Subordination And Convolution Of Multivalent Functions And Starlikeness Of Integral Transforms

Loading...
Thumbnail Image
Date
2012-01
Authors
Badghaish, Abeer Omar
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
This thesis deals with analytic functions as well as multivalent functions de- ned on the unit disk U. In most cases, these functions are assumed to be normalized, either of the form f(z) = z + 1X k=2 akzk; or f(z) = zp + 1X k=1 ak+pzk+p; p a xed positive integer. Let A be the class of functions f with the rst normalization, while Ap consists of functions f with the latter normalization. Five research problems are discussed in this work. First, let f(q) denote the q-th derivative of a function f 2 Ap. Using the theory of di erential subordination, su cient conditions are obtained for the following di erential chain to hold: f(q)(z) (p; q)zpô€€€q Q(z); or zf(q+1)(z) f(q)(z) ô€€€ p + q + 1 Q(z): Here Q is an appropriate superordinate function, (p; q) = p!=(p ô€€€ q)!, and denotes subordination. As important consequences, several criteria for univalence and convexity are obtained for the case p = q = 1: The notion of starlikeness with respect to nô€€€ply points is also generalized to the case of multivalent functions. These are functions f 2 Ap satisfying
Description
Keywords
Analytic functions
Citation