## 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

##### Collections

Perpustakaan Tuanku Syed Faizuddin Putra (PTSFP)
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