On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator

Abstract

This paper introduces and studies a new class of analytic p-valent functions in the open symmetric unit disc involving the Sălăgean-type q-difference operator. Furthermore, we present several interesting subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems.

1. Introduction

As a result of Euler and Heine’s pioneering work, Frank Hilton Jackson developed q-calculus in a systematic manner at the beginning of the previous century. In his work, Jackson systematically developed the concepts of the q-derivative (Jackson [1]), as well as the q-integral (Jackson [2]). Calculus without limits is called q-calculus. Due to its applications in mathematics, mechanics, and physics, symmetric q-calculus is experiencing rapid growth. Ismail et al. [3] were the first to apply q-calculus to geometric function theory (GFT) by generalizing the set of starlike functions into q-analogs, called q-starlike functions. Several authors have extensively investigated the q-difference operator in GFT based on the same idea. Some recent works related to this operator on analytic functions include [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Several properties of certain analytic multivalent functions are considered in this paper using the q-analog of the Sălăgean differential operator. Let ${\mathcal{A}}_p\left(j\right)$ denote the class of functions that have the form

$$f\left(z\right)=z^p+{\displaystyle \sum _{l=p+j}^{\infty }}{a}_l{z}^l,(p,j\in \mathbb{N}:=\{1,2,\dots \}),$$

(1)

that are analytic in the open unit disc $\mathbb{E}=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}.$ Let $\mathcal{A}={\mathcal{A}}_1\left(1\right).$ In [1,2], the q-derivative operator ${\partial }_q$ of a function f was defined by Jackson as follows:

$${\partial }_{q}f\left(z\right)=\left\{\begin{array}{cc}\frac{f\left(qz\right)-f\left(z\right)}{(q-1)z}\hfill & (z\ne 0),\hfill \\ f^{{}^{\prime }}\left(0\right)\hfill & (z=0).\hfill \end{array}\right.$$

(2)

For a function $f\left(z\right)\in {\mathcal{A}}_p\left(j\right)$, we deduce that

$${\partial }_{q}f\left(z\right)={\left[p\right]}_q{z}^{p-1}+\sum _{l=2}^{\infty }{\left[l\right]}_q{a}_l{z}^{l-1},$$

(3)

where

$${\left[l\right]}_q=\frac{1-q^l}{1-q}.$$

As $q\to 1^{-},$${\left[l\right]}_q\to l.$ Jackson [1] introduced the q-integral

$$\underset{0}{\overset{z}{\int }}f\left(t\right)d_{q}t=z(1-q)\sum _{l=0}^{\infty }{q}^{l}f\left(z{q}^l\right),$$

as long as the series converges. For a function $f\left(z\right)=z^l,$ one can observe that

$$\underset{0}{\overset{z}{\int }}f\left(t\right)d_{q}t=\underset{0}{\overset{z}{\int }}{t}^l{d}_{q}t=\frac{1}{{[l+1]}_q}{z}^{l+1}(l\ne -1).$$

For a function $f\left(z\right)\in {\mathcal{A}}_p\left(j\right),$ El-Qadeem and Mamon [23] defined the p-valent q-Sălăgean operator by

$$\begin{array}{ccc}\hfill D_{p,q}^{0}f\left(z\right)& =& f\left(z\right),\hfill \\ \hfill D_{p,q}^{1}f\left(z\right)& =& D_{p,q}f\left(z\right)=\frac{z{\partial }_{q}f\left(z\right)}{{\left[p\right]}_q}=z^p+\sum _{l=p+j}^{\infty }{a}_l\frac{{\left[l\right]}_q}{{\left[p\right]}_q}{z}^l,\hfill \\ \hfill D_{p,q}^{2}f\left(z\right)& =& D_{p,q}\left(D_{p,q}f\left(z\right)\right)=z^p+\sum _{l=p+j}^{\infty }{a}_l{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^2{z}^l,\hfill \end{array}$$

therefore,

$$D_{p,q}^{n}f\left(z\right)=D_{p,q}\left(D_{p,q}^{n-1}f\left(z\right)\right)=z^p+{\displaystyle \sum _{l=p+j}^{\infty }}{a}_l{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n{z}^l.$$

(4)

When $p=1$, the q-Sălăgean operator was introduced by Govindaraj and Sivasubramanian [24]. The q-shifted factorial, see [25], is defined for $a\in \mathbb{C}$ by

$${(a;q)}_n=\left\{\begin{array}{c}\hspace{1em} 1\mathrm{if}n=0,\\ (1-a)(1-aq)(1-a{q}^2)\dots (1-a{q}^{n-1}),\mathrm{if}n\in \mathbb{N}=\{1,2,\dots \},\end{array}\right.$$

let ${(a;q)}_{\infty }={\displaystyle \prod _{n=0}^{\infty }}(1-a{q}^n).$ Recalling the q-analog definitions given by Gasper and Rahman [26], the q-Gamma function is given by

