Radius and Subordination Results on Analytic and Meromorphic Functions

Abstract

This thesis studies various subclasses of the class of normalized analytic functions defined newlineon D B {and#119911; and#8712; C : |and#119911;| lt 1} by solving the radius problems as well as obtaining sufficient newlineconditions for functions to belong to these classes. Main tools are the estimates for and#119901;, newlineand#119911;and#119901;and#8242;(and#119911;)/and#119901;(and#119911;) and 1 + and#119911;and#119901;and#8242;and#8242;(and#119911;)/and#119901;and#8242;(and#119911;) for function with positive real part and the theory of newlinesecond order differential subordination. newlineIn the first chapter, various fundamental concepts of univalent function theory are introduced. newlineIt introduces the class of normalized analytic functions A B { and#119891; analytic in D : newlineand#119891; (0) = 0, and#119891; and#8242;(0) = 1} and its subclass of univalent functions S. It discusses the concept newlineof subordination and several subclasses of the class S defined using subordination. A newlinebrief summary of radius problems and differential subordination theory is provided. newlineIn Chapter 2, we investigate radius problems for three classes of normalized analytic newlinefunctions and#119891; and#8712; A characterized by its ratio with certain normalized analytic functions and#119892;. newlineThe first two classes consists of functions and#119891; satisfying Re( and#119891; (and#119911;)/and#119892;(and#119911;)) gt 0 with either newlineRe(and#119892;(and#119911;)/and#119904;(and#119911;)) gt 0 or Re(and#119892;(and#119911;)/and#119904;(and#119911;)) gt 1/2 and the other class consists of functions newlineand#119891; satisfying | ( and#119891; (and#119911;)/and#119892;(and#119911;)) and#8722; 1| lt 1 with Re(and#119892;(and#119911;)/and#119904;(and#119911;)) gt 0 for some starlike function newlineand#119904; of order and#120572;, 0 and#8804; and#120572; lt 1. For functions in these classes, we compute various radii of newlinestarlikeness, including the radius of starlikeness of order and#120572;, and radius of Ma-Minda newlinestarlikeness associated with parabola, lemniscate of Bernoulli, exponential function, newlinecardioid, sine, lune, rational function and nephroid. newlineLet H0 (and#119886;) be the class of analytic functions and#119891; in D such that and#119891; (0) = and#119891; and#8242;(0) and#8722; and#119886; = 0. For newlineand#119886; and#8712; (0, 1), let B0 (and#119886;) be the class of functions in H0(and#119886;) satisfying | and#119891; (and#119911;) | and#8804; 1 and and#119891; has newlineno nonzero zero. In Chapter 3, we determine various starlikeness radii for the class newlineB0 (and#119886;). The class and#937;and#119886; B { and#119891; and#8712; H0(and#119886;) : |and#119911; and#119891; and#8242;(and#119911;) and#8722; and#119891; (and#119911;) | lt 1/2} is considered