Inclusion and Neighborhood Properties for Certain Classes of Multivalently Analytic Functions

Serap Bulut

doi:10.1155/2013/754598

Abstract

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the classes introduced here.

Highlights

Introduction and DefinitionsLet R = (−∞, ∞) be the set of real numbers, let C be the set of complex numbers, let N = {1, 2, . . .} be the set of positive integers, and let N0 = N ∪ {0}
We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions
If we set λ = μ = 0 and β = 1 in Theorem 17, we have [2, Theorem 6]

Summary

Introduction and Definitions

Let Tp denote the class of functions of the form. Let the function f ∈ Tp. one says that f is in the class Pg(p, k, λ, μ, b, β, m, n; γ) if it satisfies the condition ((p)n Setting λ = μ = 0, β = 1 in Definition 2, we have the special class (which generalizes the class defined by Prajapat et al [3]) introduced by Srivastava et al [2]. Following a recent investigation by Frasin and Darus [6], if f ∈ Tp and δ ≥ 0, we define the (q, δ)-neighborhood of the function f by. Apart from deriving coefficient bounds and coefficient inequalities for each of these classes, we establish several inclusion relationships involving the (q, δ)-neighborhoods of functions belonging to the general classes which are introduced above

Coefficient Bounds and Coefficient Inequalities

A Set of Inclusion Relationships

Neighborhood Properties