Duality on q-Starlike Functions Associated with Fractional q-Integral Operators and Applications

Abstract

In this paper, we make use of the Riemann–Liouville fractional q-integral operator to discuss the class $S_{q,\delta }^{*}\left(\alpha \right)$ of univalent functions for $\delta >0,\alpha \in \mathbb{C}-\left\{0\right\}$, and $0<\left|q\right|<1$. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q-difference operator. Moreover, we derive the normalized classes ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )$ and ${\mathcal{P}}_{\delta ,q}\left(\beta \right)$ ($0<\left|q\right|<1$, $\delta \ge 0,0\le \beta \le 1,\zeta >0)$ of analytic functions on a unit disc and provide conditions for the parameters $q,\delta ,\zeta ,\beta$, and $\gamma$ so that ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )\subset S_{q,\delta }^{*}\left(\alpha \right)$ and ${\mathcal{P}}_{\delta ,q}\left(\beta \right)\subset S_{q,\delta }^{*}\left(\alpha \right)$ for $\alpha \in \mathbb{C}-\left\{0\right\}$. Finally, we also propose an application to symmetric q-analogues and Ruscheweh’s duality theory.

1. Introduction

In recent decades, the theory of q-calculus has been applied to various areas of science and computational mathematics. The concept of q-calculus was used in quantum groups, q-deformed super algebras, q-transform analysis, q-integral calculus, optimal control, and many other fields, to mention but a few [1,2,3,4]. Soon after the concept of q-calculus was furnished, many basic q-hypergeometric functions, q-hypergeometric symmetric functions, and q-hypergeometric and hypergeometric symmetric function polynomials were discussed in geometric function theory [5]. Jackson [6] was the first to introduce and analyze the q-derivative and the q-integral operator. Later, various researchers applied the concept of the q-derivative to various sub-collections of univalent functions. Srivastava [7] used the q-derivative operator to describe some properties of a subclass of univalent functions. Agrawal et al. [8] extended a class of q-starlike functions to certain subclasses of q-starlike functions. Kanas et al. [9] used convolutions to define a q-analogue of the Ruscheweyh operator and studied some useful applications of their operator. Srivastava et al. [10] defined the q-Noor integral operator by following the concept of convolution. Purohit [11] introduced a subclass of univalent functions by using a certain operator of a fractional q-derivative. Aouf et al. [12] employed subordination results to discuss analytic functions associated with a new fractional q-analogue of certain operators. However, many extensions of different operators can be found in [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] and the references cited therein. Here, we will make use of definitions and notations used in the literature [30,31]. For $a,q\in \mathbb{C}$, the q-analogue of the Pochhammer symbol is defined by

$$\begin{array}{c}\hfill {(a;q)}_k=\left\{\begin{array}{cc}\prod _{j=0}^{k-1}(1-a{q}^j),\hfill & \mathrm{if}k>0,\hfill \\ 1,\hfill & \mathrm{if}k=0,\hfill \\ \prod _{j=0}^{\infty }(1-a{q}^j),\hfill & \mathrm{if}k\to \infty ,\hfill \end{array}\right.\end{array}$$

and, hence, it is very natural to write ${(a;q)}_k=\frac{{(a;q)}_{\infty }}{{(a{q}^k,q)}_{\infty }},(k\in \mathbb{N}\cup \{\infty \}).$ The extension of the Pochhammer symbol to a real number $\delta$ is given as

$$\begin{array}{c}\hfill {(a;q)}_{\delta }=\frac{{(a;q)}_{\infty }}{{(a{q}^{\delta };q)}_{\infty }},(\delta \in \mathbb{R}).\end{array}$$

Therefore, for any real number $\delta >0$, the q-analogue of the gamma function is defined by

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_q\left(\alpha \right)=\frac{{(q;q)}_{\infty }}{(q^{\delta };q)\infty }{(1-q)}^{1-\delta }.\end{array}$$

The q-analogue of the natural number n and the multiple q-shifted factorial for complex numbers $a_1,\cdots ,a_k$ are, respectively, defined by

$$\begin{array}{c}\hfill {\left[n\right]}_q=\frac{1-q^n}{1-q},0<\left|q\right|<1,\mathrm{and}{(a_1,\cdots ,a_k;q)}_n=\prod _{j=1}^k{(a_j;q)}_n.\end{array}$$

Let $a_1,...,a_r,b_1,...b_s$ be complex numbers; then, the q-hypergeometric series ${}_r{\varphi }_s$ is denoted as

$$\begin{array}{c}\hfill {}_r{\varphi }_s\left(\begin{array}{ccc}{a}_1,& \cdots ,& a_r\\ b_1,& \cdots ,& b_s\end{array};q,z\right)=\sum _{n=0}^{\infty }\frac{(a_1,\cdots ,a_r;q)n}{(b_1,\cdots ,b_s;q)n}{z}^n{\left(-q^{\frac{n-1}{2}}\right)}^{n(s-r+1)}.\end{array}$$

