Some geometric properties of multivalent functions associated with a new generalized $q$-Mittag-Leffler function

1.   Introduction and preliminaries

In 1903, Mittag-Leffler ^[22] provided the function $E_{\sigma }(z)$ defined by

where $\Gamma$ is the gamma function and $\mathcal{R}$ means the real part.

Wiman ^[34] introduced the following generalized Mittag-Leffler function

Prabhakar ^[25] introduced the following function $E^{\rho }_{\sigma, \mu }\left(z\right)$ in the form

Later, Shukla and Prajapati ^[27] (see also ^[32]) defined another generalized Mittag-Leffler function

where $k\in (0, 1)\bigcup \mathbb{N}$ and $(\rho)_{kj} = \frac{\mathrm{\Gamma }(\rho +kj)}{\mathrm{\Gamma }(\rho)}$ is the generalized Pochhammer symbol defined as

Bansal and Prajapat ^[5] and Srivastava and Bansal ^[31] investigated geometric properties of the Mittag-Leffler function $E_{\sigma, \mu }(z)$, including starlikeness, convexity, and close-to-convexity (see ^{[1,4,6,8,12,13,17,28,29]}). In reality, the generalized Mittag-Leffler function $E_{\sigma, \mu }(z)$ and its extensions are still widely used in geometric function theory and in a variety of applications (see, for details, ^{[2,3,7,16,24]}).

Let $\mathcal{S}(p)$ be the class of functions of the form

where $f$ is holomorphic and multivalent in the open unit disk $\mathcal{O} = \left\{z:\left|z\right| < 1\right\}.$

Let $f$ and $\mathcal{F}$ be two functions in $\mathcal{S}(p)$. Then the convolution (or Hadamard product), denoted by $f*\mathcal{F}$, is defined as

where $f(z)$ is in (1.1) and $\mathcal{F}\left(z\right) = z^p+\sum\limits_{j = p+1}^{\infty }{d_jz^j}.$

Let $f(z)$ and $h(z)$ be two analytic functions defined in $\mathcal{O}.$ The function $f(z)$ is called subordinate to $h(z),$ or $h(z)$ is superordinate to $f(z)$, denoted by $f\left(z\right)\prec h\left(z\right)$ and $h\left(z\right)\prec f\left(z\right)$, respectively, if there is a Schwarz function $\varphi$ with $\varphi(z) = 0, \left|\varphi(z)\right| < 1$ and $f\left(z\right) = h(\varphi(z))$. If the function $h$ is univalent in $\mathcal{O}$, then the following equivalence is true if

Definition 1.1. (^[18]). Let $0 < q < 1$. Then ${\left[j\right]}_q!$ denotes the $q$-factorial, which is defined as follows:

where $[j]_q = \frac{1-q^{j}}{1-q} = 1+\sum\nolimits_{m = 1}^{j-1}{\ }q^m\mathrm{\ and\ }[0]_q = 0$.

Definition 1.2 (^[18]). The $q$-generalized Pochhammer symbol ${[\rho]}_{j, q}$, $\rho \in \mathbb{C}$, is given as

and the $q$-Gamma function is defined as

It follows that ${\Gamma}_q\left(j+1\right) = [j]_q!.$

Lately, many results have been given for some related special functions such as the Wright function ^[3] and multivalent functions (see ^[10,23,26]).

Here, we propose a $q$-extension of specific extensions of the Mittag-Leffler function, motivated by the success of Mittag-Leffler function applications in physics, biology, engineering, and applied sciences. We generalize the Mittag-Leffler function given by Shukla and Prajapati ^[27] and obtain a new generalized $q$-Mittag-Leffler function.

Now, we present a new generalized $q$-Mittag-Leffler function as follows

It is obvious that, when $q\to 1^-$, the resulting function is the generalized Mittag-Leffler function, which is given by Shukla and Prajapati ^[27].

