On Sandwich Results of Meromorphic Multivalent Functions Defined by a New Hadamard Product Operator

The goal of this research is to establish differential subordination and superordination findings for meromorphic multivalent functions defined by a new operator in a punctured open unit disk. We get a number of sandwich-type results

Definition 2 : [16] Let : ℂ 3 ×  → ℂ and let ℎ be univalent in .If  is holomorphic in  and satisfies the secondorder differential subordination, then  is called a solution of the differential subordination (1.3).The univalent function  is called a dominant of the solution of the differential subordination (1.3), or more simply dominant if  ≺  for all  satisfying (1.3).A univalent dominant  ̂() that satisfies  ̂≺  for all dominant  of (1.3) is said to be the best dominant.
In a recent paper, E-Ashwah [12 ] defined the multiplier transform  ,

2-Preliminaries :
The definitions and lemmas given below will assist us in proving our basic results.

3-Results of Differential Subordinations
Now, we discuss some differential subordination results using a new Hadamard product operator  ,, ,, .
Proof : Define the () function as follows: If  ∈ ∑  is satisfy the following subordination condition: when () given by (3.3) , then where the best dominating is 1+ 1+ .
Taking the function () = ( As a result, we may deduce the following conclusion. If the function () in (3.3) is univalent and the superordination criterion is fulfilled: holds, then where the best subordinant is q(z).If () in (3.3) is univalent in , and  ∈ ∑  fulfills the superordination condition, The proof is therefore completed by utilizing the Lemma 2.3.where  1   2 are the best subordinant and dominant of the pair, respectively (5.1).