Differential Subordination and Superordination Results Associated with Mittag–Leffler Function

Abstract

In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension of the Mittag–Leffler function.

1. Definitions and Preliminaries

Let $\mathbb{H}$ be the class of analytic functions in the open unit disc $\mathbb{U}=\left\{z:\left|z\right|<1\right\}$. Also, let $\mathbb{H}[a,n]$ denote the subclass of the functions $f\in \mathbb{H}$ of the form

$$f\left(z\right)=a+a_n{z}^n+a_{n+1}{z}^{n+1}+\dots (a\in \mathbb{C}).$$

(1)

Furthermore, let

$$A_m=\left\{f\in \mathbb{H}|f\left(z\right)=z+a_{m+1}{z}^{m+1}+a_{m+2}{z}^{m+2}+\dots \right\}.$$

Moreover, assume that $A=A_1$ which is the subclass of the functions $f\in \mathbb{H}$ of the form

$$f\left(z\right)=z+a_2{z}^2+\dots .$$

(2)

For $f,g\in \mathbb{H}$, we say that the function f is subordinate to g, written symbolically as follows:

$$f\prec g\mathrm{or}f\left(z\right)\prec g\left(z\right),$$

if there exists a Schwarz function w, which (by definition) is analytic in $\mathbb{U}$ with $w\left(0\right)=0$ and $\left|w\left(z\right)\right|<1$ $(z\in \mathbb{U})$, such that $f\left(z\right)=g\left(w\right(z\left)\right)$ for all $z\in \mathbb{U}$. In particular, if the function g is univalent in $\mathbb{U}$, then we have the following equivalence relation (cf., e.g., [1,2]; see also [3]):

$$f\left(z\right)\prec g\left(z\right)\iff f\left(0\right)\prec g\left(0\right)andf\left(\mathbb{U}\right)\subset g\left(\mathbb{U}\right).$$

Let $\lambda$ and h be two analytic functions in $\mathbb{U}$, suppose

$$\Phi (r,s,t;z):{\mathbb{C}}^3\times \mathbb{U}\to \mathbb{C}.$$

If $\lambda$ and $\Phi (\lambda \left(z\right),z{\lambda }^{\prime }\left(z\right),z^2{\lambda }^{″}\left(z\right);z)$ are univalent functions in $\mathbb{U}$ and if $\lambda$ satisfies the second-order superordination

$$h\left(z\right)\prec \Phi (\lambda \left(z\right),z{\lambda }^{\prime }\left(z\right),z^2{\lambda }^{″}\left(z\right);z),$$

(3)

then $\lambda$ is called to be a solution of the differential superordination (3). (If f is subordinate to F, then F is superordination to f). An analytic function $\mu$ is called a subordinant of (3), if $\mu \prec \lambda$ for all the functions $\lambda$ satisfying (3). A univalent subordinant $\tilde{\mu }$ that satisfies $\mu \prec \tilde{\mu }$ for all of the subordinants $\mu$ of (3), is called the best subordinant (cf., e.g., [2], see also [3]).

Miller and Mocanu [4] obtained sufficient conditions on the functions h, $\mu$ and $\Phi$ for which the following statement holds:

$$h\left(z\right)\prec \Phi (\lambda \left(z\right),z{\lambda }^{\prime }\left(z\right),z^2{\lambda }^{″}\left(z\right);z)\Rightarrow \mu \left(z\right)\prec \lambda \left(z\right).$$

(4)

The results of Miller and Mocanu [4] and Bulboaca [5] considered certain families of first-order differential superordination whenever superordination preserves integral operators [6]. Moreover, Ali et al. [7], used Bulboaca’s results [5] and obtained the sufficient conditions for normalized analytic functions f to satisfy

$${\mu }_1\left(z\right)\prec \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec {\mu }_2\left(z\right),$$

