Neighborhoods of certain p−valent analytic functions defined by using generalized differential operators

In this paper, by use of the familiar concept of neighborhoods of p−valently analytic functions, the authors show several inclusion relations associated with the (n, τ)−neighborhood of certain subclasses of analytic functions.


Introduction and preliminaries
Let A (p) denote the class of functions f of the form: f(z) = z p + ∞ n=p+1 a n z n (z ∈ U, p, n ∈ N = {1, 2, 3, · · · }) (1.1) which are analytic and univalent in the open unit disc U = {z ∈ C : |z| < 1}. The function f(z) in A (p) is said to be p−valent starlike functions of order α and univalent convex of order α in U if it satisfies: Re zf (z) f(z) > α (p ∈ N, z ∈ U, 0 α < p) (1.2) and respectively.
This generalises various operators as follows.
µ+λ m a n z n ,the operator introduced and studied by   [4].
a n z n ,the operator introduced and studied by Darus and Ibrahim (2009) [12].
[n] m a n z n ,the operator introduced and studied by Sãlãgean (1983) [13].
Further, let T * (p) denoted the supclass of A (p) consisting functions of the form a n z n (a n 0, n, p ∈ N, z ∈ U).
We define and study some (n, τ)− neighborhood properties. The classes G * (b, p) and G * (b, p, χ) are defined as follows: and m ∈ N 0 and for all z ∈ U.
and m ∈ N 0 and for all z ∈ U.

Neighborhoods for the classes
In our investigation of the inclution relations involving N n,δ (h), we shall require theorem 2.1 and theorem 2.2 below: Proof. We first suppose that f ∈ G * (b, p). Then by appealing to the condition (1.10), we readily obtain or, equivalently, µ+λ m a n z n−p We now choose values of z on the real axis and let z → 1 − through real values. Then the inequality (2.2) immediately yields the desired condition (2.1).
Conversely, by applying the hypothesis (2.1) and letting |z| = 1, we find from (1.10) that µ+λ m a n z n−p µ+λ m a n z n−p Similarly, we can prove the following result: Proof. We first suppose that f ∈ G * (b, p, χ). Then by appealing to the condition (1.11), we readily obtain We now choose values of z on the real axis and let z → 1 through real values. Then the inequality (2.8) immediately yields the desired condition (2.7).
Our first inclusion relation involving N k,δ (h) is given by theorem 2.3 below.

Neighborhoods for the classes G
In this section, we determine the neighborhood for the each classes G * ε (b, p) and G * ε (b, p, χ), which we define as follows. A function f(z) ∈ A (p) is said to be in the class Analogously, a function f(z) ∈ A (p) is said to be in the class G * ε (b, p, χ) if there exists a function g(z) ∈ G * ε (b, p, χ) such that the inequality (3.1) holds true. Theorem 3.1. If g(z) ∈ G * ε (b, p) and
Our proof of theorem 3.2 below is much akin to that of theorem 3.1. then N n,δ (g) ⊂ G * ε (b, p, χ).