Properties for a Certain Subclass of Analytic Functions Associated with the Salagean q-Differential Operator

Abstract

Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions.

1. Introduction

Let $\mathcal{A}$ be the class of functions regular in $\mathbb{U}=\{z\in \mathbb{C}:\left|z\right|<1\}$ with the following Taylor series representation

$$\xi \left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k,(a_k\ge 0,k\in N).$$

(1)

We shall denote by S the class of all functions $\xi \in \mathcal{A}$ that are univalent in $\mathbb{U}$. A function $\xi \in S$ is said to be starlike of order $\alpha$ if and only if

$$\Re \left(\frac{z{\xi }^{\prime }\left(z\right)}{\xi \left(z\right)}\right)>\alpha (0\le \alpha <1,z\in \mathbb{U}),$$

and we denote by $S^{*}\left(\alpha \right)$ to the class of all starlike functions of order $\alpha$. Furthermore, a function $\xi$ in S is said to be convex and of order $\alpha$ if and only if

$$\Re \left(1+\frac{z{\xi }^{″}\left(z\right)}{{\xi }^{\prime }\left(z\right)}\right)>\alpha ,(0\le \alpha <1,z\in \mathbb{U}),$$

and we denote by $K\left(\alpha \right)$ as the class of all convex functions of order $\alpha$. We note that

$$\xi \left(z\right)\in K\left(\alpha \right)\iff z{\xi }^{\prime }\left(z\right)\in S^{*}\left(\alpha \right).$$

It was Robertson [1] who introduced the classes $S^{*}\left(\alpha \right)$ and $K\left(\alpha \right)$, followed by Schild [2], MacGregor [3], and Pinchuk [4]. Let $\mathcal{R}$ be the family of functions $\xi \in A$, which satisfy the condition

$$\Re \left({\xi }^{\prime }\left(z\right)+z{\xi }^{″}\left(z\right)\right)>0(z\in \mathbb{U}).$$

Chichra [5] proved that if $\xi \in \mathcal{R},$ then $\Re \left({\xi }^{\prime }\left(z\right)\right)>0,z\in \mathbb{U},$ and hence $\xi$ is univalent in $\mathbb{U}.$ We denote by $\xi \ast \eta$ the convolution (Hadamard composition) for two functions, $\xi$ and $\eta$, that are analytic in $\mathbb{U}$; that is, if $\xi$ is given by (1) and

$$\eta \left(z\right)=z+\sum _{k=2}^{\infty }{b}_k{z}^k,(b_k\ge 0,k\in N),$$

then

$$(\xi \ast \eta )\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{b}_k{z}^k,(a_k\ge 0,k\in N).$$

(2)

For additional details on the Hadamard composition, see, for example, [6,7,8]. As a result of Euler and Heine’s pioneering work, Frank Hilton Jackson [9,10] developed, at the beginning of the 20th century, the theory of basic hypergeometric series in a systematic manner, studying q-differentiation and q-integration and deriving the q-analogues of the hypergeometric summation and transformation formulas. It was Ismail et al. [11] who applied q-calculus to geometric function theory (GFT) by extending the family of starlike functions into q-analogues, called q-starlike functions. Based on the same idea, many authors have extensively studied the q-calculus operators (q-differential and q-integral operators) in GFT. A recent study on these operators acting on analytic functions can be found in [12,13,14,15,16,17,18,19]. For $0<q<1$, Jackson [9,10] defined the q-differential operator, $D_q$, of a function, $\xi$, as the following:

$$D_q\xi \left(z\right)=\left\{\begin{array}{c}\frac{\xi \left(z\right)-\xi \left(qz\right)}{(1-q)z}z\ne 0,\\ {\xi }^{\prime }\left(0\right)z=0.\end{array}\right.$$

(3)

The q-derivative is the ordinary derivative without limits; if $q\to 1^{-}$, then $D_q\xi \left(z\right)$ tends to ${\xi }^{\prime }\left(z\right).$ It follows from (3) that if $\xi \in \mathcal{A}$ is of form (1), then

$$\begin{array}{ccc}\hfill D_q\xi \left(z\right)& =& D_q\left(z+\sum _{k=2}^{\infty }{a}_k{z}^k\right)\hfill \\ & =& 1+\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k{z}^{k-1}(a_k\ge 0,k\in N),\hfill \end{array}$$

where

$${\left[k\right]}_q=\frac{1-q^k}{1-q},(\underset{q\to 1}{lim}{\left[k\right]}_q=k).$$