(5)

where ${\mu }_1$ and ${\mu }_2$ are given univalent functions in $\mathbb{U}$ with ${\mu }_1\left(0\right)=1.$ Also, Shanmugam et al. [8] obtained sufficient conditions for normalized analytic functions f to satisfy

$${\mu }_1\left(z\right)\prec \frac{f\left(z\right)}{z{f}^{\prime }\left(z\right)}\prec {\mu }_2\left(z\right),$$

and

$${\mu }_1\left(z\right)\prec \frac{z^2{f}^{\prime }\left(z\right)}{{\left(f\left(z\right)\right)}^2}\prec {\mu }_2\left(z\right),$$

where ${\mu }_1$ and ${\mu }_2$ are given univalent functions in $\mathbb{U}$ with ${\mu }_1\left(0\right)=1$ and ${\mu }_2\left(0\right)=1$, while Obradovic and Owa [9] obtained some results of subordinations associated with ${\left(\frac{f\left(z\right)}{z}\right)}^{\delta }$.

Let $f\in A.$ Attiya [10] introduced the operator $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)$, where $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right):A\to A$ is defined by

$$H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)={\mu }_{\alpha ,\beta }^{\gamma ,k}*f\left(z\right)(z\in \mathbb{U}),$$

with $\beta ,\gamma \in \mathbb{C},Re\left(\alpha \right)>max\{0,Re(k)-1\}$ and $Re\left(k\right)>0$. Also, $Re\left(\alpha \right)=0$ when $Re\left(k\right)=1$; $\beta \ne 0.$ Here, ${\mu }_{\alpha ,\beta }^{\gamma ,k}$ is the generalized Mittag–Leffler function defined by [11], see also [10] and the symbol $(*)$ denotes the Hadamard product or convolution.

Due to the importance of Mittag–Leffler function, it is involved in many problems in natural and applied science.

A detailed investigation of Mittag–Leffler function has been studied by many authors see e.g., [11,12,13,14,15,16].

Attiya [10] noted that

$$H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)=z+\sum _{n=2}^{\infty }\frac{\Gamma (\gamma +nk)\Gamma (\alpha +\beta )}{\Gamma (\gamma +k)\Gamma (\beta +\alpha n)n!}{a}_n{z}^n.$$

(6)

From (6) follows (see [10])

$$z{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^{\prime }=\left(\frac{\gamma +k}{k}\right)\left(H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)\right)-\frac{\gamma }{k}\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)$$

(7)

and

$$\alpha z{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^{\prime }=(\alpha +\beta )\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)-\beta \left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right).$$

(8)

In order to derive our results, we will use the following known definitions and lemmas.

Definition 1.

Ref [4]. Denote by μ the set of all functions$f$that are analytic and injective on$\overline{\mathbb{U}}\backslash E\left(f\right)$, where

$$E\left(f\right)=\{\zeta :\zeta \in \partial \mathbb{U}and\underset{z\to \zeta }{lim}f\left(z\right)=\infty \},$$

(9)

with $f^{\prime }\left(\zeta \right)\ne 0$ for $\zeta \in \partial \mathbb{U}\backslash E\left(f\right)$.

Lemma 1.

Ref [3]. Let the function μ be univalent in the unit disc $\mathbb{U}$, and let θ and φ be analytic in a domain D containing $\mu \left(\mathbb{U}\right)$, with $\phi \left(w\right)\ne 0$ when $w\in \mu \left(\mathbb{U}\right)$. Set $\mu \left(z\right)=z{\mu }^{\prime }\left(z\right)\phi \left(\mu \left(z\right)\right)$, $h\left(z\right)=\theta \left(\mu \right(z\left)\right)+\mu \left(z\right)$ and suppose that

(i) 

μ is a starlike function in $\mathbb{U}$ (i.e, $Re\left({\displaystyle \frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}}\right)>0forz\in \mathbb{U}$),

(ii) 

$Re\left({\displaystyle \frac{z{h}^{\prime }\left(z\right)}{\mu \left(z\right)}}\right)>0forz\in \mathbb{U}$

If λ is analytic in $\mathbb{U}$ with $\lambda \left(0\right)=\mu \left(0\right)$, $\lambda \left(\mathbb{U}\right)\subseteq D$ and

$$\theta \left(\lambda \left(z\right)\right)+z{\lambda }^{\prime }\left(z\right)\phi \left(\lambda \left(z\right)\right)\prec \theta \left(\mu \left(z\right)\right)+z{\mu }^{\prime }\left(z\right)\phi \left(\mu \left(z\right)\right),$$

(10)

then $\lambda \left(z\right)\prec \mu \left(z\right)$, and μ is the best dominant.

