Study of Some Properties on Analytic Function Theory

Abstract

The purpose of this thesis is to study some of the properties on analytic function theory. Some results third-order differential subordination and superordination results for analytic functions in the open unit disk were obtained. Also, it aims to make use of convolution operator N_(λ,η)^(s,k) f(z) in the class K_(λ,η)^(s,k) (γ,D,E) and shed light on some geometric properties, such as coefficient inequality, distortion, growth theorems closure theorems , integral operators, radii of close to convexity, convexity, and starlikeness for functions in class K_(λ,η)^(s,k) (γ,D,E). Here, two new subclasses of a class m-fold symmetric bi-univalent functions in U have been studied. Coefficient bounds for the Taylor-Maclaurin coefficients|a_(m+1) | and |a_(2m+1) | are obtained. Further , "Fekete-Szeg" "o" ̈ functional problems for functions in F_∑▒〖,m〗 (γ, ,ϑ) and M_∑▒〖,m〗 (κ ,η,ϑ) are solved. The study also sheds light on the investigation of two new subclasses M^γ (κ) and N_η (κ) of meromorphic functions which are defined by terms of a quasi-subordination are introduced. The coefficient estimates inclusive the classical" Fekete" "-Szeg" "o" ̈ functional |a_1-λa_0^2 | for functions belonging to this class are then derived. Investigating a majorazation problem for the classes meromorphic functions associated with these classes are also pointed out. The discussion was shown for quasi subordination of bi-univalent functions involving convolution operators. Finally, we introduced and investigated two new subclasses of class bi univalent functions in U, which includes the convolution between Hurwitz-Lerch Zeta and the generalized derived operator and satisfies quasi subordination conditions. The coefficient estimates |a_2 | and |a_3 |were determined in these subclasses. Also, certain new subclass of univalent functions with sigmoid function is introduced and applications with "Fekete-Szeg" "o" ̈ problems in G_q^β (H,2u,v) are obtained.