PARTIAL SUMS OF GENERALIZED CLASS OF HARMONIC UNIVALENT FUNCTIONS INVOLVING A GAUSSIAN HYPERGEOMETRIC FUNCTION
The purpose of the present paper is to establish some new results giving the sharp bounds of the real parts of ratios of harmonic univalent functions to its sequences of partial sums by involving the Gaussian hypergeometric function. Relevant connections of the results presented here with various known results are briefly indicated. We also mention results which are associated with certain classical orthogonal polynomials (deduced from some of the main results) exhibiting the usefulness of the results presented in this paper.
Some properties of a subclass of harmonic univalent functions defined by the multiplier transformations
The main object of this article is to present a systematic investigation of a new class of harmonic univalent functions SH (n, λ, α) defined by the multiplier transformations. We obtain coefficient bounds, extreme points, distortion theorem and covering result for this class. Further, we give a sufficient condition for a function defined by Srivastava-Owa fractional calculus operator belonging to this class. Apart of these results, many interesting properties on convolution, partial sums and neighborhoods are also obtained. Relevant connections of the results presented herewith various well-known results are briefly indicated.
Planar harmonic convolution operators generated by hypergeometric functions
Let S Ĥ be the family of all planar harmonic, univalent and sense-preserving mappings f=h+g¯ where h and g are analytic funtions in the open unit disk. The purpose of this article is to investigate connections between the theory of harmonic mappings in the plane and hypergeometric functions by applying certain planar harmonic convolution operators on various subclasses of S Ĥ . Dedicated to Professor Eveyln M. Silvia, late Professor of Mathematics, UC Davis February 8, 1948–January 21, 2006