Lemma 2.

Ref [6].Let μ be a convex univalent function in the unit disc $\mathbb{U}$ and let ϑ and φ be analytic in a domain D containing $\mu \left(\mathbb{U}\right)$. Suppose that

(i) 

$Re\left\{{\displaystyle \frac{{\vartheta }^{\prime }\left(\mu \left(z\right)\right)}{\phi \left(\mu \right(z\left)\right)}}\right\}>0$ for $z\in \mathbb{U}$;

(ii) 

$z{\mu }^{\prime }\left(z\right)\phi \left(\mu \left(z\right)\right)$ is starlike in $\mathbb{U}$.

If $\lambda \in \mathbb{H}\left[\mu \right(0),1]\cap \mu$ with $\lambda \left(\mathbb{U}\right)\subseteq D$, and $\vartheta \left(\lambda \left(z\right)\right)+z{\lambda }^{\prime }\left(z\right)\phi \left(\lambda \left(z\right)\right)$ is univalent in $\mathbb{U}$, and

$$\vartheta \left(\mu \left(z\right)\right)+z{\mu }^{\prime }\left(z\right)\phi \left(\mu \left(z\right)\right)\prec \vartheta \left(\lambda \left(z\right)\right)+z{\lambda }^{\prime }\left(z\right)\phi \left(\lambda \left(z\right)\right),$$

then $\mu \left(z\right)\prec \lambda \left(z\right)$, and μ is the best subordinant.

Lemma 3.

Ref [4]. Let μ be a convex function in $\mathbb{U}$ and let $\psi \in C$ with $\varkappa \in {\mathbb{C}}^{*}=\mathbb{C}\backslash \left\{0\right\}$ with

$$Re\left(1+{\displaystyle \frac{z{\mu }^{″}\left(z\right)}{{\mu }^{\prime }\left(z\right)}}\right)>max\left\{0;-Re\left({\displaystyle \frac{\psi }{\varkappa }}\right)\right\}\left(z\in \mathbb{U}\right).$$

If λ is analytic in $\mathbb{U}$, and

$$\psi \lambda \left(z\right)+\delta z{\lambda }^{\prime }\left(z\right)\prec \psi \mu \left(z\right)+\varkappa z{\mu }^{\prime }\left(z\right),$$

(11)

then $\lambda \left(z\right)\prec \mu \left(z\right)$, and μ is the best dominant.

Lemma 4.

Ref [17] Let μ be convex univalent in $\mathbb{U}$ and let $\delta \in C$, with $Re\left(\delta \right)>0$. If $\lambda \in \mathbb{H}\left[\mu \right(0),1]\cap \mu$ and $\lambda \left(z\right)+\delta z{\lambda }^{\prime }\left(z\right)$ is univalent in $\mathbb{U}$, then

$$\mu \left(z\right)+\delta z{\mu }^{\prime }\left(z\right)\prec \lambda \left(z\right)+\delta z{\lambda }^{\prime }\left(z\right),$$

(12)

implies

$$\mu \left(z\right)\prec \lambda \left(z\right)(z\in \mathbb{U})$$

and μ is the best subordinant.

In this paper we drive a number of interesting results concerning subordination and superordination relations for the operator $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)$. Also, some of interesting sandwich results of the operator $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)$ have been obtained.

2. Subordination and Superordination Results with ${\mathit{H}}_{\mathit{\alpha },\mathit{\beta }}^{\mathit{\gamma },\mathit{k}}\left(\mathit{f}\right)\left(\mathit{z}\right)$

Theorem 1.

