Some Classes of Analytic Function Involving Linear Operators

Abstract

The purpose of this thesis is to explore some concepts onto a survey on some analytical and geometrical properties of classes included univalent and multivalent functions in the open unit disk U, and meromorphic univalent functions in the punctured open unit disk U^*. By making use of linear operator J_((λ_p ),(μ_q ),b)^(s,a,λ) in the class〖 I〗_((λ_p),(μ_q),b)^(s,a,λ) (δ,d) and shading light on some geometric properties, such as coefficient inequality distortion and growth theorems closure theorems and integral operators, radii of close to convexity, convexity and starlikeness for functions in the class I_((λ_p),(μ_q),b)^(s,a,λ) (δ,d) . The study also, has shed the light on meromorphically univalent functions by using a linear operator G_μ^λ (a_l,b_m;q)in the punctured open unit disk . In addition to that, how to make use of Mittag-Lefflerr function certain subclass of analytic univalent functions to investigate inclusion properties associated with the concept of differential subordination . Moreover, it discusses the applications of q-Ruscheweyh differen-tial operator〖 R〗_q^k on some certain subclass J_(k,j) (q,β,A,B) of analytic univalent functions have been obtained a necessary condition for the ffunction to be in the J_(k,j) (q,β,A,B), like coefficient estimates, radii of starlikeness, distortion theorem, close-to-convexity, convexity, extreme points, neighborhoods, and the integral mean inequalities of functions affiliation to these classes. Also, we have installed some interesting results by studying a certain subclass TU_H^* (γ,n,μ,l,β) of univalent harmonic functions of the from f=h+g̅ defined by the Catas operator J_(μ,l)^m . Finally, by using the Srivastava-Attiya operator, we give some of the results which have tackled the class of SJ_(α,b) (γ) that consisting the family of harmonic functions f=h+g ̅ on the open unit disk.