(1)

It is clear that the series representation of the function ${}_r{\varphi }_s$ converges absolutely for all $z\in \mathbb{C}$ if $r\le s$ and converges only for $\left|z\right|<1$ if $r=s+1$. Now, let $\mathcal{A}$ be the collection of all analytic functions in the open unit disc $\mathcal{U}=\{z\in \mathbb{C};|z|<1\}$ expressed in the normalized form

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,$$

(2)

and let ${\mathcal{A}}_0$ be a collection comprising all functions g such that $zg\in \mathcal{A}$ and $g\left(0\right)=1$, $z\in \mathbb{C}$. Then, the sub-collection of $\mathcal{A}$ of functions that are univalent in $\mathcal{U}$ is denoted by S. However, in geometric function theory, a variety of sub-collections of univalent functions have been discussed. See the monographs published by [32,33] for details.

Let us consider the Riemann–Liouville fractional q-integral operator of a non-integer of order $\delta$ defined by [34]

$$\begin{array}{c}\hfill I_q^{\delta }f\left(z\right)=\frac{1}{{\mathsf{\Gamma }}_q\left(\delta \right)}{\int }_0^z{(x-\left[qt\right])}^{\delta -1}f\left(t\right)d_{q}t.\end{array}$$

(3)

Then, $I_q^{\delta }f⟶I_q$ when $\delta ⟶1$, where $I_{q}f$ is the q-Jackson integral defined by [6]

$$\begin{array}{c}\hfill I_{q}f\left(z\right)={\int }_0^{z}f\left(t\right)d_q\left(t\right)z\in \mathcal{U},z\ne 0,\left|q\right|<1.\end{array}$$

With the concept of the Riemann–Liouville fractional q-integral of the non-integer order $\delta$, we recall some rules associated with $I_q^{\delta }$ by (3):

(i)

$I_q^{\delta }\left(cf\right)=c{I}_q^{\delta }f,c\in \mathbb{C}-\left\{0\right\},f\in \mathcal{A},$

(ii)

$I_q^{\delta }(f+g)=I_q^{\delta }f+I_q^{\delta }g,f,g\in \mathcal{A},$

(iii)

$I_q^{\delta }\left|f\right|\le |I_q^{\delta }f|.$

Agarwal [34] defined the q-analogue difference operator of a non-integer order $\delta$ as follows:

$$\begin{array}{c}\hfill D_q^{\delta }f\left(z\right)=\frac{1}{{(1-q)}^{\delta }{z}^{\delta }}\sum _{n=0}^{\infty }\frac{{(q^{-\delta };q)}_n}{{(q;q)}_n}{q}^{n}f\left(q^{n}z\right).\end{array}$$

(4)

Note that $D_q^{\delta }f⟶D_{q}f$ when $\delta ⟶1$. $D_{q}f$ is the q-derivative of the function f introduced in [6] in the subsequent form:

$$\begin{array}{c}\hfill D_{q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{z(1-q)},z\in \mathcal{U},z\ne 0,\left|q\right|<1.\end{array}$$

(5)

Thus, for $n\in \mathbb{N}$, through simple computations, we obtain

$$\begin{array}{c}\hfill D_q^{\delta }{z}^n=\frac{z^{n-\delta }}{{(1-q)}^{\delta }}\frac{{(q^{1+n-\delta };q)}_{\infty }}{{(q^{1+n};q)}_{\infty }}\mathrm{and}{I}_q^{\delta }{z}^n=\frac{{(q^{n+1+\delta };q)}_{\infty }}{{(q^{n+1};q)}_{\infty }}{z}^{n+\delta }.\end{array}$$

Let $0<\left|q\right|<1$, $\delta \ge 0$$\zeta >0,0\le \beta \le 1$, and $0<\gamma \le 1$. By the definition of the q-analogue difference operator with the non-integer order $\delta$, the following rules of $D_q^{\delta }$ hold:

(i)

$D_q^{\delta }\left(cf\right)=c{D}_q^{\delta }f,c\in \mathbb{C}-\left\{0\right\},f\in \mathcal{A},$

(ii)

$D_q^{\delta }(f+g)=D_q^{\delta }f+D_q^{\delta }g,f,g\in \mathcal{A}.$

We define ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )$ as the class of all functions $f\in \mathcal{A}$ satisfying the following condition:

$$Re\left\{\frac{{(1-q)}^{\delta }\left(D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)+\frac{1-\gamma }{\zeta \gamma }z{D}_q^{\delta +2}{I}_q^{\delta }f\left(z\right)\right)-\beta }{1-\beta }\right\}>0,\left|z\right|<1.$$