The q-derivative of the product of two functions is given by the following rule:

$$D_q\left(\xi \left(z\right).g\left(z\right)\right)=g\left(z\right)D_q\xi \left(z\right)+\xi \left(qz\right)D_{q}g\left(z\right).$$

Letting $\xi \in \mathcal{A}$ and $0<q<1,$ analogous to the Salagean differential operator, Govindaraj and Givasubramanian [20] introduced ${\mathcal{S}}_q^n$ as follows:

$${\mathcal{S}}_q^0\xi \left(z\right)=\xi \left(z\right),{\mathcal{S}}_q^1\xi \left(z\right)=z{D}_q\xi \left(z\right)\dots {\mathcal{S}}_q^n\xi \left(z\right)=z{D}_q\left(S_q^{n-1}\xi \left(z\right)\right).$$

(4)

We note that

$${\mathcal{S}}_q^n\xi \left(z\right)=(\xi \ast {\mathcal{G}}_q^n)\left(z\right)(n\in N_0=\{0,1,\cdots \}),$$

where

$${\mathcal{G}}_q^n\left(z\right)=z+\sum _{k=2}^{\infty }{\left[k\right]}_q^n{z}^k(n\in N_0,z\in \mathbb{U}),$$

and the differential operator ${\mathcal{S}}_q^n$ is called the Salagean q-differential operator.

Definition 1.

A function, $\xi \in \mathcal{A}$, is said to be in the class ${\mathcal{R}}_q^n\left(\alpha \right)$if it satisfies the following inequality

$$\Re \left(\frac{{\mathcal{S}}_q^n\xi \left(z\right)}{z}\right)>\alpha \left(z\in \mathbb{U}\right),$$

(5)

where $q$ ∈ $(0,1),0\le \alpha <1,$ and ${\mathcal{S}}_q^n$ is the Salagean q-differential operator.

We note that

• The class ${\mathcal{R}}_q^n\left(\alpha \right)$ is not empty since the function $\xi \left(z\right)=z+\frac{1-\alpha }{{\left[2\right]}_q^n}{z}^2$ belongs to ${\mathcal{R}}_q^n\left(\alpha \right).$
• ${\mathcal{R}}_q^2\left(\alpha \right)$ is the class of functions satisfying the following inequality:

  $$\Re [D_q\xi \left(z\right)+qz{D}_q^2\xi \left(z\right)]>\alpha ,$$

  where $D_q^2\xi =D_q\left(D_q\xi \right)$ and $0\le \alpha <1.$
• ${\displaystyle \underset{q\to 1}{lim}}{\mathcal{R}}_q^2\left(\alpha \right)$ is the class of functions satisfying the following inequality:

  $$\Re \left({\xi }^{\prime }\left(z\right)+z{\xi }^{″}\left(z\right)\right)>\alpha ,0\le \alpha <1,$$

  which was introduced and studied by Ponnusamy [21] (see also [22]).

In this paper, we discuss various properties and characteristics of functions in the class ${\mathcal{R}}_q^n\left(\alpha \right)$, including the inclusion relations, coefficient conditions, convolution properties, and applications of the q-fractional calculus operators. Furthermore, we extend Miller and Mocanu’s result [23] to the q-theory of analytic functions.

2. Inclusion Relations

We need the following lemmas to prove our main result.

A sequence of nonnegative real numbers, ${\left\{a_n\right\}}_{n=0}^{\infty }$, is said to be a convex null sequence if ${\displaystyle \underset{n\to \infty }{lim}}$$a_n$$=0$ and

$$a_0-a_1\ge a_1-a_2\ge ...\ge a_n-a_{n+1}\ge ...\ge 0.$$

Lemma 1 ([24]). 

If ${\left\{c_k\right\}}_{k=0}^{\infty }$ is a convex null sequence, then the function $p\left(z\right)=\frac{c_0}{2}+{\displaystyle \sum _{k=1}^{\infty }}{c}_k{z}^k$ is regular in $\mathbb{U}$ and satisfies the following inequality:

$$\Re \left(p\left(z\right)\right)>0.$$

Lemma 2 ([22]). 

Let p be an analytic function in $\mathbb{U}$ with $p\left(0\right)=1$ and $\Re \left(p(z\right))>\frac{1}{2},$$z\in \mathbb{U}$. If F is any analytic function in $\mathbb{U}$, then the function $p\ast F$ takes its values in the convex hull of $F\left(\mathbb{U}\right)$.

