On Study of Certain Subclasses of Analytic and Multivalent Functions

Abstract

In this thesis, we study and define certain families of regular and multivalent functions. In particular, we study, the set𝒮ℒ of lemniscate of Bernouli, the set of Janowski Bazilevič functions, the set ℒ𝒟𝜂 𝑘 (𝑎, 𝑐, 𝜌) of regular functions of reciprocal order and the set 𝒮𝒯𝑝 (𝑘, 𝛿, 𝜂, 𝑞) of multivalent functions. Our new defined families generalize the set of Bazilevič functions, the set of convex and starlike functions of reciprocal order and the families of uniformly convex and starlike functions. The family 𝒮𝒯𝑝 (𝑘, 𝛿, 𝜂, 𝑞) of multivalent functions is defined by using newly defined𝑞-analogue of p-valent Ruscheweyh operator. We employ various tools and techniques to study certain geometric properties of these families of functions. We mainly study fourth Hankel determinant of the functions belongs to set 𝒮ℒ. Further we investigate coefficient bounds, arc-length problem, growth result, integral preservance and some other interesting properties of these classes of functions.