For $0<\left|q\right|<1$, $\delta \ge 0$, and $0\le \beta \le 1$, the class ${\mathcal{P}}_{\delta ,q}\left(\beta \right)$ consists of functions satisfying the following condition:

$$Re\left\{\frac{{(1-q)}^{\delta }\left(D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)+qz{D}_q^{\delta +2}{I}_q^{\delta }f\left(z\right)\right)-\beta }{1-\beta }\right\}>0,\left|z\right|<1.$$

Now, for two functions

$$f\left(z\right)=\sum _{n=0}^{\infty }{a}_n{z}^n\mathrm{and}g\left(z\right)=\sum _{n=0}^{\infty }{b}_n{z}^n,$$

we recall the convolution (or the Hadamard product) of f and g, denoted by $f\ast g$, which is given by

$$\left(f\ast g\right)\left(z\right)=\sum _{n=0}^{\infty }{a}_n{b}_n{z}^n,z\in \mathcal{U}.$$

For a set $\mathcal{V}\subseteq {\mathcal{A}}_0$, the dual set ${\mathcal{V}}^{*}$ is defined by

$${\mathcal{V}}^{*}=\{g\in {\mathcal{A}}_0:\left(f\ast g\right)\left(z\right)\ne 0,\forall f\in \mathcal{V},z\in \mathcal{U}\}.$$

However, the second dual of $\mathcal{V}$ is defined as ${\mathcal{V}}^{**}={\left({\mathcal{V}}^{*}\right)}^{*}$. However, $\mathcal{V}\subseteq {\mathcal{V}}^{**}$. For basic reference to this theory, we may refer to the book by Ruscheweyh [35] (see also [36,37,38]).

In this paper, we define the class $S_{q,\delta }^{*}$ for $\delta >0,0<\left|q\right|<1$, and establish the convolution condition of this class. Furthermore, we find conditions for $q,\delta ,\zeta ,\beta$, and $\gamma$ so that ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )\subset S_{q,\delta }^{*}\left(\alpha \right)$ and ${\mathcal{P}}_{\delta ,q}\left(\beta \right)\subset S_{q,\delta }^{*}\left(\alpha \right)$.

2. Preliminary Lemmas

The following lemmas are very useful in our investigation.

Lemma 1

(Duality principle; see [35]). Let $\mathcal{V}\subseteq {\mathcal{A}}_0$ be compact; it has the following property:

$$f\in \mathcal{V}⟹\forall \left|x\right|\le 1:f_x\in \mathcal{V},$$

(6)

where $f_x\left(z\right)=f\left(xz\right)$. Then,

$$\phi \left(\mathcal{V}\right)=\phi \left({\mathcal{V}}^{**}\right),$$

for all continuous linear functionals φ on $\mathcal{A}$, and

$$\overline{co}\left(\mathcal{V}\right)\subseteq \overline{co}\left({\mathcal{V}}^{**}\right),$$

where $\overline{co}$ stands for the closed convex hull of a set.

Lemma 2

([35]). Let $0\le \gamma <1$ and $\beta \in \mathbb{R},\beta \ne 1$. If

$$V_{\beta ,\gamma }=\left\{\gamma (1-\beta )\frac{1+xz}{1-xz}+(1-\gamma )(1-\beta )\frac{1+yz}{1-yz}+\beta ,\left|x\right|=\left|y\right|=1,z\in \mathcal{U}\right\},$$

(7)

then

$$\begin{array}{c}\hfill V_{\beta ,\gamma }^{*}=\left\{f\in {\mathcal{A}}_0:\exists \zeta \in \mathbb{R},Re\left\{g\left(z\right)-\frac{1-2\beta }{2(1-\beta )}\right\}>0,g\left(z\right)=f_x\left(z\right),\left|x\right|\le 1\right\},\end{array}$$

and

$$\begin{array}{c}\hfill V_{\beta ,\gamma }^{**}=\left\{f\in {\mathcal{A}}_0;Re\left\{\frac{g\left(z\right)-\beta }{1-\beta }\right\}>0,g\left(z\right)=f_x\left(z\right),\left|x\right|\le 1\right\}.\end{array}$$

We see that the set $V_{\beta ,\gamma }$ in (7) does not satisfy the property (6), i.e., if $f\in V_{\beta ,\gamma }$, then $f\left(xz\right)\in V_{\beta ,\gamma }$ for all $\left|x\right|\le 1$, as is required in the Duality Principle. However, the Duality Principle can be stated with a slightly weaker but more complicated condition that $V_{\beta ,\gamma }$ can be seen to satisfy (see [35] for more details).

3. Main Results

Definition 1.

Let $f\in \mathcal{A}$, $\delta >0$, and $\alpha \in \mathbb{C}-\left\{0\right\}$. Then, a function f is said to be in the class $S_{q,\delta }^{*}\left(\alpha \right)$ if it satisfies the following inequality:

$$\begin{array}{c}\hfill Re\left\{1+\frac{1}{\alpha }\left(\frac{z{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)}{f\left(z\right)}-\frac{1}{{(1-q)}^{\delta }}\right)\right\}>0,\end{array}$$

where the operators $D_q^{\delta }$ and $I_q^{\delta }$ are given by (4) and (3), respectively.

