A new subclass of univalent functions

Abstract

Complex analysis is an old and vulnerable subject. Geometric function theory is a branch of complex analysis that deals and studies the geometric properties of the analytic functions. The geometric function theory studies the classes of analytic functions in a domain lying in the complex plane $C$ subject to various conditions. The cornerstone of the Geometric function theory is the theory of univalent and multivalent functions which is considered as one of the active fields of the current research. Most of this field is concerned with the class $S$ of functions analytic and univalent in the unit disc $E=\left\{z:\mid z \mid<1\right\}$. One of the most famous problem in this field was Bieberbach Conjecture. For many years this problem stood as a challenge to the mathematicians and inspired the development of many new techniques in complex analysis. In the course of tackling Bieberbach Conjecture, new classes of analytic and univalent functions such as classes of convex and starlike functions were defined and some nice properties of these classes were widely studied. In the present study, we introduce an interesting subclass of analytic and close-to-convex functions in the open unit disc $E$. For functions belonging to this class, we derive several properties such as coefficient estimates, distortion theorems, inclusion relation, radius of convexity and Fekete-Szegö Problem. The various results presented here would generalize some known results.