Let μ be convex univalent in $\mathbb{U}$, with $\mu \left(0\right)=1$,$\rho \in {\mathbb{C}}^{*},\delta >0$. Suppose $\mu$ satisfies

$$Re\left(1+\frac{z{\mu }^{″}\left(z\right)}{{\mu }^{\prime }\left(z\right)}\right)>max\left\{0;-Re\left(\frac{\delta }{\rho }\right)\right\}.$$

(13)

If$f\in A$satisfies the following subordination relation

$$\begin{array}{cc}\hfill & {\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }+\frac{\rho (\gamma +k)}{k}{\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}-1\right).\hfill \\ \hfill & \prec \mu \left(z\right)+\frac{\rho }{\delta }z{\mu }^{\prime }\left(z\right)\hfill \end{array}$$

(14)

then

$${\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\prec \mu \left(z\right)$$

(15)

and$\mu \left(z\right)$is the best dominant of (14).

Proof. 

Define the function $\lambda$ by

$$\lambda \left(z\right)={\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }(z\in \mathbb{U}).$$

(16)

The function $\lambda$ is analytic in $\mathbb{U}$ and $\lambda \left(0\right)=1$. Differentiating the function $\lambda$ with respect to z logarithmically, we have

$$\frac{z{\lambda }^{\prime }\left(z\right)}{\lambda \left(z\right)}=\delta \left[\frac{z{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^{\prime }}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}-1\right].$$

In the resulting equation by using the identity (7), we have

$$\frac{z{\lambda }^{\prime }\left(z\right)}{\lambda \left(z\right)}=\delta \left(\frac{\gamma +k}{k}\right)\left[\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}-1\right].$$

Therefore,

$$\frac{z{\lambda }^{\prime }\left(z\right)}{\delta }=\frac{(\gamma +k)}{k}\left[\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}-1\right]{\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }.$$

It follows from (14) that

$$\lambda \left(z\right)+\frac{\rho }{\delta }z{\lambda }^{\prime }\left(z\right)\prec \mu \left(z\right)+\frac{\rho }{\delta }z{\mu }^{\prime }\left(z\right).$$

Thus, an application of Lemma 3 with $\psi =1$ and $\varkappa =\frac{\rho }{\delta },$ we obtain (15). □

In view of (8), and by using the similar method of proof the Theorem 1, we get the proof of Theorem 2.

Theorem 2.

Let μ be convex univalent in $\mathbb{U}$, with $\mu \left(0\right)=1$,$\rho \in {\mathbb{C}}^{*},\delta >0$. Suppose $\mu$ satisfies (13). If $f\in A$ satisfies the subordination

$$\begin{array}{cc}\hfill & {\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }+\frac{\rho (\alpha +\beta )}{\alpha }{\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}-1\right).\hfill \\ \hfill & \prec \mu \left(z\right)+\frac{\rho }{\delta }z{\mu }^{\prime }\left(z\right)\hfill \end{array}$$

(17)

then

$${\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\prec \mu \left(z\right)$$

(18)

and $\mu \left(z\right)$ is the best dominant of (18).

Theorem 3.

Let ${\zeta }_i\in \mathbb{C}(i=1,2,3,4),\delta >0,\xi >0$ (ξ is a real number) and μ be convex univalent in $\mathbb{U}$, with $\mu \left(0\right)=1$,$\mu \left(z\right)\ne 0(z\in \mathbb{U})$ and assume that $\mu$ satisfies

$$\Re \left\{1+\frac{{\zeta }_2}{\xi }\mu \left(z\right)+\frac{2{\zeta }_3}{\xi }{\mu }^2\left(z\right)+\frac{3{\zeta }_4}{\xi }{\mu }^3\left(z\right)+\frac{z{\mu }^{″}\left(z\right)}{{\mu }^{\prime }\left(z\right)}-\frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}\right\}>0.$$

(19)

Suppose that $\frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}$ is starlike univalent in $\mathbb{U}$. Also, if $f\in A$ satisfies the following subordination relation:

$$\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)\prec {\zeta }_1+{\zeta }_2\mu \left(z\right)+{\zeta }_3{\mu }^2\left(z\right)+{\zeta }_4{\mu }^3\left(z\right)+\xi \frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)},$$

(20)

where

$$\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)\prec {\zeta }_1+{\zeta }_2{\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }+{\zeta }_3{\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{2\delta }$$

$$\begin{array}{cc}\hfill & +{\zeta }_4{\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{3\delta }+\xi \delta \frac{(\gamma +k)}{k}\left[\frac{H_{\alpha ,\beta }^{\gamma +2,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}-\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right]\hfill \\ \hfill & +\frac{\xi \delta }{k}\left[\frac{H_{\alpha ,\beta }^{\gamma +2,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}-1\right],\hfill \end{array}$$

(21)

then

$${\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }\prec \mu \left(z\right)$$

and $\mu \left(z\right)$ is the best dominant of (20).