Lemma 3 ([25]). 

Let ${\chi }_{\alpha }$ be the class of functions analytic in $\mathbb{U}$, normalized by the condition $h\left(z\right)=1+{\displaystyle \sum _{k=1}^{\infty }}{c}_k{z}^k$, and satisfy the following inequality:

$$\Re \left\{h\left(z\right)\right\}>\alpha ,(0\le \alpha <1).$$

If $h_i\left(z\right)\in {\chi }_{{\alpha }_i}(0\le {\alpha }_i<1;i=1,2),$ then

$$(h_1\ast h_2)\left(z\right)\in {\chi }_{{\alpha }_3},{\alpha }_3=1-2(1-{\alpha }_1)(1-{\alpha }_2).$$

The result is sharp.

The following theorem can be proved using Lemmas 1 and 2.

Theorem 1.

For $0\le \alpha \le 1,$ we have the following inclusion relation:

$${\mathcal{R}}_q^{n+1}\left(\alpha \right)\subset {\mathcal{R}}_q^n\left(\alpha \right).$$

Proof. 

Let $\xi$ given by (1) belong to the class ${\mathcal{R}}_q^{n+1}\left(\alpha \right)$. Then, we have

$$\Re \left(\frac{{\mathcal{S}}_q^{n+1}\xi \left(z\right)}{z}\right)=\Re \left(1+\sum _{k=2}^{\infty }{\left[k\right]}_q^{n+1}{a}_k{z}^{k-1}\right)>\alpha ,$$

which is equivalent to

$$\Re \left(1+\frac{1}{2(1-\alpha )}\sum _{k=2}^{\infty }{\left[k\right]}_q^{n+1}{a}_k{z}^{k-1}\right)>\frac{1}{2}.$$

Now,

$$\begin{array}{ccc}\hfill \frac{{\mathcal{S}}_q^n\xi \left(z\right)}{z}& =& 1+\sum _{k=2}^{\infty }{\left[k\right]}_q^n{a}_k{z}^{k-1}\hfill \\ & =& \left(1+\frac{1}{2(1-\alpha )}\sum _{k=2}^{\infty }{\left[k\right]}_q^{n+1}{a}_k{z}^{k-1}\right)\ast \left(1+2(1-\alpha )\sum _{k=2}^{\infty }\frac{1}{{\left[k\right]}_q}{z}^{k-1}\right).\hfill \end{array}$$

Since the sequence of real numbers ${\left(\frac{1}{{\left[k+1\right]}_q}\right)}_{k=0}^{\infty }$ is decreasing and converges to the limit $(1-q)$, then the sequence ${\left(\frac{1}{{\left[k+1\right]}_q}-(1-q)\right)}_{k=0}^{\infty }$ is a null sequence. By applying Lemma 1 with $c_0=q$ and $c_k=\frac{1}{{\left[k+1\right]}_q}-(1-q)(k=1,2,\dots )$, we obtain

$$\Re \left(\frac{q}{2}+\sum _{k=1}^{\infty }\left(\frac{1}{{\left[k+1\right]}_q}-(1-q)\right)z^k\right)>0$$

or, equivalently,

$$\Re \left(\frac{q}{2}+\sum _{k=2}^{\infty }\left(\frac{1}{{\left[k\right]}_q}-(1-q)\right)z^{k-1}\right)>0.$$

(6)

From (6), we have

$$\begin{array}{ccc}\hfill \Re \left(\frac{q}{2}+\sum _{k=2}^{\infty }\frac{1}{{\left[k\right]}_q}{z}^{k-1}\right)& >& (1-q)\Re \left(\sum _{k=2}^{\infty }{z}^{k-1}\right),\hfill \\ & \Rightarrow & \Re \left(\frac{q}{2}+\sum _{k=2}^{\infty }\frac{1}{{\left[k\right]}_q}{z}^{k-1}\right)>(1-q)\Re \left(\frac{z}{1-z}\right)\hfill \\ & \Rightarrow & \Re \left(\frac{q}{2}+\sum _{k=2}^{\infty }\frac{1}{{\left[k\right]}_q}{z}^{k-1}\right)>\frac{-1}{2}(1-q)\hfill \\ \hfill & \Rightarrow & \Re \left(q(1-\alpha )+2(1-\alpha )\sum _{k=2}^{\infty }\frac{1}{{\left[k\right]}_q}{z}^{k-1}\right)>-(1-q)(1-\alpha )\hfill \\ & \Rightarrow & \Re \left(1+2(1-\alpha )\sum _{k=2}^{\infty }\frac{1}{{\left[k\right]}_q}{z}^{k-1}\right)>\alpha .\hfill \end{array}$$

Applying Lemma 2, we obtain

$$\Re \left(\frac{S_q^{n}f\left(z\right)}{z}\right)>\alpha ,$$

which ends the proof of Theorem 1. □

Remark 1.

