A study in the theory of geometric functions of a complex variable

Abstract

This thesis deals with various types of analytic geometric functions in the open unit disk, such as normalized, meromorphic, p-valent, harmonic, and fractional analytic functions. Five problems are discussed. First, the class of analytic functions of fractional power is suggested and used to define a generalized fractional differential operator, which corresponds to the Srivastava–Owa operator. The upper and lower bounds for fractional analytic functions containing this operator are discussed by employing the first-order subordination and superordination. Coefficient bounds for the new subclass of multivalent ( p-valent) analytic functions containing a certain linear operator are then presented. Other geometric properties of this class are studied. A new subclass of meromorphic valent functions defined by subordination and convolution is also established, and some of its geometric properties are studied. For a normalized function, the extended Gauss hypergeometric functions, which are generalized integral operators involving the Noor integral operator, are posed and examined. New subclasses of analytic functions containing the generalized integral operator are defined and established. In addition, some sandwich results are obtained. Third-order differential subordination outcomes for the linear operator convoluting the fractional integral operator with the incomplete beta function related to the Gauss hypergeometric function, are investigated. The dual concept of the third-order differential superordination is also considered to obtain third-order differential sandwich-type outcomes. Results are acquired by determining the appropriate classes of admissible functions for third-order differential functions. The final phase of this dissertation introduces two subclasses of S'h , which are denoted by LH(r) and H(a,B) . Coefficient bounds, extreme points, convolution, convex combinations, and closure under an integral operator are investigated for harmonic univalent functions in the subclasses H(a,B) and Lh (r) . Connections between harmonic univalent and hypergeometric functions are also fully investigated