Proof. 

Define the function $\lambda$ by

$$\lambda \left(z\right)={\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }(z\in \mathbb{U}).$$

(22)

The function $\lambda$ is analytic in $\mathbb{U}$ and we note that $\lambda \left(0\right)=1$.

After some computation and using (7), we have

$${\zeta }_1+{\zeta }_2\lambda \left(z\right)+{\zeta }_3{\lambda }^2\left(z\right)+{\zeta }_4{\lambda }^3\left(z\right)+\xi \frac{z{\lambda }^{\prime }\left(z\right)}{\lambda \left(z\right)}=\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z),$$

(23)

where $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is given by (21).

From (20) and (23) we obtain

$${\zeta }_1+{\zeta }_2\lambda \left(z\right)+{\zeta }_3{\lambda }^2\left(z\right)+{\zeta }_4{\lambda }^3\left(z\right)+\xi \frac{z{\lambda }^{\prime }\left(z\right)}{\lambda \left(z\right)}\prec {\zeta }_1+{\zeta }_2\mu \left(z\right)+{\zeta }_3{\mu }^2\left(z\right)+{\zeta }_4{\mu }^3\left(z\right)+\xi \frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}.$$

By setting

$$\theta \left(w\right)={\zeta }_1+{\zeta }_{2}w+{\zeta }_3{w}^2+{\zeta }_4{w}^{3}and\varphi \left(w\right)=\frac{\xi }{w},w\ne 0,$$

we see that $\theta$ is analytic in the complex plane $\mathbb{C}$ and $\varphi$ is analytic in ${\mathbb{C}}^{*},$ also, $\varphi \left(w\right)\ne 0$, $w\in {\mathbb{C}}^{*}$. Moreover

$$\mu \left(z\right)=z{\mu }^{\prime }\left(z\right)\varphi \left(\mu \left(z\right)\right)=\xi \frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}$$

and

$$h\left(z\right)=\theta \left(\mu \left(z\right)\right)+\mu \left(z\right)={\zeta }_1+{\zeta }_2\mu \left(z\right)+{\zeta }_3{\mu }^2\left(z\right)+{\zeta }_4{\mu }^3\left(z\right)+\xi \frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}.$$

It is clear that $\mu \left(z\right)$ is starlike univalent in $\mathbb{U},$

$$Re\left({\displaystyle \frac{z{h}^{\prime }\left(z\right)}{\mu \left(z\right)}}\right)=Re\left\{1+\frac{{\zeta }_2}{\xi }\mu \left(z\right)+\frac{2{\zeta }_3}{\xi }{\mu }^2\left(z\right)+\frac{3{\zeta }_4}{\xi }{\mu }^3\left(z\right)+\frac{z{\mu }^{″}\left(z\right)}{{\mu }^{\prime }\left(z\right)}-\frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}\right\}>0.$$

Thus, from Lemma 1, we have $\lambda \left(z\right)\prec \mu \left(z\right).$ By using (22), we obtain the required result. □

In view of (8), and by using the similar method of proof of Theorem 3, we get the proof of Theorem 4

Theorem 4.

Let ${\zeta }_i\in \mathbb{C}(i=1,2,3,4),\delta >0,\xi >0$ (ξ is a real number) and μ be convex univalent function in $\mathbb{U}$, with $\mu \left(0\right)=1$,$\mu \left(z\right)\ne 0(z\in \mathbb{U})$ and assume that the function $\mu$ satisfies (19). Also, let $\frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}$ be starlike univalent in $\mathbb{U}$. If $f\in A$ satisfies (20), where

$$\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)\prec {\zeta }_1+{\zeta }_2{\left(\frac{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }+{\zeta }_3{\left(\frac{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{2\delta }$$

$$\begin{array}{cc}\hfill & +{\zeta }_4{\left(\frac{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{3\delta }+\xi \delta \frac{\alpha +\beta }{\alpha }\left[\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}-\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right]\hfill \\ \hfill & +\frac{\xi \delta }{\alpha }\left[\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}-1\right],\hfill \end{array}$$

(24)

then

$${\left(\frac{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }\prec \mu \left(z\right)$$

and$\mu \left(z\right)$is the best dominant of (20).

