Study of Some Applications in Univalent Function Theory

Abstract

The main aim of this thesis is to study some of applications in univalent function theory and obtain expected results . It includes studying class T(m)of normalized analytic functions in the open unit disk U={z∈C:│z│<1 } and class ∑ of meromorphic functions in the unit punctured disk〖 U〗^*={z∈C:0<│z│<1 } . We introduced and studied the subclasses R_λ^(γ,δ) (β,τ,σ,φ) of univalent function with negative coefficients and K^* (σ,τ,S) of univalent meromorphic functions defined by new linear operator I^(γ,δ) (c,b,λ) and differential operator D_(γ,δ)^S f(z) respectively.We obtain some geometric properties from classes R_λ^(γ,δ) (β,τ,σ,φ) and K^* (σ,τ,S), such as coefficient inequality , growth theorem and distortion theorem , we get radii of starlikeness , convexity and close-to-convexity of class R_λ^(γ,δ) (β,τ,σ,φ) the concept of convolution investigate and neighborhoods of the elements for the K^* (σ,τ,S) are obtained . Also, we studied of some results for differential subordination by generalized differential operatorA_(μ,λ,τ)^η (α,β)f(z) . Also, we get the concept of Quasi-Subordination and the estimates of the Fekete- Szego coefficient function│a_3-σa_2^2│ for functions belonging to classes N_(ρ,δ)^q (Φ) and 〖NH〗^q (δ,θ,ρ,Φ). By making use of the general liner operator J_p^(s,σ) (d_i,k_j )f(z) on the KT_s (ξ,τ,δ,) harmonic univalent function we give some properties like coefficient condition, distortion bounds ,extreme points, convolution and convex combinations. Also ,by utilizing generalization of the Srivastava- Attiya operator T_((δ_m 〖),(ϑ〗_(k)),d)^(t,a,δ) f(z) on two classes NK(δ,ϑ,t,η,σ) and NR(δ,ϑ,t,η,σ)for harmonic univalent functions we get sufficient coefficient conditions of the classes ,in addition to that ,we obtain properties like distortion bounds and extremepoints, so this proves that the classes studied in this section one is closed under convolution and convex combination.