Putting $\delta =0$ into Definition 1 leads to the following definition.

Definition 2.

The function $f\in \mathcal{A}$ is said to be in the class of q-starlike functions of order α, $S_q^{*}\left(\alpha \right)$, if it satisfies the following inequality:

$$\begin{array}{c}\hfill Re\left\{1+\frac{1}{\alpha }\left(\frac{z{D}_{q}f\left(z\right)}{f\left(z\right)}-1\right)\right\}>0,\alpha \in \mathbb{C}-\left\{0\right\},\end{array}$$

where $D_{q}f\left(z\right)$ is given by (5).

Theorem 1.

Let $f\in \mathcal{A}$, $\delta >0$, $\alpha \in \mathbb{C}-\left\{0\right\}$, and $\left|z\right|<R<1$. Then, $f\in S_{q,\delta }^{*}\left(\alpha \right)$ if and only if

$$\begin{array}{c}\hfill \frac{f\left(z\right)}{z}\ast \frac{1+qz\left(\frac{x+1}{2\alpha {(1-q)}^{\delta }}-1\right)}{(1-z)(1-qz)}\ne 0,\end{array}$$

where $\left|x\right|=1$ and $x\ne -1$.

Proof. 

Since $\frac{z{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)}{f\left(z\right)}-\frac{1}{{(1-q)}^{\delta }}=0$ at $z=0$, we have

$$\begin{array}{c}\hfill 1+\frac{1}{\alpha }\left(\frac{z{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)}{f\left(z\right)}-\frac{1}{{(1-q)}^{\delta }}\right)\ne \frac{x-1}{x+1},\left|x\right|=1,x\ne -1.\end{array}$$

By following simple computations, we can rewrite this as

$$\begin{array}{c}\hfill (x+1){(1-q)}^{\delta }z{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)-\left(2\alpha {(1-q)}^{\delta }-x-1\right)f\left(z\right)\ne 0.\end{array}$$

(8)

Since the function f satisfies (2), we obtain

$$\begin{array}{c}\hfill z{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)=\frac{1}{{(1-q)}^{\delta }}\left(z+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{z}^n\right)=\frac{1}{{(1-q)}^{\delta }}\left(f\left(z\right)\ast \frac{z}{(1-z)(1-qz)}\right).\end{array}$$

Now, as Equation (8) is equivalent to

$$\begin{array}{c}\hfill \left(f\left(z\right)\ast \frac{(x+1)z}{(1-z)(1-qz)}\right)+\left(f\left(z\right)\ast \frac{z(2\alpha {(1-q)}^{\delta }-x-1)}{1-z}\right)\ne 0,\end{array}$$

it simplifies to

$$\begin{array}{c}\hfill f\left(z\right)\ast \frac{(x+1)z+z(1-qz)(2\alpha {(1-q)}^{\delta }-x-1)}{(1-z)(1-qz)}\ne 0.\end{array}$$

Hence, the required result has been proven. □

Putting $\delta =0$ into Theorem 1, we get the following corollary.

Corollary 1.

Let $\alpha \in \mathbb{C}-\left\{0\right\}$, $\left|x\right|=1$, and $x\ne -1$. Then, the function f is a q-starlike function of order α if and only if

$$\begin{array}{c}\hfill \frac{f\left(z\right)}{z}\ast \frac{1+qz\left(\frac{x+1}{2\alpha }-1\right)}{(1-z)(1-qz)}\ne 0,\left|z\right|<R\le 1.\end{array}$$

(9)

Theorem 2.

Let $\delta >0$, $0<q<1$, $\alpha \in \mathbb{C}-\left\{0\right\}$, $\zeta >0$, $0\le \beta <1$, $0<\gamma <1$, and $\left|x\right|=1$ with $x\ne -1$. Then, ${\mathcal{P}}_{q,\delta }^{\zeta }(\beta ,\gamma )\subseteq S_{q,\delta }^{*}\left(\alpha \right)$ if and only if

$$Re\left\{F(x,z)\right\}>-\frac{{(1-q)}^{\delta }}{\zeta \gamma (1-\beta )},$$

(10)

where

$$F(x,z)=\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{(x+1){[n+1]}_q+2\alpha {(1-q)}^{\delta }-(x+1)}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}{z}^n,\left|z\right|<R\le 1,\left|z\right|<R\le 1.$$

(11)

Proof. 