Theorem 5.

Let ${\zeta }_i\in \mathbb{C}(i=1,2,3,4),\xi >0$(ξ is a real number) and μ be convex univalent in $\mathbb{U}$, with $\mu \left(0\right)=1$,$\mu \left(z\right)\ne 0(z\in \mathbb{U})$ and assume that $\mu$ satisfies (19). Also, if $\frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}$ is starlike univalent in $\mathbb{U}$. Moreover, if $f\in A$ satisfies (20), where

$$\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)\prec {\zeta }_1+{\zeta }_2\frac{z{H}_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}+{\zeta }_3\frac{z^2{\left(H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)\right)}^2}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^4}$$

$$\begin{array}{cc}\hfill & +{\zeta }_4\frac{z^3{\left(H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)\right)}^3}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^6}+\xi \frac{\gamma +k}{k}\left[1+\frac{H_{\alpha ,\beta }^{\gamma +2,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}-2\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right]\hfill \\ \hfill & +\frac{\xi }{k}\left[\frac{H_{\alpha ,\beta }^{\gamma +2,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}-1\right],\hfill \end{array}$$

(25)

then

$$\frac{z{H}_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}\prec \mu \left(z\right)$$

and$\mu \left(z\right)$is the best dominant of (20).

Proof. 

Define the function $\lambda$ by

$$\lambda \left(z\right)=\frac{z{H}_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}(z\in \mathbb{U}).$$

(26)

Then the function $\lambda$ is analytic in $\mathbb{U}$ and $\lambda \left(0\right)=1$.

We note that

$${\zeta }_1+{\zeta }_2\lambda \left(z\right)+{\zeta }_3{\lambda }^2\left(z\right)+{\zeta }_4{\lambda }^3\left(z\right)+\xi \frac{z{\lambda }^{\prime }\left(z\right)}{\lambda \left(z\right)}=\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z),$$

(27)

where $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is given by (25).

From (20) and (27) we obtain

$${\zeta }_1+{\zeta }_2\lambda \left(z\right)+{\zeta }_3{\lambda }^2\left(z\right)+{\zeta }_4{\lambda }^3\left(z\right)+\xi \frac{z{\lambda }^{\prime }\left(z\right)}{\lambda \left(z\right)}\prec {\zeta }_1+{\zeta }_2\mu \left(z\right)+{\zeta }_3{\mu }^2\left(z\right)+{\zeta }_4{\mu }^3\left(z\right)+\xi \frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}.$$

The remaining part of the proof of Theorem 5 is similar to that of Theorem 3 and hence we omit it. □

In view of (8), and by using the similar method of proof of Theorem 5, we get the proof Theorem 6.

Theorem 6.

Let ${\zeta }_i\in \mathbb{C}(i=1,2,3,4),\xi >0;real$ and μ be convex univalent function in $\mathbb{U}$, with $\mu \left(0\right)=1$,$\mu \left(z\right)\ne 0(z\in \mathbb{U})$ and assume that $\mu$ satisfies (19). Also, let $\frac{z{\mu }^{\prime }\left(z\right)}{\mu \left(z\right)}$ be starlike univalent in $\mathbb{U}$. If $f\in A$ satisfies (20), where

$$\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)\prec {\zeta }_1+{\zeta }_2\frac{z{H}_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}+{\zeta }_3\frac{z^2{\left(H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}{{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^4}$$

$$\begin{array}{cc}\hfill & +{\zeta }_4\frac{z^3{\left(H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^3}{{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^6}+\xi \frac{\alpha +\beta }{\alpha }\left[\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}-2\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right]\hfill \\ \hfill & +\frac{\xi }{\alpha }\left[\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}-1\right],\hfill \end{array}$$

(28)

then

$$\frac{z{H}_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}\prec \mu \left(z\right)$$

and$\mu \left(z\right)$is the best dominant of (20).

