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

Let $\Sigma$ denote the class of functions of the form which are analytic in the punctured open unit disk ${\mathbb{U}}^{\ast }=\left\{z:z\in \mathbb{C},0<\left|z\right|<1\right\}=\mathbb{U}/\left\{0\right\}$.

Let ${\Sigma }^{\ast }\left(\rho \right)$ and ${\Sigma }_k\left(\rho \right)$ denote the subclasses of $\Sigma$ that are meromorphically starlike functions of order $\rho$ and meromorphically convex functions of order $\rho$ respectively. Analytically, a function f of the form (1) is in the class ${\Sigma }^{\ast }\left(\rho \right)$ if it satisfies and $f\in {\Sigma }_k\left(\rho \right)$ if satisfies

The Hadamard product for two functions $f\in \Sigma$, defined by (1) and is given by

For the two functions $f\left(z\right)$ and $g\left(z\right)$ analytic in $\mathbb{U}$, we say that $f\left(z\right)$ is subordinate to $g\left(z\right)$, written $f\prec g$ or $f\left(z\right)\prec g\left(z\right)$ $(z\in \mathbb{U})$, if there exists a Schwarz function $w\left(z\right)$ in $\mathbb{U}$ with $w\left(0\right)=0$ and $\left|w\right(z\left)\right|<1$ $(z\in \mathbb{U})$, such that $f\left(z\right)=g\left(w\right(z\left)\right),(z\in \mathbb{U})$.

For complex parameters $a_i,b_k,q(i=1,\dots ,l,k=1,\dots ,r,b_k\in \mathbb{C}\backslash \{0,-1,-2,\dots \})$ the basic hypergeometric function (or q-hypergeometric function) ${}_l{\mathrm{\Psi }}_r\left(z\right)$ is defined by: with $\left(\genfrac{}{}{0pt}{}{j}{2}\right)=j(j-1)/2$, where $q\ne 0$ when $l>r+1$ $(l,r\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},\mathbb{N}=\{1,2,\dots \})$, and ${(a,q)}_j$ is the q-analogue of the Pochhammer symbol ${\left(a\right)}_j$ defined by:

The hypergeometric series defined by (3) was initially introduced by Heine in 1846 and referred to as the Heines series. More details on q-theory are available in [1,2,3] for readers to refer to.

It is clear that where ${}_l{F}_r(a_1,\dots ,a_l;b_1,\dots .,b_r;z)$ 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 $z\in \mathbb{U},$ $\left|q\right|<1$, and $l=r+1$, the q-hypergeometric function defined in (3) takes the following form: which converges absolutely in the open unit disk $\mathbb{U}$.

In reference to the function ${}_l{\mathrm{\Psi }}_r(a_1,\dots ,a_l;b_1,\dots .,b_r;q,z)$ for meromorphic functions $f\in \Sigma$ 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

Before going further, we state the well-known Mittag–Leffler function $E_{\alpha }\left(z\right)$, put forward by Mittag–Leffler [9,10], as well as Wiman’s generalisation [11] $E_{\alpha ,\beta }\left(z\right)$ given respectively as follows: and where $\alpha ,\beta \in \mathbb{C}$, $Re\left(\alpha \right)>0$ and $Re\left(\beta \right)>0$.

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 $E_{\alpha ,\beta }\left(z\right)$ Mittag–Leffler function can be seen in [19]. Additionally, Ref. [20] presents findings related to partial sums for $E_{\alpha ,\beta }\left(z\right)$.

The function given by (7) is not within the class $\Sigma$. Based on this, the function is then normalised as follows:

Having use of the function ${\mathrm{\Omega }}_{\alpha ,\beta }\left(z\right)$ given by (8), a new operator ${\mathfrak{D}}_{\beta }^{\alpha ,m}[a_l,b_r,\lambda ]:\Sigma \to \Sigma$ is defined, in terms of Hadamard product, as follows:

If $f\in \Sigma$, then from (9) we deduce that where

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 ${\mathcal{T}}_{\alpha ,\beta }^m(a_l,b_r,\lambda ;D,H,d)$ of $\Sigma$, which involves the operator ${\mathfrak{D}}_{\beta }^{\alpha ,m}[a_l,b_r,\lambda ]f\left(z\right)$, and is shown as follows:

Let ${\Sigma }^{\ast }$ denote the subclass of $\Sigma$ consisting of functions of the form:

Now, we define the class ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$ by

This section presents work to acquire sufficient conditions in which (14) gives the function f within the class ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$, 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 ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$.

In our first theorem, we begin with the necessary and sufficient conditions for functions f in ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$.

Growth and distortion bounds for functions belonging to the class ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$ will be given in the following result:

Next, we determine the radii of meromorphic starlikeness and convexity of order $\rho$ for functions in the class ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$.

The closure theorems and extreme points of the class ${\mathcal{T}}_{\alpha ,\beta }^{m,\ast }(a_l,b_r,\lambda ;D,H,d)$ will now be determined.