A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions

We introduce a new class ofmeromorphically analytic functions, which is defined bymeans of aHadamard product (or convolution) involving some suitably normalized meromorphically functions related to Cho-Kwon-Srivastava operator. A characterization property giving the coefficient bounds is obtained for this class of functions.The other related properties, which are investigated in this paper, include distortion and the radii of starlikeness and convexity. We also consider several applications of our main results to generalized hypergeometric functions.


Introduction
A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities).A simpler definition states that a meromorphic function () is a function of the form where () and ℎ() are entire functions with ℎ() ̸ = 0 (see [1, page 64]).A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain.An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere.For example, the gamma function is meromorphic in the whole complex plane C.
In the present paper, we initiate the study of functions which are meromorphic in the punctured disk  * = { : 0 < || < 1} with a Laurent expansion about the origin; see [2].
Let  be the class of analytic functions ℎ() with ℎ(0) = 1, which are convex and univalent in the open unit disk  =  * ∪ {0} and for which R {ℎ ()} > 0, ( ∈  * ) . ( For functions  and  analytic in , we say that  is subordinate to  and write  ≺  in  or  () ≺  () , ( ∈  * ) if there exists an analytic function () in  such that Furthermore, if the function  is univalent in , then This paper is divided into two sections; the first introduces a new class of meromorphically analytic functions, which is defined by means of a Hadamard product (or 2 International Journal of Mathematics and Mathematical Sciences convolution) involving linear operator.The second section highlights some applications of the main results involving generalized hypergeometric functions.

Preliminaries
Let Σ denote the class of meromorphic functions () normalized by which are analytic in the punctured unit disk  * = { : 0 < || < 1}.For 0 ≤ , we denote by  * () and () the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order  and convex of order  in .
For functions   () ( = 1; 2) defined by we denote the Hadamard product (or convolution) of  1 () and  2 () by Cho et al. [3] and Ghanim and Darus [4] studied the following function: Corresponding to the function  , () and using the Hadamard product for () ∈ Σ, we define a new linear operator (, ) on Σ by The Hadamard product or convolution of the functions  given by (10) where As for the second result of this paper on applications involving generalized hypergeometric functions, we need to utilize the well-known Gaussian hypergeometric function.One denotes (, ; ) the class of the function given by for  ̸ = 0, −1, −2, . .., and  ∈ C\{0}, where () = ( + 1) +1 is the Pochhammer symbol.We note that where is the well-known Gaussian hypergeometric function.

Characterization and Other Related Properties
In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function  ∈ Σ of the form (6)  ( The equality is attained for the function   () given by Proof.Let  of the form (6) belong to the class Σ   (, , ).Then, in view of (12), we find that Putting || =  (0 ≤  < 1) and noting the fact that the denominator in the above inequality remains positive by virtue of the constraints stated in (13) for all  ∈ [0, 1), we easily arrive at the desired inequality (20) by letting  → 1.
Conversely, if we assume that the inequality (20) holds true in the simplified form (22), it can readily be shown that which is equivalent to our condition of theorem, so that  ∈ Σ   (, , ), hence the theorem. Hence This completes the proof of Theorem 5.
We next determine the radii of meromorphic starlikeness and meromorphic convexity of the class Σ   (, , ), which are given by Theorems 6 and 7 below.Theorem 6.If the function  defined by (6) is in the class Σ   (, , ), then  is meromorphic starlike of order  in the disk || <  1 , where The equality is attained for the function   () given by (21). Proof.
The equality is attained for the function   () given by (21).
Proof.By using the same technique employed in the proof of Theorem 6, we can show that For || <  1 and with the aid of Theorem 3, we have the assertion of Theorem 7.

Applications Involving Generalized Hypergeometric Functions
Proof.By using the same technique employed in the proof of Theorem 3 along with Definition 2, we can prove Theorem 8.
The following consequences of Theorem 8 can be deduced by applying (39) and ( 40 ( The equality is attained for the function   () given by (40).( The equality is attained for the function   () given by (40).