By letting $q\to 1$, Theorem 1 gives the corresponding result given by Al-Oboudi [22].

Putting $n=2$ in Theorem 1, we obtain the following corollary.

Corollary 1.

Let ξ given by (1). If $\Re [D_q\xi \left(z\right)+qz{D}_q^2\xi \left(z\right)]>\alpha ,$ $0\le \alpha \le 1,$ then we have $\Re \left(D_q\xi \left(z\right)\right)>\alpha .$

Remark 2.

Let $q\to 1$. Then, Corollary 1 gives the result obtained by Singh and Singh [26].

Recently, many authors have used the q-analogue differential subordinations in the q-theory of analytic functions without proving them, for example, Aldweby and Darus [27] and Alb Lupas [28] and Khan et al. [29] used the following q-analogue from Miller and Mocanu’s result [23] without proving it. If $p=1+c_{1}z+c_2{z}^2+\dots$ is analytic in $\mathbb{U},$ then

$$\Re [p\left(z\right)+z{D}_{q}p\left(z\right)]>0\Rightarrow \Re \left[p\left(z\right)\right]>0.$$

Now, the result given in Corollary 2 below extends the result given by Miller and Mocanu [23] to the q-theory of analytic functions by setting $p\left(z\right)=D_q\xi \left(z\right)$ in Corollary 1.

Corollary 2.

If $p=1+c_{1}z+c_2{z}^2+\dots$ satisfies $\Re [p\left(z\right)+qz{D}_{q}p\left(z\right)]>\alpha ,$ then we have $\Re \left(p\left(z\right)\right)>\alpha .$

3. Convolution Conditions

In the following theorem, we obtain convolution conditions for functions whose q-derivatives have a positive real part.

Theorem 2.

If $\xi ,\eta \in \mathcal{A}$ satisfy $Re\left\{D_q\xi \left(z\right)\right\}>\alpha$ and $Re\left\{D_q\eta \left(z\right)\right\}>\beta ,$ then

$$Re\frac{(\xi \ast \eta )}{z}>1-2(1-\alpha )(1-\beta ).$$

Proof. 

Let $\phi =\xi \ast \eta ;$ from (2), it is easy to see that

$$D_q\phi \left(z\right)+qz{D}_q^2\phi \left(z\right)=1+\sum _{k=2}^{\infty }{\left[k\right]}_q^2{a}_k{b}_k{z}^{k-1}.$$

Therefore,

$$D_q\xi \left(z\right)\ast D_q\eta \left(z\right)=D_q\phi \left(z\right)+qz{D}_q^2\phi \left(z\right).$$

(7)

From (7) and Lemma 3, we conclude that

$$\Re \left(D_q\phi \left(z\right)+qz{D}_q^2\phi \left(z\right)\right)>1-2(1-\alpha )(1-\beta ),$$

(8)

and as a result of Corollary 1, we obtain

$$\Re \left(D_q\phi \left(z\right)\right)>1-2(1-\alpha )(1-\beta ).$$

(9)

Furthermore, let $p\left(z\right)=\frac{\phi \left(z\right)}{z};$ then, we obtain

$$\Re \left[p\left(z\right)+qz{D}_{q}p\left(z\right)\right]>1-2(1-\alpha )(1-\beta ),$$

(10)

and by applying Corollary 2, inequality (10) gives

$$\Re \left\{p\left(z\right)\right\}>1-2(1-\alpha )(1-\beta ).$$

This finishes the proof of the theorem. □

Remark 3.

Letting $q\to 1$ in the above theorem, we get the corresponding result introduced by Lashin [30].

In Theorem 3 below, the class $\Re [D_q\xi \left(z\right)+qz{D}_q^2\xi \left(z\right)]>\alpha$ can be characterized in terms of a convolution using a technique similar to that described by Silverman et al. [31].

Theorem 3.