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

and the q-binomial expansion is given by

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

For functions f and g analytic in $\mathbb{E}$, one can say that f is subordinate to g, written as f$\prec g$ or $f\left(z\right)\prec g\left(z\right)(z\in \mathbb{E}),$ if there exists a Schwarz function $\omega$, that is analytic in $\mathbb{E}$ with $\omega \left(0\right)=0,$$\left|\omega \left(z\right)\right|<1$ and $f\left(z\right)=g\left(\omega \left(z\right)\right)$ $(z\in \mathbb{E}).$ In addition, if the function g is univalent in $\mathbb{E}$, then the following equivalence will occur

$$f\left(z\right)\prec g\left(z\right)\iff f\left(0\right)\prec g\left(0\right),$$

and

$$f\left(\mathbb{E}\right)\subset g\left(\mathbb{E}\right).$$

For functions $f_j\left(z\right)={\sum }_{l=0}^{\infty }{a}_{l,j}{z}^l$$\left(j=1,2\right)$ analytic in $\mathbb{E}$, the Hadamard product (or convolution) of $f_1\left(z\right)$ and $f_2\left(z\right)$ is defined by

$$\left(f_1\ast f_2\right)\left(z\right)=\sum _{l=0}^{\infty }{a}_{l,1}{a}_{l,2}{z}^l=\left(f_2\ast f_1\right)\left(z\right)\left(z\in \mathbb{E}\right).$$

A function $f\in \mathcal{A}$ is convex, if and only if $f\left(\mathbb{E}\right)$ is a convex domain. We denote this subclass of $\mathcal{A}$ by K. Analytically, a function $f\in \mathcal{A}$ belongs to the class K if and only if

$$\Re \left\{1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)}\right\}>0\left(z\in \mathbb{E}\right).$$

The proof can be found in [27]. A similar characterization can be made for the class $S^{*}$ of functions starlike in $\mathbb{E}$. A function $f\in \mathcal{A}$ belongs to the class $S^{*}$ if and only if

$$\Re \left\{\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}\right\}>0\left(z\in \mathbb{E}\right).$$

More details on the classes of starlike and convex functions can be found in [28,29]. A univalent function $f:\mathbb{E}\to \widehat{C}=\mathbb{C}\cup \{\infty \}$ is said to be concave if the complement $\widehat{C}\backslash f$ is convex (functions mapping on the exterior of a convex curve). An analytic, univalent function $f\in \mathcal{A}$ is said to be in the class $C_o\left(\alpha \right)$, if it is concave, satisfies $f\left(1\right)=\infty$ with an opening angle of $f\left(\mathbb{E}\right)$ at ∞ less than or equal to $\alpha \pi$ with $\alpha \in (1,2].$ Due to the similarity with convex functions, sometimes the inequality

$$\Re \left\{1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)}\right\}<0\left(z\in \mathbb{E}\right),$$

is also used as a definition for concave functions (see e.g., [30]) see also, Avkhadiev et al. [31], Cruz and Pommerenke [32], and the references within. Recently, Nishiwaki and Owa [33] defined and studied the subclasses $\mathcal{M}\left(\beta \right)$ and $\mathcal{N}\left(\beta \right)$ of $\mathcal{A}$ as follows: for some $\beta \left(\beta >1\right),$ let $\mathcal{M}\left(\beta \right)$ be the subclass of $\mathcal{A}$ consisting of functions $f\left(z\right)$, which satisfy

$$\Re \left\{\frac{z{f}^{{}^{\prime }}\left(z\right)}{f\left(z\right)}\right\}<\beta \left(z\in \mathbb{E}\right),$$

let $\mathcal{N}\left(\beta \right)$ be the subclass of $\mathcal{A}$ consisting of functions $f\left(z\right)$, which satisfy

$$\Re \left\{1+\frac{z{f}^{{}^{″}}\left(z\right)}{f^{{}^{\prime }}\left(z\right)}\right\}<\beta \left(z\in \mathbb{E}\right),$$

(see [33,34,35,36]). With the use of the differential operator $D_{p,q}^n$, we introduce class ${\mathcal{A}}_{p,q}(n,j,\beta )$, which generalizes the above-mentioned classes $\mathcal{M}\left(\beta \right)$ and $\mathcal{N}\left(\beta \right)$.

Definition 1.

We say that a function $f\left(z\right)\in$${\mathcal{A}}_p\left(j\right)$ belongs to the class ${\mathcal{A}}_{p,q}(n,j,\beta )$, if it satisfies the condition

$$\Re \left\{\frac{z{\partial }_q\left(D_{p,q}^{n}f\left(z\right)\right)}{D_{p,q}^{n}f\left(z\right)}\right\}<\beta \left(z\in \mathbb{E}\right),$$

(5)

where $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},p\in \mathbb{N},\beta >{\left[p\right]}_q,$ and $0<q<1$.

