Properties of Certain Subclasses of Meromorphically p-Valent Functions Associated with Certain Integral Operator

Fatma El-Emam Delta Higher Institute for Engineering and Technology, Mansoura, Egypt Correspondence should be addressed to Fatma El-Emam; fatma_elemam@yahoo.com Received 8 August 2020; Accepted 27 November 2020; Published 16 December 2020 Academic Editor: Elena Guardo Copyright © 2020 Fatma El-Emam. 'is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 'e object of the this paper is to derive some interesting properties of certain subclasses of meromorphically p-valent functions which are defined by using an integral operator.

For functions f(z) ∈ p,m given by (1) and g(z) ∈ p,m defined by the Hadamard product (or convolution) of f(z) and g(z) is given by (3) Let f and g be analytic in U. e function f is said to be subordinate to g, or g is superordinate to f, written f≺g (z ∈ U), if there exists a Schwarz function w(z) in U with w(0) � 0 and is univalent in U, then the equivalence (cf., e.g., [1,2]) For 0 ≤ μ, α ≤ 1, m > − p, p ∈ N, and f ∈ p,m , Saleh et al. [3] introduced the p-valent Rafid operator S α μ,p : p,m ⟶ p,m as follows: where (]) k is the Pochhammer symbol defined, in terms of the Gamma function Γ, by [4]). It follows from (5) that By using the integral operator S α μ,p f(z), we define a subclasses of p,m as follows.
Definition 1. For fixed parameters A and B, we say that a function f(z) ∈ p,m is in the class S p,m (α, μ, λ, A, B) if it satisfies the following condition: ere are many papers about some subclasses of meromorphic functions associated with several families of linear operators (see, for example, [5][6][7][8][9][10][11]). In this paper, we obtain some properties of the class S p,m (α, μ, λ, A, B).

Preliminary Lemmas
To establish our main results, in this paper, we shall need the following lemmas.
Let P(c) be the class of analytic in U of the form which satisfies the following inequality: Lemma 2 (see [13]). Let the function φ(z), given by (12), be in the class P(c). en, Lemma 3 (see [14]).
e result is the best possible. Let a, b, and c be any real or complex numbers with and consider the function given by is function, called the Gauss hypergeometric function, is analytic and converges absolutely for z ∈ U (see [15]).
Lemma 4 (see [15]). Let a, b, and c any real or complex numbers with c ∉ Z − 0 . en,

Main Results
Unless otherwise mentioned, we shall assume throughout the sequel that where the function q 1 (z) given by is the best dominant of (22). Furthermore, where e result is the best possible.
Proof. Set en, the function ϕ(z) is of form (9) and is analytic in U. Differentiating (26) and with the aid of identity (7), we obtain Now, by using Lemma 1 for c � (α + 1)/λ, we deduce that where q 1 (z) is the best dominant of (22) given by by change of variables followed by the use of identities (17) and (18) (with a � 1, b � (α + 1)/(λ(p + m)), and c � b + 1). is proves assertion (22) of eorem 1. Next, in order to prove assertion (24) of eorem 1, it suffices to show that Indeed, for |z| ≤ r < 1, Setting Journal of Mathematics which is a positive measure on [0, 1], we obtain Letting r ⟶ 1 − in the above inequality, we obtain assertion (30). e result in (24) is best possible as the function q 1 (z) is the best dominant of (22).
Corollary 2. If f(z) ∈ p,2− p satisfies the following inequality e result is the best possible.
Remark 2. e result (asserted by Corollary 2) was also obtained by Srivastava and Patel [18].
e result is the best possible.
(45) e result is the best possible. Replacing ϕ(z) by z p S α μ,p f(z) in (26) and applying the same method and technique as the proof of eorem 1, we can prove the following result.