If $0\le \alpha <1$, then the function $\xi \in \mathcal{A}$ belongs to the class ${\mathcal{R}}_q^2\left(\alpha \right)$ if and only if

$$\frac{1}{z}(\xi \ast \mathcal{G})\ne 0,$$

where $\mathcal{G}=\xi \left(z\right)\ast \left[\frac{z(1+qz)\left(e^{-i\theta }-1\right)}{(1-z)(1-qz)(1-q^{2}z)}-z[e^{-i\theta }+(1-2\alpha )]\right].$

Proof. 

It is known that (see Piejko et al. [18] and El-Emam [32])

$$\xi \left(z\right)=\xi \left(z\right)\ast \frac{z}{(1-z)}$$

(11)

and

$$z{D}_q\xi \left(z\right)=\xi \left(z\right)\ast \frac{z}{(1-z)(1-qz)}=z+\sum _{k=2}^{\infty }{\left[k\right]}_q{z}^k.$$

(12)

Note that $\xi \in {\mathcal{R}}_q^2\left(\alpha \right)$ if and only if

$$D_q\xi \left(z\right)+qz{D}_q^2\xi \left(z\right)\ne \frac{1+(1-2\alpha )e^{i\theta }}{1-e^{i\theta }},$$

which simplifies to

$$\frac{1}{z}\left(z{D}_q\left(z{D}_q\xi \right)[1-e^{i\theta }]-z[1+(1-2\alpha )e^{i\theta }]\right)\ne 0.$$

(13)

Using (12) and the identity $z{D}_q(\xi \ast g)=\xi \ast z{D}_{q}g,$ we can write

$$z{D}_q\left(z{D}_q\xi \right)=\xi \left(z\right)\ast \frac{z+q{z}^2}{(1-z)(1-qz)(1-q^{2}z)}.$$

Using the above formula, (13) becomes as follows:

$$\begin{array}{ccc}& & \frac{1}{z}\left(z{D}_q\left(z{D}_q\xi \right)[1-e^{i\theta }]-z[1+(1-2\alpha )e^{i\theta }]\right)\hfill \\ & =& \frac{1}{z}\left(\xi \left(z\right)\ast \left[\frac{z(1+qz)\left(e^{-i\theta }-1\right)}{(1-z)(1-qz)(1-q^{2}z)}-z[e^{-i\theta }+(1-2\alpha )]\right]\right)\ne 0.\hfill \end{array}$$

□

4. Application of q-Fractional Calculus Operators

Many problems in applied sciences are solved using fractional calculus operators. An extension of fractional calculus is q-fractional calculus. It has been used in optimal control problems, q-analysis, and geometric function theory, as well as to solve q-difference and integral equations (see [33]). Gasper and Rahman [33] defined the q-Gamma function for $z\in \mathbb{C},z\ne -n,n\in N_0,$ by

$${\Gamma }_q\left(z\right)=\frac{{(q,q)}_{\infty }}{{(q^z,q)}_{\infty }}{(1-q)}^{1-z}(0<q<1)$$

and the q-binomial expansion by

$${(x-y)}_{\nu }=x^{\nu }{\left(\frac{y}{x},q\right)}_{\nu }=x^{\nu }\prod _{n=0}^{\infty }\frac{1-\left(\frac{y}{x}\right)q^n}{1-\left(\frac{y}{x}\right)q^{n+\nu }},$$

where

$${(\alpha ,q)}_n=\prod _{m=0}^{n-1}(1-\alpha q^m)(n\ge 1)$$

and ${(\alpha ,q)}_{\infty }$ is defined as

$${(\alpha ,q)}_{\infty }=\prod _{m=0}^{\infty }(1-\alpha q^m)(0<q<1).$$

We now recall the definitions of q-calculus operators given by Purohit and Raina [34] (see also [35]).

Definition 2.

The fractional q-integral operator $D_{q,z}^{-\mu }\xi \left(z\right)$ of order $\mu (\mu >0)$ is defined, for a function, $\xi \in \mathcal{A}$, by

$$\begin{array}{ccc}\hfill I_{q,z}^{\mu }\xi \left(z\right)& =& D_{q,z}^{-\mu }\xi \left(z\right),\hfill \\ & =& \frac{1}{{\Gamma }_q\left(\mu \right)}\underset{0}{\overset{z}{\int }}{(z-qt)}_{\mu -1}\xi \left(t\right)d(t,q),\hfill \end{array}$$

where, ${\left(z-qt\right)}_{m-1}$ is single-valued when $\left|arg\left(\frac{-t{q}^m}{z}\right)\right|<\pi ,\left|\frac{t{q}^m}{z})\right|<1$ and $\left|argz\right|<\pi$.