Let the function f be in the class ${\mathcal{P}}_{q,\delta }^{\zeta }(\beta ,\gamma ),\left|z\right|<R\le 1$. If we denote

$$\begin{array}{c}\hfill g\left(z\right)={(1-q)}^{\delta }\left(D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)+\frac{1-\gamma }{\zeta \gamma }z{D}_q^{\delta +2}{I}_q^{\delta }f\left(z\right)\right),\end{array}$$

then we have $g\in V_{\beta ,\gamma }^{**}$. If f satisfies (2), then we obtain

$$\begin{array}{ccc}\hfill g\left(z\right)& =& 1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{z}^{n-1}+\sum _{n=2}^{\infty }\frac{1-\gamma }{\zeta \gamma }{\left[n\right]}_q{[n-1]}_q{a}_n{z}^{n-1}\hfill \\ & =& 1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n\left(1+\frac{1-\gamma }{\zeta \gamma }{[n-1]}_q\right)z^{n-1}\hfill \\ & =& 1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n\left(\frac{\zeta \gamma +(1-\gamma ){[n-1]}_q}{\zeta \gamma }\right)z^{n-1}.\hfill \end{array}$$

Therefore,

$$\frac{f\left(z\right)}{z}=1+\sum _{n=2}^{\infty }{a}_n{z}^{n-1}=g\left(z\right)\ast \left(1+\sum _{n=2}^{\infty }\frac{\zeta \gamma }{{\left[n\right]}_q(\zeta \gamma +(1-\gamma ){[n-1]}_q)}{z}^{n-1}\right).$$

We now obtain a one-to-one correspondence between ${\mathcal{P}}_{q,\delta }^{\zeta }(\beta ,\gamma )$ and $V_{\beta ,\gamma }^{**}$. Thus, by Theorem 1, ${\mathcal{P}}_{q,\delta }^{\zeta }(\beta ,\gamma )\subseteq S_{q,\delta }^{*}\left(\alpha \right)$ if and only if

$$\begin{array}{c}\hfill g\left(z\right)\ast \left(1+\sum _{n=2}^{\infty }\frac{\zeta \gamma }{{\left[n\right]}_q(\zeta \gamma +(1-\gamma ){[n-1]}_q)}{z}^{n-1}\right)\ast \frac{1+qz\left(\frac{x+1}{2\alpha {(1-q)}^{\delta }}-1\right)}{(1-z)(1-qz)}\ne 0.\end{array}$$

(12)

For $z\in \mathcal{U}$, consider the continuous linear functional ${\lambda }_z:{\mathcal{A}}_0⟶\mathbb{C}$ such that

$${\lambda }_z\left(h\right)=h\left(z\right)\ast \left(1+\sum _{n=2}^{\infty }\frac{\zeta \gamma }{{\left[n\right]}_q(\zeta \gamma +(1-\gamma ){[n-1]}_q)}{z}^{n-1}\right)\ast \frac{1+qz\left(\frac{x+1}{2\alpha {(1-q)}^{\delta }}-1\right)}{(1-z)(1-qz)}\ne 0.$$

By the Duality Principle, we have ${\lambda }_z\left(V\right)={\lambda }_z\left(V_{\beta ,\gamma }^{**}\right)$. Therefore, (12) holds if and only if

$$\begin{array}{ccc}\hfill \left(1+2(1-\beta )\sum _{k=1}^{\infty }{z}^k\right)& \ast & \left(1+\sum _{n=1}^{\infty }\frac{\zeta \gamma }{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}{z}^n\right)\hfill \\ & \ast & \left(1+\sum _{n=1}^{\infty }\left({[n+1]}_q+\left(\frac{(x+1)q}{2\alpha {(1-q)}^{\delta }}-q\right){\left[n\right]}_q\right)z^n\right)\ne 0.\hfill \end{array}$$

Using the properties of convolution, we obtain

$$\begin{array}{c}\hfill 1+\frac{2(1-\beta )\zeta \gamma }{2\alpha {(1-q)}^{\delta }}\sum _{n=1}^{\infty }\frac{2\alpha {(1-q)}^{\delta }{[n+1]}_q+\left(q(x+1)-2\alpha q{(1-q)}^{\delta }\right){\left[n\right]}_q}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}{z}^n\ne 0.\end{array}$$

Since ${[n+1]}_q=1+q{\left[n\right]}_q$, we get

$$\begin{array}{c}\hfill 1+\frac{2(1-\beta )\zeta \gamma }{2\alpha {(1-q)}^{\delta }}\sum _{n=1}^{\infty }\frac{(x+1){[n+1]}_q+2\alpha {(1-q)}^{\delta }-(x+1)}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}{z}^n\ne 0.\end{array}$$

Hence, we have

$$\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{(x+1){[n+1]}_q+2\alpha {(1-q)}^{\delta }-(x+1)}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}{z}^n\ne -\frac{{(1-q)}^{\delta }}{\zeta \gamma (1-\beta )},$$

(13)

where $z\in \mathcal{U}$, $\left|x\right|=1$. The equality on the right side of Equation (13) takes its value on the line $Rew\ne -\frac{{(1-q)}^{\delta }}{\zeta \gamma (1-\beta )}$, and so (13) is equivalent to (10). □

Remark 1.

