On A Subclass Of Analytic Functions Satisfying A Differential Inequality - Chung, Yao Liang (2022)
The present dissertation investigates complex-valued analytic functions in the open unit disk D := {z ∈ C : |z| < 1}. A brief survey of the basic concepts and results from the classical theory of analytic univalent functions are given. For λ ∈ (0,1], the class of normalized analytic functions f satisfying | f ′(z)(z/ f (z))2 −1| < λ has been actively investigated and is shown to be univalent in D. Motivated by this class, a class of normalized analytic functions f satisfying | f ′(z)(z/ f (z))2 −μ| < λ is introduced. Conditions on λ and μ are chosen suitably to ensure f is univalent in D. This family is shown to be preserved under a number of elementary transformations. The necessary and sufficient condition (in terms of integral representation) of the function f is derived. Several important results such as finding the coefficient estimate and the bound for the second and third Hankel determinant are determined. Lastly, some radius problems are investigated. Connection are made with earlier results.