Definition 3.

The fractional q-derivative operator $D_{q,z}^{\mu }\xi \left(z\right)$ of order $\mu (\mu >0)$ is defined, for a function, $\xi \in \mathcal{A}$, by

$$\begin{array}{ccc}\hfill D_{q,z}^{\mu }\xi \left(z\right)& =& D_{q,z}{I}_{q,z}^{1-\mu }\xi \left(z\right),\hfill \\ & =& \frac{1}{{\Gamma }_q(1-\mu )}{D}_{q,z}\underset{0}{\overset{z}{\int }}{(z-qt)}_{-\mu }\xi \left(t\right)d(t,q),(0\le \mu <1),\hfill \end{array}$$

where, the multiplicity of ${(z-qt)}_{-\mu }$ is removed as in Definition 2.

Lemma 4.

Let $\mu >0$ and $k>-1.$ Then,

$$I_{q,z}^{\mu }{z}^k=\frac{{\Gamma }_q(k+1)}{{\Gamma }_q(k+\mu +1)}{z}^{\mu +k}.$$

Lemma 5.

Let $\mu \ge 0$ and $k>-1.$ Then,

$$D_{q,z}^{\mu }{z}^k=\frac{{\Gamma }_q(k+1)}{{\Gamma }_q(k-\mu +1)}{z}^{k-\mu }.$$

Hence,

$$D_{q,z}^{\mu }\xi \left(z\right)=\frac{{\Gamma }_q\left(2\right)}{{\Gamma }_q(2-\mu )}{z}^{1-\mu }+\sum _{k=2}^{\infty }\frac{{\Gamma }_q(k+1)}{{\Gamma }_q(k+1-\mu )}{a}_k{z}^{k-\mu },(\mu >0)$$

and

$$I_{q,z}^{\mu }\xi \left(z\right)=\frac{{\Gamma }_q\left(2\right)}{{\Gamma }_q(2+\mu )}{z}^{1+\mu }+\sum _{k=2}^{\infty }\frac{{\Gamma }_q(k+1)}{{\Gamma }_q(k+1+\mu )}{a}_k{z}^{k+\mu }.$$

We now prove the following theorem to obtain the coefficient estimates for functions in the class ${\mathcal{R}}_q^n\left(\alpha \right).$

Theorem 4.

The sufficient condition for the function $\xi \in \mathcal{A}$ to belong to the class ${\mathcal{R}}_q^n\left(\alpha \right)$ is

$$\sum _{k=2}^{\infty }{\left[k\right]}_q^n\left|a_k\right|\le 1-\alpha .$$

(14)

The result is sharp.

Proof. 

It suffices to show that the values for $\frac{{\mathcal{S}}_q^n\xi \left(z\right)}{z}$ lie in a circle centered on $w=1$ and whose radius is $1-\alpha$. We have

$$\begin{array}{ccc}& & \left|\frac{{\mathcal{S}}_q^n\xi \left(z\right)}{z}-1\right|\hfill \\ & =& \left|\sum _{k=2}^{\infty }{\left[k\right]}_q^n{a}_k{z}^{k-1}\right|\hfill \\ & \le & \sum _{k=2}^{\infty }{\left[k\right]}_q^n\left|a_k\right|.\hfill \end{array}$$

Because condition (14) is satisfied, the last expression is bounded above by 1 − $\alpha$, so $\xi \in {\mathcal{R}}_q^n\left(\alpha \right)$. Finally, the result is sharp, with the extremal function $\xi \left(z\right)$ given by

$$\xi \left(z\right)=z+\frac{1-\alpha }{{\left[2\right]}_q^n}{z}^2.$$

(15)

□

Let ${\mathcal{R}}_q^{\ast ,n}\left(\alpha \right)$ be the class of functions $\xi \in \mathcal{A}$, whose coefficients satisfy the condition (14). In Theorems 5 and 6, we prove two distortion theorems for functions in the class ${\mathcal{R}}_q^{\ast ,n}\left(\alpha \right)$, involving the q-fractional calculus operators.

Theorem 5.