Under the hypothesis of Theorem 2, the inequality (10) can be written in the form

$$\begin{array}{ccc}& & \frac{{(1-q)}^{\delta }}{\zeta \gamma (1-\beta )}+Re\left\{\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{\zeta \gamma +(1-\gamma ){\left[n\right]}_q}\right\}\hfill \\ & & +{(1-q)}^{\delta }Re\left\{\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}\right\}-Re\left\{\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}\right\}\hfill \\ & & \ge Re\left\{\frac{x}{\alpha }\left(\sum _{n=1}^{\infty }\frac{z^n}{\zeta \gamma +(1-\gamma ){\left[n\right]}_q}-\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}\right)\right\}.\hfill \end{array}$$

Therefore, for more clarification, we can see that this satisfies the inequality when

$$\begin{array}{}& \frac{{(1-q)}^{\delta }}{\zeta \gamma (1-\beta )}+Re\left\{\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{\zeta \gamma +(1-\gamma ){\left[n\right]}_q}\right\}\hfill & +{(1-q)}^{\delta }Re\left\{\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}\right\}-Re\left\{\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}\right\}\hfill \mathrm{(14)}& \ge \left|\frac{1}{\alpha }\left(\sum _{n=1}^{\infty }\frac{z^n}{\zeta \gamma +(1-\gamma ){\left[n\right]}_q}-\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}\right)\right|.\hfill \end{array}$$

Assume that the function ψ is given by

$$\begin{array}{c}\hfill \psi \left(z\right)=\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}.\end{array}$$

Then, inequality (14) can be written in the form

$$\begin{array}{c}\hfill \frac{{(1-q)}^{\delta }}{\zeta \gamma (1-\beta )}+Re\{D_{q}z\psi \left(z\right)+(2\alpha {(1-q)}^{\delta }-1)\psi \left(z\right)\}\ge |D_{q}z\psi \left(z\right)+\psi \left(z\right)|.\end{array}$$

(15)

Hence, $\psi \left(z\right)\in S_{q,\delta }^{*}\left(\alpha \right)$ if and only if (15) is satisfied.

Putting $\delta =0$ into Theorem 2 leads to the following corollary.

Corollary 2.

Let $0<q<1$, $\alpha \in \mathbb{C}-\left\{0\right\}$, $\zeta >0$, $0\le \beta <1$, $0<\gamma <1$, and $\left|x\right|=1$ with $x\ne -1$. Then, ${\mathcal{P}}_{q,0}^{\zeta }(\beta ,\gamma )\subseteq S_q^{*}\left(\alpha \right)$ if and only if

$$Re\left\{F_1(x,z)\right\}>\frac{1}{\zeta \gamma (1-\beta )},$$

where

$$F_1(x,z)=\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{(x+1){[n+1]}_q+2\alpha -(x+1)}{{[n+1]}_q(\zeta \gamma +(1-\gamma ){\left[n\right]}_q)}{z}^n,\left|z\right|<R\le 1.$$

Similarly, from Theorem 2, we get the following theorem.

Theorem 3.

Let $\delta >0$, $0<q<1$, $\alpha \in \mathbb{C}-\left\{0\right\}$, $0\le \beta <1$, and $\left|x\right|=1$ with $x\ne -1$. Then, ${\mathcal{P}}_{q,\delta }\left(\beta \right)\subseteq S_{q,\delta }^{*}\left(\alpha \right)$ if and only if

$$Re\left\{F_1(x,z)\right\}>-\frac{{(1-q)}^{\delta }}{(1-\beta )},$$

(16)

where

$$F_1(x,z)=\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{(x+1){[n+1]}_q+2\alpha {(1-q)}^{\delta }-(x+1)}{{[n+1]}_q^2}{z}^n,\left|z\right|<R\le 1.$$

(17)

Putting $\delta =0$ into Theorem 3 leads to the following corollary.

Corollary 3.

Let $0<q<1$, $\alpha \in \mathbb{C}-\left\{0\right\}$, $0\le \beta <1$, and $\left|x\right|=1$ with $x\ne -1$. Then, ${\mathcal{P}}_{q,0}(\beta ,)\subseteq S_q^{*}\left(\alpha \right)$ if and only if

$$Re\left\{F_2(x,z)\right\}>-\frac{1}{(1-\beta )},$$

where

$$F_2(x,z)=\frac{1}{\alpha }\sum _{n=1}^{\infty }\frac{(x+1){[n+1]}_q+2\alpha -(x+1)}{{[n+1]}_q^2}{z}^n,\left|z\right|<R\le 1.$$

Remark 2.

The function $F_1(x,z)$ can be represented in terms of a q-hypergeometric function as follows:

$$\begin{array}{c}\hfill F_1(x,z)=\frac{x+1}{\alpha }{}_2{\varphi }_1\left(\begin{array}{cc}q& q\\ q^2\end{array};q,z\right)+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }{}_2{\varphi }_1\left(\begin{array}{ccc}q& q& q\\ q^2& q^2\end{array};q,z\right).\end{array}$$

Proof. 

