On Meromorphic Functions Defined by a New Operator Containing the Mittag–Leffler Function

This study defines a new linear differential operator via the Hadamard product between a q-hypergeometric function and Mittag–Leffler function. The application of the linear differential operator generates a new subclass of meromorphic function. Additionally, the study explores various properties and features, such as convex properties, distortion, growth, coefficient inequality and radii of starlikeness. Finally, the work discusses closure theorems and extreme points.


Introduction
Let Σ denote the class of functions of the form which are analytic in the punctured open unit disk U * = {z : z ∈ C, 0 < |z| < 1} = U/{0}.Let Σ * (ρ) and Σ k (ρ) denote the subclasses of Σ that are meromorphically starlike functions of order ρ and meromorphically convex functions of order ρ respectively.Analytically, a function f of the form (1) The Hadamard product for two functions f ∈ Σ, defined by (1) and is given by For the two functions f (z) and g(z) analytic in U, we say that f (z) is subordinate to g(z), written f ≺ g or f (z) ≺ g(z) (z ∈ U), if there exists a Schwarz function w(z) in U with w(0) = 0 and |w(z)| < 1 (z ∈ U), such that f (z) = g(w(z)), (z ∈ U).
It is clear that lim q→1 − l Ψ r (q a 1 , ..., q a l ; q b 1 , ...., q b r ; q, (q − 1) 1+r−l z) = l F r (a 1 , ..., a l ; b 1 , ...., b r ; z), where l F r (a 1 , ..., a l ; b 1 , ...., b r ; z) represents the generalised hypergeometric function (as shown in [4]).Riemann, Gauss, Euler and others have conducted extensive studies of hypergeometric functions some hundreds years ago.The focus on this area is based mostly on the structural beauty and distinctive applications that this theory has, which include dynamic systems, mathematical physics, numeric analysis and combinatorics.Based on this, hypergeometric functions are utilised in various disciplines and this includes geometric function theory.One example that can be associated with the hypergeometric functions is the well-known Dziok-Srivastava operator [5,6] defined via the Hadamard product.Now for z ∈ U, |q| < 1, and l = r + 1, the q-hypergeometric function defined in (3) takes the following form: l Ψ r (a 1 , ..., a l ; b 1 , ...., b r ; q, z) = ∞ ∑ j=0 (a 1 , q) j ...(a l , q) j (q, q) j (b 1 , q) j ...(b r , q) j z j , ( which converges absolutely in the open unit disk U. In reference to the function l Ψ r (a 1 , ..., a l ; b 1 , ...., b r ; q, z) for meromorphic functions f ∈ Σ that consist of functions in the form of (1), (see Aldweby and Darus [7], Murugusundaramoorthy and Janani [8]), as illustrated below, have recently introduced the q-analogue of the Liu-Srivastava operator For convenience, we write l Υ r (a 1 , ..., a l ; b 1 , ..., b r ; q, z) f (z) = l Υ r (a i , b k ; q, z) f (z).
There has been a growing focus on Mittag-Leffler-type functions in recent years based on the growth of possibilities for their application for probability, applied problems, statistical and distribution theory, among others.Further information about how the Mittag-Leffler functions are being utilised can be found in [12][13][14][15][16][17][18].In most of our work related to Mittag-Leffler functions, we study the geometric properties, such as the convexity, close-to-convexity and starlikeness.Recent studies on the E α,β (z) Mittag-Leffler function can be seen in [19].Additionally, Ref. [20] presents findings related to partial sums for E α,β (z).
The function given by ( 7) is not within the class Σ.Based on this, the function is then normalised as follows: Having use of the function Ω α,β (z) given by ( 8), a new operator D α,m β [a l , b r , λ] : Σ → Σ is defined, in terms of Hadamard product, as follows: If f ∈ Σ, then from (9) we deduce that where Remark 1.It can be seen that, when specialising the parameters λ, l, r, m, α, β, q, a 1 , .., a l and b 1 , ..., b r , it is observed that the defined operator D α,m β [a l , b r , λ] f (z) leads to various operators.Examples are presented for further illustration.
., r, l = r + 1) and q → 1 we get the operator H l,r [a i , b k ] f (z) which was investigated by Liu and Srivastava [22].
was introduced by Ganigi and Uralegaddi [24], and then it was generalised by Yang [25].
A range of meromorphic function subclasses have been explored by, for example, Challab et al. [26], Elrifai et al. [27], Lashin [28], Liu and Srivastava [22] and others.These works have inspired our introduction of the new subclass T m α,β (a l , b r , λ; D, H, d) of Σ, which involves the operator , and is shown as follows: or, equivalently, to: Let Σ * denote the subclass of Σ consisting of functions of the form:

Main Result
This section presents work to acquire sufficient conditions in which (14) gives the function f within the class T m, * α,β (a l , b r , λ; D, H, d), as well as demonstrates that this condition is required for functions which belong to this class.In addition, linear combinations, growth and distortion bounds, closure theorems and extreme points are presented for the class T m, * α,β (a l , b r , λ; D, H, d).In our first theorem, we begin with the necessary and sufficient conditions for functions f in T m, * α,β (a l , b r , λ; D, H, d).
Theorem 1.Let the function f (z) be of the form (14). Then the function f (z Proof.Suppose that the inequality (15) holds true, we obtain Then, by the maximum modulus theorem, we have H, d) with f (z) of the form ( 14), then we find from (13) that since the above inequality is genuine for all z ∈ U * , choose values of z on the real axis.After clearing the denominator in (16) and letting z → 1 − through real values, we get Thus, we obtain the desired inequality (15) of Theorem 1.

Corollary 1.
If the function f of the form ( 14) is in the class T m, * α,β (a l , b r , λ; D, H, d) then the result is sharp for the function Growth and distortion bounds for functions belonging to the class T m, * α,β (a l , b r , λ; D, H, d) will be given in the following result: Theorem 2. If a function f given by ( 14) is in the class T m, * α,β (a l , b r , λ; D, H, d) then for |z| = r, we have: and which yields: and |z|.
Now, by differentiating both sides of ( 14) with respect to z, we get: and .
Next, we determine the radii of meromorphic starlikeness and convexity of order ρ for functions in the class T m, * α,β (a l , b r , λ; D, H, d).
Theorem 3. Let the function f given by ( 14) be in the class T m, * α,β (a l , b r , λ; D, H, d).Thus, we have: (i) f is meromorphically starlike of order ρ in the disc |z| < r 1 , that is where (ii) f is meromorphically convex of order ρ in the disc |z| < r 2 , that is where Proof.(i) From the definition ( 14), we can get: Then, we have: Thus, by Theorem 1, the inequality (20) will be true if .
The last inequality leads us immediately to the disc |z| < r 1 , where r 1 is given by ( 18).
(ii) In order to prove the second affirmation of Theorem 3, we find from ( 14) that: Thus, we have the desired inequality: Thus, by Theorem 1, the inequality (21) will be true if .
The last inequality readily yields the disc |z| < r 2 , where r 2 is given by (19).