complex analysts only after the appearance of a basic paper by Clunie and Sheil-Small [4] in 1984. These researchers laid the foundation for the study of harmonic univalent mappings over the unit disk as a generalization of analytic univalent functions. Interestingly, almost at the same time, the famous Bieberbach conjecture which was posed in 1916 by L. Bieberbach [3] was settled by Louiz de Branges [5] in 1985. Surprisingly, the use of hypergeometric functions in the proof of Bieberbach conjecture has been motivated several researchers to make renewed interest in applying the hypergeometric functions and its related other special functions in the Geometric Function Theory including the theory of Harmonic Mappings. This thesis deals with some classes of univalent, multivalent and p-harmonic mappings by involving the Wright generalized hypergeometric (Wgh) functions and in particular the generalized hypergeometric functions which are analytic in the unit disk U: A family of all harmonic complex-valued, orientation-preserving univalent functions f = u+ iv de…ned in the unit disk U normalized with the condition f(0) = 0 = fz(0) 1 is denoted by SH : Thus, a function f = u+ iv 2 SH admits the representation f = h+ g; where h(z) and g(z) are of the form