Remark 1.

Superordination results associated with $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)$ can be done analogously by using Lemmas 2 and 4.

3. Sandwich Results

Combining results of differential subordinations and superordinations, we get the following sandwich theorem.

Theorem 7.

Let ${\mu }_1$ and ${\mu }_2$ be convex univalent in $\mathbb{U}$, with ${\mu }_1\left(0\right)={\mu }_2\left(0\right)=1$. Suppose ${\mu }_2$ satisfies (13),$\delta >0$ and $Re\left\{\rho \right\}>0.$ Let $f\in A$ satisfies

$${\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\in \mathbb{H}[1,1]\cap \mu$$

and

$${\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }+\frac{\rho (\gamma +k)}{k}{\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}-1\right)$$

be univalent in$\mathbb{U}$. If

$$\begin{array}{cc}\hfill {\mu }_1\left(z\right)+\frac{\rho }{\delta }z{\mu }_1^{\prime }\left(z\right)& \prec {\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }+\frac{\rho (\gamma +k)}{k}{\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}-1\right)\hfill \\ \hfill & \prec {\mu }_2\left(z\right)+\frac{\rho }{\delta }z{\mu }_2^{\prime }\left(z\right)\hfill \end{array}$$

then

$${\mu }_1\left(z\right)\prec {\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\prec {\mu }_2\left(z\right)$$

and${\mu }_1$ and ${\mu }_2$ are respectively the best subordinate and best dominant.

Theorem 8.

Let ${\mu }_1$ and ${\mu }_2$ be convex univalent in $\mathbb{U}$, with ${\mu }_1\left(0\right)={\mu }_2\left(0\right)=1$. Suppose ${\mu }_2$ satisfies (13),$\delta >0$ and $Re\left\{\rho \right\}>0.$ Let $f\in A$ satisfies

$${\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\in \mathbb{H}[1,1]\cap \mu$$

and

$$(1-\rho \frac{\beta }{\alpha }){\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }+\frac{\rho (\beta +\alpha )}{\alpha }{\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right)$$

be univalent in$\mathbb{U}$. If

$$\begin{array}{cc}\hfill {\mu }_1\left(z\right)+\frac{\rho }{\delta }z{\mu }_1^{\prime }\left(z\right)& \prec (1-\rho \frac{\beta }{\alpha }){\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }+\frac{\rho (\beta +\alpha )}{\alpha }{\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\left(\frac{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}\right)\hfill \\ \hfill & \prec {\mu }_2\left(z\right)+\frac{\rho }{\delta }z{\mu }_2^{\prime }\left(z\right),\hfill \end{array}$$

then

$${\mu }_1\left(z\right)\prec {\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{z}\right)}^{\delta }\prec {\mu }_2\left(z\right)$$

and${\mu }_1$ and ${\mu }_2$ are respectively the best subordinate and best dominant.

Theorem 9.

Let ${\mu }_1$ and ${\mu }_2$ be convex univalent functions in $\mathbb{U}$, with ${\mu }_1\left(0\right)={\mu }_2\left(0\right)=1$. Suppose ${\mu }_1$ satisfies

$$\Re \left\{\frac{{\zeta }_2}{\xi }{\mu }_1\left(z\right)+\frac{2{\zeta }_3}{\xi }{\mu }_1^2\left(z\right)+\frac{3{\zeta }_4}{\xi }{\mu }_1^3\left(z\right)\right\}>0.$$

(29)

and ${\mu }_2$ satisfies (19). Let $f\in A$ satisfies ${\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }\in \mathbb{H}[1,1]\cap \mu ,$ and $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is univalent in $\mathbb{U},$ where $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is given by (21). If

$$\begin{array}{cc}\hfill {\zeta }_1+{\zeta }_2{\mu }_1\left(z\right)+{\zeta }_3{\mu }_1^2\left(z\right)+{\zeta }_4{\mu }_1^3\left(z\right)+\xi \frac{z{\mu }_1^{\prime }\left(z\right)}{{\mu }_1\left(z\right)}& \prec \Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)\hfill \\ \hfill & \prec {\zeta }_1+{\zeta }_2{\mu }_2\left(z\right)+{\zeta }_3{\mu }_2^2\left(z\right)+{\zeta }_4{\mu }_2^3\left(z\right)+\xi \frac{z{\mu }_2^{\prime }\left(z\right)}{{\mu }_2\left(z\right)},\hfill \end{array}$$

(30)

then

$${\mu }_1\left(z\right)\prec {\left(\frac{H_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }\prec {\mu }_2\left(z\right)$$

and${\mu }_1$ and ${\mu }_2$ are respectively the best subordinate and best dominant.