Corresponding to the function ${\mathcal{E}}^{\rho }_{\sigma ,\mu}\left(q;z\right)$ in (1.2), we establish the following generalized $q$-Mittag-Leffler function ${\mathcal{E}}^{\rho }_{\sigma ,\mu}\left(p,q;z\right)$ in multivalent functions $\mathcal{S}\left(p\right)$, as given below

Again, using the new function (1.3), we define the following function:

Definition 1.3. For $f\in \mathcal{S}(p)$, we define the new linear operator ${\mathcal{A}}^{\mu, \rho; k}_{\sigma; p, q}f(z):\mathcal{S}(p)\to \mathcal{S}(p)$ by

where ${\chi }_j = \frac{{\mathrm{\Gamma }}_q\left(\mu\right)(\rho)_{kj}}{{\mathrm{\Gamma }}_q\left(\mu+\sigma j\right)j!}.$

We now define a subclass $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ of the family $\mathcal{S}(p)$ using the multivalent linear operator in (1.5) and the subordination concept.

Definition 1.4. Let ${\mathcal{A}}^{\mu, \rho; k}_{\sigma; p, q}f(z)$ be an operator in (1.5). A function $f(z)\in \mathcal{S}(p)$ is said to be in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ if satisfies the following subordination condition:

or equivalently

and

where $-1\le \mathcal{M} < \mathcal{N}\le 1, \ 0\le \tau < p,$ and $p\in \mathbb{N}.$

Remark 1.1. Some well-known special classes of the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ can be obtained by choosing the values of the parameters $\varsigma, \mu, \rho; \tau, k, p, q, \ \mathcal{M},$ and $\mathcal{N}.$

$(1)$ $Q^{0, 0, 1}_{0, 1}\left(\mathcal{M}, \mathcal{N}; \tau, p\right) = S^*_p(\mathcal{M}, \mathcal{N}; \tau, p)$ was provided by Aouf ^[2].

$(2)$ $Q^{0, 0, 1}_{0, 1}\left(\mathcal{M}, \mathcal{N}; 0, p\right) = S^*_p(\mathcal{M}, \mathcal{N}; p)$ was provided by Goel and Sohi ^[16].

In this work, we introduce a new subclass of multivalent functions $Q^{\mu ,\rho ;k}_{\sigma;q}(\mathcal{M},\mathcal{N};\tau,p)$ defined by the new linear operator ${\mathcal{A}}^{\mu,\rho ;k}_{\sigma ;p,q}f(z)$. And we study some geometric properties for the class $Q^{\mu ,\rho ;k}_{\sigma;q}(\mathcal{M},\mathcal{N};\tau,p)$ such as the coefficient estimates, convexity and convex linear combination. Finally, the radius theorems associated with the generalized Srivastava-Attiya integral operator will be investigated.

2.   Main results

The first theorem in this section presents the necessary and sufficient condition for the function $f(z)$ in (1.1) belong to the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$

Theorem 2.1. A function $f(z)$ is in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ if and only if

where $1\le \mathcal{M} < \mathcal{N}\le 1, \ 0\le \tau < p,$ and $p\in \mathbb{N}.$

Proof. Assume that the condition (2.1) is true. Then by (1.7), we have

By maximum modulus theorem ^[11], we get $f(z)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$

Conversely, suppose that $f(z)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$. Then

Since $\mathcal{R}(z)\le \left|z\right|$, we get

Taking $z\to 1^-$, we have

This completes the proof.

Theorem 2.2. Let $f_1$ and $f_2$ be analytic functions in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$ Then $f_1*f_2\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$, where

where ${\mathrm{\chi}}_1 = \frac{{\mathrm{\Gamma }}_q\left(\mu \right)(\rho)_k}{{\mathrm{\Gamma }}_q\left(\mu +\varsigma \right)}.$

Proof. We will show that ${\tau }_1$ is the largest satisfying

Since $f_1, f_2\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$, by the condition (2.1) and the Cauchy-Schwarz inequality, we get

From (2.3) and (2.4), we observe that

From (2.4), it is necessary to prove

Furthermore, from the inequality (2.5) it follows that

Now, set

We observe that the function $E(j)$ is increasing for $j\in \mathbb{N}.$ Putting $j = 1$, we have

This completes the proof.