From the definition of $F_1(x,z)$ introduced in (11), we infer that

$$\begin{array}{}& \hfill F_1(x,z)& =& \frac{x+1}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q}+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }\sum _{n=1}^{\infty }\frac{z^n}{{[n+1]}_q^2}\hfill \mathrm{(18)}& & =& \frac{x+1}{\alpha }\sum _{n=0}^{\infty }\frac{z^n}{{[n+1]}_q}+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }\sum _{n=0}^{\infty }\frac{z^n}{{[n+1]}_q^2}-2(1-q).\hfill \end{array}$$

Since ${[n+1]}_q=\frac{{(q^2;q)}_n}{{(q;q)}_n}$, we have

$$\begin{array}{ccc}\hfill F_1(x,z)& =& \frac{x+1}{\alpha }\sum _{n=0}^{\infty }\frac{{(q;q)}_n}{{(q^2;q)}_n}{z}^n+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }\sum _{n=0}^{\infty }\frac{{(q;q)}_n{(q;q)}_n}{{(q^2;q)}_n{(q^2;q)}_n}{z}^n-2(1-q).\hfill \end{array}$$

Hence, by using the definition of ${}_r{\varphi }_s$ from (1), the proof of the corollary is complete. □

Putting $\delta =0$ into Remark 2 leads to the following corollary.

Corollary 4.

The function $F_2(x,z)$ can be expressed in terms of the q-hypergeometric function as follows:

$$\begin{array}{c}\hfill F_2(x,z)=\frac{x+1}{\alpha }{}_2{\varphi }_1\left(\begin{array}{cc}q& q\\ q^2\end{array};q,z\right)+\frac{2\alpha -(x+1)}{\alpha }{}_2{\varphi }_1\left(\begin{array}{ccc}q& q& q\\ q^2& q^2\end{array};q,z\right).\end{array}$$

We now consider the Riemann–Liouville fractional q-integral and obtain the following corollary.

Remark 3.

The function $F_1(x,z)$ can be expressed in terms of the Riemann–Liouville fractional q-integral as follows:

$$\begin{array}{c}\hfill F_1(x,z)=\frac{x+1}{\alpha }{\int }_0^1\frac{1}{1-tz}{d}_{q}t+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }{\int }_0^1{\int }_0^1\frac{1}{1-vtz}{d}_{q}v{d}_{q}t-2(1-q)\end{array}$$

Proof. 

Since Equation (18) is satisfied, we have

$$\begin{array}{ccc}\hfill F_1(x,z)& =& \frac{x+1}{\alpha }\sum _{n=0}^{\infty }{\int }_0^1{t}^n{d}_{q}t{z}^n+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }\sum _{n=0}^{\infty }{\int }_0^1{v}^n{d}_{q}v{\int }_0^1{t}^n{d}_t{z}^n\hfill \\ & =& \frac{x+1}{\alpha }{\int }_0^1\left(\sum _{n=0}^{\infty }{t}^n{z}^n\right)d_{q}t+\frac{2\alpha {(1-q)}^{\delta }-(x+1)}{\alpha }{\int }_0^1{\int }_0^1\left(\sum _{n=0}^{\infty }{v}^n{t}^n{z}^n\right)d_{q}v{d}_{q}t.\hfill \end{array}$$

This completes the proof of the corollary. □

Putting $\delta =0$ into Remark 3 leads to the following corollary.

Corollary 5.

The function $F_2(x,z)$ can be expressed in terms of the Riemann–Liouville fractional q-integral in the following form:

$$\begin{array}{c}\hfill F_2(x,z)=\frac{x+1}{\alpha }{\int }_0^1\frac{1}{1-tz}{d}_{q}t+\frac{2\alpha -(x+1)}{\alpha }{\int }_0^1{\int }_0^1\frac{1}{1-vtz}{d}_{q}v{d}_{q}t-2(1-q).\end{array}$$

Theorem 4.

Let $0<\left|q\right|<1,\delta >0$, $0\le \beta \le 1$, and $f\in {\mathcal{P}}_{q,\delta }\left(\beta \right)$. We define

$$\begin{array}{c}\hfill K_q={\int }_0^1\frac{d_{q}t}{1-t}.\end{array}$$

(19)

If

$$\begin{array}{c}\hfill \beta \ge \frac{1-2K_q}{2(1-K_q)},\end{array}$$

then $f\in {\mathcal{P}}_{q,\delta }\left(0\right)$, and, hence, it is univalent.

Proof. 