Let the function ξ have form (1). If $\xi \in {\mathcal{R}}_q^{\ast ,n}\left(\alpha \right),$ then

$$\left|I_{q,z}^{\mu }\xi \left(z\right)\right|\ge \left(\frac{{\Gamma }_q\left(2\right)}{{\Gamma }_q(2+\mu )}-\frac{{\Gamma }_q\left(3\right)}{{\left[2\right]}_q^n{\Gamma }_q(3+\mu )}\left|z\right|\right)\left|z\right|,(\mu >0)$$

(16)

and

$$\left|I_{q,z}^{\mu }\xi \left(z\right)\right|\le \left(\frac{{\Gamma }_q\left(2\right)}{{\Gamma }_q(2+\mu )}+\frac{{\Gamma }_q\left(3\right)}{{\left[2\right]}_q^n{\Gamma }_q(3+\mu )}\left|z\right|\right)\left|z\right|,(\mu >0).$$

(17)

Proof. 

Consider the function $\Omega \left(z\right)$ defined in $\mathbb{U}$ by

$$\begin{array}{ccc}\hfill \Omega \left(z\right)& =& \frac{{\Gamma }_q(2+\mu )}{{\Gamma }_q\left(2\right)}{z}^{-\mu }{I}_{q,z}^{\mu }\xi \left(z\right),\hfill \\ & =& z+\sum _{k=2}^{\infty }\frac{{\Gamma }_q(k+1){\Gamma }_q(2+\mu )}{{\Gamma }_q(k+1+\mu ){\Gamma }_q\left(2\right)}{a}_k{z}^k,(\mu >0,k\ge 2),\hfill \\ & =& z+\sum _{k=2}^{\infty }\Phi \left(k\right)a_k{z}^k,\hfill \end{array}$$

(18)

where

$$\Phi \left(k\right)=\frac{{\Gamma }_q(k+1){\Gamma }_q(2+\mu )}{{\Gamma }_q(k+1+\mu ){\Gamma }_q\left(2\right)},(\mu >0,k\ge 2).$$

Since $\Phi \left(k\right)$ is a decreasing function of$k(k\ge 2)$, when $\mu >0$, we have

$$0<\Phi \left(k\right)\le \Phi \left(2\right)=\frac{{\Gamma }_q\left(3\right){\Gamma }_q(2+\mu )}{{\Gamma }_q(3+\mu ){\Gamma }_q\left(2\right)}.$$

(19)

Now, (14) gives

$${\left[2\right]}_q^n\sum _{k=2}^{\infty }{a}_k\le \sum _{k=2}^{\infty }{\left[k\right]}_q^n{a}_k\le 1-\alpha ,a_k\ge 0,$$

that is, that

$$\sum _{k=2}^{\infty }{a}_k\le \frac{1}{{\left[2\right]}_q^n}.$$

(20)

Therefore, by using (19) and (20), we see that

$$\begin{array}{ccc}\hfill \left|\Omega \left(z\right)\right|& \ge & \left|z\right|-\Phi \left(2\right){\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k\hfill \\ & \ge & \left|z\right|-\frac{\Phi \left(2\right)}{{\left[2\right]}_q^n}{\left|z\right|}^2\hfill \\ & =& \left|z\right|-\frac{{\Gamma }_q\left(3\right){\Gamma }_q(2+\mu )}{{\left[2\right]}_q^n{\Gamma }_q(3+\mu ){\Gamma }_q\left(2\right)}{\left|z\right|}^2\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left|\Omega \left(z\right)\right|& \le & \left|z\right|+\Phi \left(2\right){\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k\hfill \\ & \le & \left|z\right|+\frac{\Phi \left(2\right)}{{\left[2\right]}_q^n}{\left|z\right|}^2\hfill \\ & =& \left|z\right|+\frac{{\Gamma }_q\left(3\right){\Gamma }_q(2+\mu )}{{\left[2\right]}_q^n{\Gamma }_q(3+\mu ){\Gamma }_q\left(2\right)}{\left|z\right|}^2.\hfill \end{array}$$

This yields the desired result. □

Theorem 6.

Let the function ξ be in form (1). If $\xi \in {\mathcal{R}}_q^{\ast ,n}\left(\alpha \right),$ then

$$\left|D_{q,z}^{\mu }f\left(z\right)\right|\ge \left(\frac{{\Gamma }_q\left(2\right)}{{\Gamma }_q(2-\mu )}-\frac{{\Gamma }_q\left(3\right)}{{\left[2\right]}_q^n{\Gamma }_q(3-\mu )}\left|z\right|\right)\left|z\right|,(\mu >0)$$

(21)

and

$$\left|D_{q,z}^{\mu }f\left(z\right)\right|\le \left(\frac{{\Gamma }_q\left(2\right)}{{\Gamma }_q(2-\mu )}+\frac{{\Gamma }_q\left(3\right)}{{\left[2\right]}_q^n{\Gamma }_q(3-\mu )}\left|z\right|\right)\left|z\right|,(\mu >0).$$

(22)

Proof. 