Theorem 2.3. Let $f_1$ and $f_2$ be analytic functions in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ of forms given in (1.1) with $a_{j, 1}$ and $a_{j, 2}$, respectively. Then

where

Proof. By Theorem 2.1, we have

From the above inequality, we obtain

Therefore, the largest $\eta$ can be obtained such that

That is,

Now, set

We observe that the function $E(j)$ is increasing for $j\in \mathbb{N}.$ Putting $j = 1$, we have

This completes the proof.

Theorem 2.4. Let $f_1, f_2\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$. Then for $\gamma \in \left[0, 1\right]$, the function $F\left(z\right) = \left(1-\gamma \right)f_1+\gamma f_2$ belongs to the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$

Proof. Since the functions $f_1$ and $f_2$ belong to the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$,

where ${\eta }_j = \left(1-\gamma \right)a_{j, 1}+\gamma a_{j, 2}$.

By (2.1), we observe that

Hence $F\left(z\right)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$

Theorem 2.5. Let $f_{s}(z) = z^{p}+\sum_{j = p+1}^{\infty} a_{j, s} z^{j}$ be in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ for $s = 1, 2, \dots, m.$ Then the function $\mathcal{P}\left(z\right) = \sum^m_{s = 1}{{\aleph }_sf_s}$, where $\sum^m_{s = 1}{{\aleph }_s = 1}$, is also in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$

Proof. By Theorem 2.1, we have

Since

Thus $\mathcal{P}\left(z\right)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p).$

3.   Radius property of the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ with differintegral operator

In this section, we investigate radii of multivalent starlikeness, multivalent convexity, and multivalent close-to-convex for the function $f(z)$ in the class $Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ with the generalized integral operator of Srivastava-Attiya.

Jung et al. ^[19] introduced an integral operator with one parameter as follows:

In 2007, Srivastava and Attiya ^[30] investigated a new integral operator, which is called Srivastava-Attiya operator, given by

Many studies are concerned with the study of the operator of Srivastava-Attiya (see ^[9,14,15,20]).

Mishra and Gochhayat ^[21] (also ^[33]) provided a fractional differintegral operator $\mathcal{J}_{u, p}^{m} f(z):\mathcal{S}(p)\to \mathcal{S}(p)$ which is called a generalized of Srivastava-Attiya integral operator, defined by

Theorem 3.1. If $f\left(z\right)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ and $0\le \tau < p$, then ${\mathcal{J}}^m_{u, p}f\left(z\right)$ in (3.1) is multivalent starlike of order $\tau$ in $\left|z\right|\le {\mathfrak{r}}_1,$ where

Proof. According to the definition of a starlike function in ^[28], we have

By (3.2), we have

By (2.1) in Theorem 2.1, it is clear that

Therefore,

This completes the proof.

Theorem 3.2. If $f\left(z\right)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ and $0\le \tau < p$, then ${\mathcal{J}}^m_{u, p}f\left(z\right)$ in (3.1) is multivalent convex of order $\tau$ in $\left|z\right|\le {\mathfrak{r}}_2,$ where

Proof. To verify (3.4), it is necessary to prove

but the result is obtained by repeating the steps in Theorem 3.1.

Corollary 3.1. If $f\left(z\right)\in Q^{\mu, \rho; k}_{\sigma; q}(\mathcal{M}, \mathcal{N}; \tau, p)$ and $0\le \tau < p$, then ${\mathcal{J}}^m_{u, p}f\left(z\right)$ in (3.1) is multivalent close-to-convex of order $\tau$ in $\left|z\right|\le {\mathfrak{r}}_3,$ where

4.   Conclusions

In this work, we established and investigated a new generalized Mittag-Leffler function, which is a generalization of $q$-Mittag-Leffler function defined by Shukla and Prajapati ^[27]. Also, we studied some of the geometric properties of a certain subclass of multivalent functions. In addition, we introduced radius theorem using a generalized Srivastava-Attiya integral operator. Since the Mittag-Leffler function is of importance, it is related to a wide range of problems in mathematical physics, engineering, and the applied sciences. The results obtained in this article may have many other applications in special functions.

Acknowledgments

The authors express many thanks to the Editor-in-Chief, handling editor, and the reviewers for their outstanding comments that improve our paper.

Conflict of interest

The authors declare that they have no competing interests concerning the publication of this article.