A NEW SUBCLASS OF MEROMORPHIC FUNCTIONS DEFINED BY RAPID OPERATOR

We present and investigate a new subclass of meromorphic univalent functions described by the Rapid operator in this study. Coe¢ cient inequalities is discussed, as well as distortion properties, closure theorems, Hadamard product. After this, integral transforms for the class (#; %; }; ; ) are obtained. 1. Introduction Let stands for the function class of the form @(~) = 1 ~ + 1 X `=1 a`~; ` 2 N = f1; 2; 3; g (1) analytic in the punctured unit disc = f~ 2 C : 0 < j~j < 1g = n f0g: A function @ 2 given by (1) is said to be meromorphically starlike of order % if it satises the following: < ~@0(~) @(~) > %; (~ 2 ) for some %(0 % < 1): We say that @ is in the class (%) of such functions. Similarly a function @ 2 given by (1) is said to be meromorphically convex of order % if it satises the following: < 1 + ~@00(~) @0(~) > %; (~ 2 ) 2020 Mathematics Subject Classication. 30C45. Keywords and phrases. Meromorphic, starlike, coe¢ cient estimates, integral operator. bvlmaths@gmail.com-Corresponding author; reddypt2@gmail.com, sujathavaishnavy@gmail.com; siri_settipalli@yahoo.co.in 0000-0003-3669-350X; 0000-0002-0034-444X; 0000-0002-2109-3328; 0000-0003-1918-6127. c 2021 Ankara University Communications Facu lty of Sciences University of Ankara-Series A1 Mathematics and Statistics 858 A NEW SUBCLASS OF MEROMORPHIC FUNCTIONS 859 for some %(0 % < 1): We say that @ is in the class `(%) of such functions. Akgul [1,2], Miller [8], Pommerenke [9], Royster [10], Aydogan and Sakar [4,5,11] and Venkateswarlu et al. [14,15,16] have all studied the class (%) and numerous other subclasses of extensively. For functions @ 2 given by (1) and g 2 given by g(~) = 1 ~ + 1 X `=1 b`~; we dene the Hadamard product of @ and g by (@ g)(~) = 1 ~ + 1 X `=1 a`b`~: Jung et al. dened the integral operator on normalised analytic functions in [6] and Lashin [7] updated their operator for meromorphic functions in the following manner: Lemma 1. For @ 2 given by (1), if the operator S  : ! is dened by S @(~) = 1 (1 ) ( + 1) 1 Z 0 t e t 1 @(t~)dt; (2) (0 < 1; 0 1 and ~ 2 ) then

If is a real number and ! is a complex number then <(!) , j! + (1 )j j! (1 + )j 0: Lemma 4. If ! is a complex number and ;`are real numbers then The key purpose of this paper is to look at some traditional geometric function theory properties for the class of geometric functions, such as coe¢ cient bounds, distortion properties, closure theorems, Hadamard product, and integral transforms.

Coefficient estimates
We obtain required and adequate conditions for a function @ to be in the class in this section.
Theorem 5. Let @ 2 be given by (1). Then @ 2 (#; %; }; ; ) i¤ Proof. Let @ 2 (#; %; }; ; ): Then by De…nition 2 and using Lemma 4, It su¢ ces to demonstrate that For convenience That is, the equation (7) is equivalent to We only need to prove that in light of Lemma 3 It is to show that 0; by the given condition (6).
Conversely suppose @ 2 (#; %; }; ; ): Then by Lemma 3, we have (7). The inequality (7) is reduced to when the values of~are chosen on the positive real axis Since <( e i ) je i j = 1; the above inequality is reduced to We obtained the inequality (6) by letting r ! 1 and using the mean value theorem.
The estimate is sharp for the function We get the following corollary by taking } = 0 in Theorem 5.

Distortion theorem
Theorem 8. If @ 2 (#; %; }; ; ) then for 0 < j~j = r < 1; This estimate is sharp for the function Since Using the above inequality in (13), we have The estimate is sharp for the function @(~) We omit the proof of the following corollary since it is similar to that of Theorem 8.
Proof. Assume that Then it follows that which implies that @ 2 (#; %; }; ; ): On the other side, assume that the function @ de…ned by (1) be in the class @ 2 (#; %; }; ; ): Then : where @ can be expressed in the form (20), as can be shown.

Modified Hadamard products
Let the functions @ j (j = 1; 2) de…ned by (14). The modi…ed Hadamard product of @ 1 and @ 2 is de…ned by The estimate is sharp for the functions @ j (j = 1; 2) given by Proof. Using the same method that Schild and Silverman [12] used earlier, we need to …nd the largest real parameter ' such that Since @ j 2 (#; %; }; ; ); j = 1; 2; we readily see that By Cauchy-Schwarz inequality, we have Then merely demonstrating that is necessary Hence, it light of the inequality (26), then merely demonstrating that is necessary It follows from (27) that : Now de…ning the function E(`); : We see that E(`) is an increasing of`;` 1: Therefore, we conclude that ; Hence the proof.

Conclusion
This research has introduced a new subclass of meromorphic functions de…ned by Rapid operator and studied some basic properties of geometric function theory. Accordingly, some results to coe¢ cient estimates, distortion properties, closure theorems, hadamard product and integral transforms have been considered, inviting further research for this …eld of study.
Author Contribution Statements All authors contributed equally to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript.

Declaration of Competing Interests
The authors declare that there is no con- ‡ict of interest regarding the publication of this.