RADIUS ESTIMATES AND CERTAIN DIFFERENTIAL SUBORDINATIONS FOR ANALYTIC FUNCTIONS

Abstract

Univalent function theory is a fascinating topic of complex analysis and deals with the geometrical aspects of analytic functions due to which it is classified under geometric function theory. Furthermore, the theory of differential subordination plays a crucial role in Univalent function theory as it is used as a tool for establishing various impli cation results. This thesis chiefly focuses on establishing results pertaining to radius estimates and differential subordination implications for certain classes of analytic functions, which are either well known or introduced and studied here. It comprises of six chapters and concluded with the future scope. The chapter wise arrangement of the content is as follows: Chapter 1 provides a brief review of the general principles of Univalent function theory. It provides an overview of the relevant literature and mentions some of the remarkable works done by various authors. The concepts and the techniques which serve as prerequisite for the main study are also discussed in this chapter. Chapter 2 introduces a new subclass of S ∗ associated with the modified sigmoid function, given by S ∗ SG = {f ∈ A : z f′ (z)/ f(z) ≺ 2/(1 + e −z )}. To describe the general behavior of the functions in this class, we study its geometrical properties and use these properties to prove certain results. Furthermore, we discuss admissibility conditions for the modified sigmoid function. The first and second order admissibility conditions are simply obtained by the general admissibility criteria given by Miller and Mocanu [61] and for the third order conditions, we derive a new criteria by modifying the previously existing theory. Chapter 3 presents differential subordination implications, involving the modified sigmoid function and other well known Ma-Minda functions, proved by using three different techniques. Using Miller Mocanu lemma, we prove several first order dif ferential subordination results involving real parameters. Later by using the method of contradiction, we extend these results for complex parameters. Finally, using the ix idea of admissible functions, we underwent tedious computations to prove differen tial subordination results upto third order, which has not been done before in the literature. In Chapter 4, we employ some remote properties of Schwarz function in order to find radius estimates for three classes, namely, G1 2 , 1 2 -the Silverman class, Ω =

f ∈ A : |z f′ (z)− f(z)| < 1/2, z ∈ D

and S ∗ SG-the class of Sigmoid starlike functions. In addition, we prove sufficient conditions in the form of differential inequalities for a general form of the Silverman class. Finally, using the concept of subordination, we develop a number of inclusion relations for the general form of Silverman class and the class Ω, involving various subclasses of starlike functions. Chapter 5deals with differential subordination results involving Pythagorean means. In the first part we prove an extremely general result involving the convex weighted harmonic mean of p(z) and p(z)Θ(z)+zp′ (z)Φ(z), where Θ, Φ are analytic functions. Furthermore, we discuss some special cases of this result. In the next part, a combina tion of arithmetic, geometric and harmonic mean of p(z) and quantities involving its derivatives has been taken into consideration. We prove certain implications for this combination and use them further for proving starlikeness and univalence criteria. Establishments in this chapter generalize many earlier known results. In Chapter 6, we introduce the following special type of differential subordination implication: p(z)Q(z)+ zp′ (z) βp(z)+α ≺ h(z) ⇒ p(z) ≺ h(z), (0.0.1) where p, Q are analytic with p(0) = 1 and 0 , β,α ∈ C. The differential subordination given above is a general form of Briot-Bouquet differential subordination. As a conse quence, we discuss some of the special cases of the aforementioned result. Moreover, we prove some results which are analogous to open door lemma and integral existence theorem. As an application, the outcomes of this chapter have been used to obtain criteria for univalence and starlikeness. The bibliography and list of publications have been given at the end of the thesis.