COEFFICIENT ESTIMATES AND SPECIAL DIFFERENTIAL SUBORDINATIONS OF CERTAIN ANALYTIC FUNCTIONS

Abstract

The thesis entitled “Coefficient Estimates and Special Differential Subordinations of Certain Analytic Functions” is divided into 6 chapters. After chapter 6, the thesis is concluded with the future scope. The aim at the beginning of each chapter provides a brief summary of the research work done and concluding remarks at the end of each chapter gives the highlights of that chapter. We enlist below chapter wise outline of the research work. Chapter 1 titled “Introduction" provides a quick overview of the topic. It covers fundamental concepts, some essential definitions, terminologies and ideas that are required further to achieve the objectives. Chapter 2 titled “First Order Differential Subordinations" deals with certain differen tial subordination implications involving certain parameters. These implications are achieved by finding conditions on the parameters. Furthermore, sufficient conditions for normalized analytic function f to belong to various sub-classes of starlike functions are obtained as an application of the derived results. Chapter 3 titled “Certain Exact Differential Subordinations" instigates the concept of exact differential subordinations, which is analogous to first order exact differential equations on the real line. Mainly, this chapter involves two special type of exact differ ential subordinations to study the newly introduced concept and obtain the dominant and best dominant for these differential subordinations. Certain applications to uni valent functions are appended to this chapter. Chapter 4 titled “A Special Type of Ma-Minda Function" deals with the extensive study on Ma-Minda functions based on its deep rooted conditions and it’s geometrical aspects. Following which, a special type of Ma-Minda function Φ is introduced and the classes S ∗ (Φ) and C(Φ) are defined. A newly defined subclass of starlike functions involving a special type of Ma-Minda function 1−log(1+z) studied here for obtaining inclusion and radius results. In addition, majorization and Bloch function norm xi related results are discussed. Chapter 5 titled “Coefficient Estimates of Certain Analytic Functions", establishes bounds of various initial coefficients, certain Hankel determinants for functions in both type of classes, involving Ma-Minda function and the special type of Ma-Minda function. Studied the special cases for each of the classes for certain coefficient es timates. The bounds obtained are all sharp, among which finding the sharp third Hankel determinant for functions in the class associated with lemniscate of Bernoulli is the key feature of this chapter, which was open until now. Chapter 6 titled “A Novel Subclass of Starlike Functions" deals with defining a new subclass S ∗ α of starlike functions involving a real part and modulus of certain expres sions, combined by way of an inequality. It is also inferred that this class reduces to a class S ∗ (qα), involving subordination and the subordinating function qα, which is well-known in the literature with certain interesting properties. Certain inclusion and radius results are deduced for functions in the classes S ∗ α and S ∗ (qα). Furthermore, various sharp coefficient estimates are obtained for functions in S ∗ (qα). Finally, the bibliography and list of author’s publications have been given at the end of the thesis.