In the present paper, we investigate a new linear operator through the Hadamard product between the fundamental hypergeometric function and the Mittag Leffler function. Furthermore, the geometric properties of a new meromorphic function subclass will be investigated.

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The 𝛶 subclasses S*(τ, λ) and C*(τ, λ), as well as the class of analytic and univalent functions of the form h ( w ) = w − ∑ i = 2 ∞ | e i | w i , were studied. Sharp coefficients, distortion, as well as a function expression formula in S*(τ, λ) are determined, as well as expression formulae for functions in C*(τ, λ). In addition, convex linear combination over arithmetic average, the classes S*(τ, λ) and C*(τ, λ) are proved to be closed.

The authors introduced and addressed  several new subclasses  of the family of meromorphically multivalent -star-like functions in the punctured unit disk  in this study, which makes use of several higher order -derivatives. Many fascinating properties and characteristics are extracted systematically for each of these newly identified function classes. Distortion theorems and radius problems are among these characteristics and functions. A number of coefficient inequalities for functions belonging to the subclasses are studied, and discussed, as well as a suitable condition for them is set. The numerous results are presented in this study and the previous works on this subject are also connected together in this study.

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