Consider the function $\Omega \left(z\right)$ defined in $\mathbb{U}$ by

$$\begin{array}{ccc}\hfill \Omega \left(z\right)& =& \frac{{\Gamma }_q(2-\mu )}{{\Gamma }_q\left(2\right)}{z}^{\mu }{D}_{q,z}^{\mu }f\left(z\right),\hfill \\ & =& z+\sum _{k=2}^{\infty }\frac{{\Gamma }_q(k+1){\Gamma }_q(2-\mu )}{{\Gamma }_q(k+1-\mu ){\Gamma }_q\left(2\right)}{a}_k{z}^k\hfill \\ & =& z+\sum _{k=2}^{\infty }\Psi \left(k\right)a_k{z}^k,\hfill \end{array}$$

where

$$\Psi \left(k\right)=\frac{{\Gamma }_q(k+1){\Gamma }_q(2-\mu )}{{\Gamma }_q(k+1-\mu ){\Gamma }_q\left(2\right)},(\mu >0,k\ge 2).$$

Let

$$g\left(k\right)=\frac{\Psi \left(k\right)}{{\left[k\right]}_q}(\mu >0,k\ge 2).$$

(23)

Since $g\left(k\right)$ is a decreasing function of$k(k\ge 2)$, it is easily seen from (23) that

$$0<g\left(k\right)\le g\left(2\right)=\frac{{\Gamma }_q\left(3\right){\Gamma }_q(2-\mu )}{{\left[2\right]}_q{\Gamma }_q(2-\mu ){\Gamma }_q\left(2\right)}.$$

(24)

From Theorem 4, we know that

$${\left[2\right]}_q^{n-1}\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\le \sum _{k=2}^{\infty }{\left[k\right]}_q^n{a}_k\le 1-\alpha ,a_k\ge 0,$$

that is, that

$$\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\le \frac{1}{{\left[2\right]}_q^{n-1}},$$

(25)

since

$$\Omega \left(z\right)=z+\sum _{k=2}^{\infty }g\left(k\right){\left[k\right]}_q{a}_k{z}^k.$$

(26)

Therefore, by using (24) and (25), we see that

$$\begin{array}{ccc}\hfill \left|\Omega \left(z\right)\right|& \ge & \left|z\right|-g\left(2\right){\left|z\right|}^2\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\hfill \\ & =& \left|z\right|-\frac{g\left(2\right)}{{\left[2\right]}_q^{n-1}}{\left|z\right|}^2\hfill \\ & =& \left|z\right|-\frac{{\Gamma }_q\left(3\right){\Gamma }_q(2+\mu )}{{\left[2\right]}_q^n{\Gamma }_q(3-\mu ){\Gamma }_q\left(2\right)}{\left|z\right|}^2\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left|\Omega \left(z\right)\right|& \le & \left|z\right|+g\left(2\right){\left|z\right|}^2\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\hfill \\ & \le & \left|z\right|+\frac{g\left(2\right)}{{\left[2\right]}_q^{n-1}}{\left|z\right|}^2\hfill \\ & =& \left|z\right|+\frac{{\Gamma }_q\left(3\right){\Gamma }_q(2+\mu )}{{\left[2\right]}_q^n{\Gamma }_q(3-\mu ){\Gamma }_q\left(2\right)}{\left|z\right|}^2.\hfill \end{array}$$

This completes the proof. □

5. Conclusions

Quantum calculus is classical calculus without limits. Due to its application in various areas of mathematics and physics, it has recently attracted the extraordinary interest of many researchers. It was Jackson [9,10], who developed, at the beginning of the 20th century, the theory of q-Calculus. In geometric function theory, various subclasses of normalized analytic functions in the open unit disc involving q-derivatives have already been investigated. The q-analogue differential subordinations have also been used in the q-theory of analytic functions without being proven. In this paper, we extend Miller and Mocanu’s result [23] to the q-theory of analytic functions. Furthermore, the Salagean q-differential operator is used to investigate a new subclass of analytic functions in the open unit disc, and some inclusion relations, coefficient conditions, convolution properties, and applications of the q-calculus are presented.