Let $\zeta >0$ and $\gamma >0$; we define

$${\varphi }_q\left(z\right)=1+\sum _{n=1}^{\infty }{[n+1]}_q{z}^n,$$

and

$$\begin{array}{ccc}\hfill {\psi }_q\left(z\right)& =& 1+\sum _{n=1}^{\infty }\frac{1}{{[n+1]}_q}{z}^n=1+\sum _{n=1}^{\infty }{\int }_0^1{t}^n{d}_{q}t{z}^n\hfill \\ & =& {\int }_0^1\left(1+\sum _{n=1}^{\infty }{t}^n{z}^n\right)d_{q}t={\int }_0^1\frac{1}{1-tz}{d}_{q}t.\hfill \end{array}$$

In view of these representations, we can write

$$\begin{array}{c}\hfill D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)+q{D}_q^{\delta +2}{I}_q^{\delta }f\left(z\right)=D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)\ast {\varphi }_q\left(z\right)\end{array}$$

and

$$\begin{array}{c}\hfill \left(D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)+q{D}_q^{\delta +2}{I}_q^{\delta }f\left(z\right)\right)\ast {\psi }_q\left(z\right)=D_q^{\delta +1}{I}_q^{\delta }f\left(z\right).\end{array}$$

Let $f\in {\mathcal{P}}_{q,\delta }\left(\beta \right)$. Then, by using Lemma 2, we may restrict our attention to the function $f\in P_{\zeta }(\beta ,\gamma )$ for which

$${(1-q)}^{\delta }\left(D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)+q{D}_q^{\delta +2}{I}_q^{\delta }f\left(z\right)\right)=\gamma (1-\beta )\frac{1+xz}{1-xz}+(1-\beta )(1-\gamma )\frac{1+yz}{1-yz}+\beta .$$

Thus, we obtain

$${(1-q)}^{\delta }{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)=\left(\gamma (1-\beta )\frac{1+xz}{1-xz}+(1-\beta )(1-\gamma )\frac{1+yz}{1-yz}+\beta \right)\ast {\psi }_q\left(z\right).$$

(20)

Hence, Equation (20) is equivalent to

$$\begin{array}{}& \hfill {(1-q)}^{\delta }{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)& =& \left(\gamma \frac{1+xz}{1-xz}+(1-\gamma )\frac{1+yz}{1-yz}\right)\ast \left((1-\beta ){\psi }_q\left(z\right)+\beta \right).\hfill & & =& \left(\gamma \frac{1+xz}{1-xz}+(1-\gamma )\frac{1+yz}{1-yz}\right)\ast \left({\int }_0^1\left((1-\beta )\frac{1}{1-tz}+\beta \right)d_{q}t\right)\hfill \mathrm{(21)}& & =& \left(\gamma \frac{1+xz}{1-xz}+(1-\gamma )\frac{1+yz}{1-yz}\right)\ast G_q\left(z\right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill G_q\left(z\right)={\int }_0^1\left((1-\beta )\frac{1}{1-tz}+\beta \right)d_{q}t.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill Rel\left(G_q\left(z\right)\right)={\int }_0^1\left((1-\beta )\frac{1}{1-t}+\beta \right)d_{q}t=(1-\beta )k_q+\beta ,\end{array}$$

where $K_q$ is defined by (19). Note that if $\beta \ge (1-2K_q)/2(1-K_q)$, then $ReG\left(z\right)\ge 1/2$. Functions with real parts greater than $1/2$ are known to preserve the closed convex hull under convolution [10], p. 23. Therefore, from (21), we have

$$\begin{array}{ccc}\hfill {(1-q)}^{\delta }{D}_q^{\delta +1}{I}_q^{\delta }f\left(z\right)& =& \gamma \left(\frac{2}{1-xz}-1\right)\ast G_q\left(z\right)+(1-\gamma )\left(\frac{2}{1-yz}-1\right)\ast G_q\left(z\right)\hfill \\ & =& 2\gamma G_q\left(xz\right)-\gamma +2(1-\gamma )G_q\left(yz\right)-(1-\gamma )\hfill \\ & =& 2\gamma G_q\left(xz\right)+2(1-\gamma )G_q\left(yz\right)-1.\hfill \end{array}$$

In addition, since $Re\left\{D_q^{\delta +1}{I}_q^{\delta }f\left(z\right)\right\}>0$, we have $f\in {\mathcal{P}}_{q,\delta }\left(0\right)$. This completes the proof of the theorem. □

4. Conclusions

In this article, a new class of univalent functions was introduced by using Riemann–Liouville fractional q-integrals and q-difference operators of non-integer orders. Then, some convolution results for such a class of univalent functions were obtained. In addition, two classes of normalized analytic functions in the unit disc were derived, and some conditions on $q,\delta ,\zeta ,\beta$, and $\gamma$ were given so that the new classes satisfied ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )\subset S_{q,\delta }^{*}\left(\alpha \right)$ and ${\mathcal{P}}_{\delta ,q}\left(\beta \right)\subset S_{q,\delta }^{*}\left(\alpha \right)$.

The result obtained during this research can be further used for writing fractional differential and integral operators in order to extend the results of analytic functions.