Analytic Functions of Complex Order Defined by New Differential Operator

Abdussalam Eghbiq,Maslina Darus

doi:10.30538/oms2018.0034

Abstract

Analytic Functions of Complex Order Defined by New Differential Operator

Highlights

Introduction and preliminariesLet H be the class of functions analytic in U = {z : |z| < 1} and let H[a,n]be the subclasses of H consisting of functions of the form f (z) = a + anzn + an+1zn+1 + an+2zn+2 +
We define (n, δ)−neighbourhood for the functions belonging to class A(n) and for identity function
For any two functions f and g analytic in U, f is said to be subordinate to g in U denoted by f ≺ g if there exists an analytic function w defined U satisfying w(0) = 0 and |w(z)| < 1 such that f (z) = g(w(z)), z ∈ U

Summary

Introduction and preliminaries

Let Rn(γ, λ, β) denote the subclass of A(n) consisting of functions f (z) which satisfy the following inequality. In the term of the generalized Salagean differential operator, let Sn,μ(γ, α, β, λ, ν, , ) denote the subclass of A(n) consisting of the functions f (z) which satisfy the inequality (μ)z(Dλ,+ν,3 (α, ω)f (z)) (μ)z(Dλ,+ν,2 (α, ω)f (z)). Our work is to investigate several new results like growth and distortion theorems, Hadamard product, extreme points, integral means inequalities and inclusion properties for the function included in the classes Sn,μ(γ, α, β, λ, ν, , , ω) and Rn,μ(γ, α, β, λ, ν, , , ω). Similar work has been seen for different subclasses done by other authors (see for example [14, 15, 16, 17, 18, 19, 20, 21])

Growth and distortion theorems

Extreme points

Integral means inequalities

Inclusion properties

Hadamard Product