Theorem 10.

Let ${\mu }_1$ and ${\mu }_2$ be convex univalent in $\mathbb{U}$, with ${\mu }_1\left(0\right)={\mu }_2\left(0\right)=1$. Suppose ${\mu }_1$ satisfies (29), and ${\mu }_2$ satisfies (19). Let $f\in A$ satisfies ${\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }\in \mathbb{H}[1,1]\cap \mu$ and $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is univalent in $\mathbb{U},$ where $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is given by (24). If (30) has been satisfied,

then

$${\mu }_1\left(z\right)\prec {\left(\frac{H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)}{H_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}\right)}^{\delta }\prec {\mu }_2\left(z\right)$$

and ${\mu }_1$ and ${\mu }_2$ are respectively the best subordinate and best dominant.

Theorem 11.

Let ${\mu }_1$ and ${\mu }_2$ be convex univalent in $\mathbb{U}$, with ${\mu }_1\left(0\right)={\mu }_2\left(0\right)=1$. Suppose ${\mu }_1$ satisfies (29), and ${\mu }_2$ satisfies (19). Let $f\in A$ satisfies $\frac{z{H}_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}\in \mathbb{H}[1,1]\cap \mu$ and $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is univalent in $\mathbb{U},$ where $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is given by (25). If (30) has been satisfied,

then

$${\mu }_1\left(z\right)\prec \frac{z{H}_{\alpha ,\beta }^{\gamma +1,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}\prec {\mu }_2\left(z\right)$$

and ${\mu }_1$ and ${\mu }_2$ are respectively the best subordinate and best dominant.

Theorem 12.

Let ${\mu }_1$ and ${\mu }_2$ be convex univalent in $\mathbb{U}$, with ${\mu }_1\left(0\right)={\mu }_2\left(0\right)=1$. Suppose ${\mu }_1$ satisfies (29), and ${\mu }_2$ satisfies (19). Let $f\in A$ satisfies $\frac{z{H}_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}\in \mathbb{H}[1,1]\cap \mu$ and $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is univalent in $\mathbb{U},$ where $\Omega ({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,\xi ,\delta ,\gamma ,k,\alpha ,\beta ;z)$ is given by (28). If (30) has been satisfied,

then

$${\mu }_1\left(z\right)\prec \frac{z{H}_{\alpha ,\beta +2}^{\gamma ,k}\left(f\right)\left(z\right)}{{\left(H_{\alpha ,\beta +1}^{\gamma ,k}\left(f\right)\left(z\right)\right)}^2}\prec {\mu }_2\left(z\right)$$

and ${\mu }_1$ and ${\mu }_2$ are respectively the best subordinate and best dominant.

Remark 2.

By specifying the function Ω and selecting the particular values of $\alpha ,\beta ,\gamma$ and k we can derive a number of known results. Some of them are given below.

(i) 

If we put $\gamma =k=1$ and $\alpha =0$ in Theorem 1, we obtain the results obtained by Murugusundaramoorthy and Magesh ([18], Corollary 3.3),

(ii) 

If we put $\gamma =k=1$ and $\alpha =0$ in Theorem 7 we obtain the results obtained by Raducanu and Nechita ([19], Corollary 3.10 ).

4. Conclusions

We obtained a number of interesting results concerning subordination and superordination relations for the operator $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)$ of analytic functions associated with an extension of the Mittag–Leffler function in the open unit disk $\mathbb{U}$. Also, some of interesting sandwich results of the operator $H_{\alpha ,\beta }^{\gamma ,k}\left(f\right)\left(z\right)$ have been obtained.