Neighborhood Properties of Multivalent Functions Defined Using an Integral Operator

In this paper, we introduce the generalized integral operator Jp(σ, λ) and using this generalized integral operator, the new subclasses H n,m(b, σ, λ), Ln,m(b, σ, λ;μ), H n,m(b, σ, λ) and L n,m(b, σ, λ;μ) of the class of multivalent functions denoted by Tp(n) are defined. Further for functions belonging to these classes, certain properties of neighborhoods of functions of complex order are studied. 2000 Mathematics Subject Classification: 30C45. 1.INTRODUCTION Let Ap(n) be the class of normalized functions f of the form f(z) = z + ∞ ∑ k=n+p akz , (n, p ∈ N), (1) which are analytic and p -valent in the open unit disc U = {z ∈ C : |z| < 1}. Let Tp(n) be the subclass of Ap(n) consisting functions f of the form f(z) = z − ∞ ∑ k=n+p akz , (ak ≥ 0, n, p ∈ N), (2) which are p valent in U . Definition 1 Let σ, λ ∈ R, σ > 0, λ > −p, p ∈ N and f ∈ Ap(n), the integral operator Jp(σ, λ) is defined as Jp(σ, λ)f(z) = (λ+ p)σ zλΓ(σ) ∫ z 0 tλ−1 ( log z t )σ−1 f(t) dt = z + ∞ ∑ k=n+p ( λ+ p λ+ k )σ akz , (3) where Γ denotes the Gamma function.