As $f\left(z\right)=z^p$ belongs to the class ${\mathcal{A}}_{p,q}(n,j,\beta )$, it is not empty. ${\mathcal{A}}_{p,q}(n,j,\beta )$ generalizes the classes $\mathcal{M}\left(\beta \right)$ and $\mathcal{N}\left(\beta \right)$ as follows

Remark 1.

1.  ${\displaystyle \underset{q\to 1}{lim}}{\mathcal{A}}_{1,q}(0,1,\beta )=\mathcal{M}\left(\beta \right);$

2. 

${\displaystyle \underset{q\to 1}{lim}}{\mathcal{A}}_{1,q}(1,1,\beta )=\mathcal{N}\left(\beta \right)$.

In this paper, we derive some interesting subordination results, coefficient inequalities, and various distortion theorems involving fractional q-calculus operators for functions in the class ${\mathcal{A}}_{p,q}(n,j,\beta ).$ Moreover, some special cases are also indicated.

2. Coefficient Estimates

Theorem 1.

If $f\left(z\right)\in$${\mathcal{A}}_p\left(j\right)$ satisfies the condition

$$\sum _{l=p+j}^{\infty }{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q+\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|\right)\left|a_l\right|\le 2\left(\beta -{\left[p\right]}_q\right),$$

(6)

for some $\beta \left(\beta >{\left[p\right]}_q\right);n\in {\mathbb{N}}_0,$then $f\left(z\right)\in {\mathcal{A}}_{p,q}(n,j,\beta ).$

Proof. 

Let condition (6) be true. Then, we have

$$\begin{array}{ccc}& & \left|\frac{\frac{z{\partial }_q\left(D_{p,q}^{n}f\left(z\right)\right)}{D_{p,q}^{n}f\left(z\right)}-{\left[p\right]}_q}{\frac{z{\partial }_q\left(D_{p,q}^{n}f\left(z\right)\right)}{D_{p,q}^{n}f\left(z\right)}-\left(2\beta -{\left[p\right]}_q\right)}\right|\hfill \\ & =& \left|\frac{{\displaystyle {\displaystyle \sum _{l=p+j}^{\infty }}}{a}_l{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q\right)z^l}{-2\left(\beta -{\left[p\right]}_q\right)z^p+{\displaystyle \sum _{l=p+j}^{\infty }}{a}_l{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q+{\left[p\right]}_q-2\beta \right)z^l}\right|\hfill \\ & \le & \frac{{\left|z\right|}^{p+j}{\displaystyle \sum _{l=p+j}^{\infty }}\left|a_l\right|{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n({\left[l\right]}_q-{\left[p\right]}_q)}{2\left(\beta -{\left[p\right]}_q\right)-{\left|z\right|}^{p+j}{\displaystyle \sum _{l=p+j}^{\infty }}\left|a_l\right|{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|}\hfill \\ & \le & \frac{{\displaystyle \sum _{l=p+j}^{\infty }}\left|a_l\right|{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q\right)}{2\left(\beta -{\left[p\right]}_q\right)-{\displaystyle \sum _{l=p+j}^{\infty }}\left|a_l\right|{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|}\hfill \end{array}$$

the last expression is bounded above by 1 if

$$\sum _{l=p+j}^{\infty }{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q+\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|\right)\left|a_l\right|\le 2\left(\beta -{\left[p\right]}_q\right).$$

This completes the proof of Theorem 1. □

Remark 2.

Letting $p=1,q\to 1$, and $n=1$ in Theorem 1, we obtain the result obtained by Nishwaki and Owa [33].

Corollary 1.

If $f\left(z\right)\in$${\mathcal{A}}_p\left(j\right)$ satisfies the condition

$$\sum _{l=p+j}^{\infty }{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-\beta \right)\left|a_l\right|\le \left(\beta -{\left[p\right]}_q\right),$$

for some $\beta \left({\left[p\right]}_q\le \beta <\frac{{\left[p+j\right]}_q+{\left[p\right]}_q}{2}\right);n\in {\mathbb{N}}_0,$ then $f\left(z\right)\in {\mathcal{A}}_{p,q}(n,j,\beta ).$

Proof. 

Since $({\left[l\right]}_q+{\left[p\right]}_q-2\beta )$ is an increasing function of $l(l\ge p+j),$ we have

$${\left[l\right]}_q+{\left[p\right]}_q-2\beta \ge 0if{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \ge 0,$$

or

$$\beta \le \frac{{\left[p+j\right]}_q+{\left[p\right]}_q}{2}.$$

□

3. Subordination Results

Definition 2

([37]). A sequence ${\left\{b_l\right\}}_{l=1}^{\infty }$ of complex numbers is said to be subordinating factor sequence if, whenever $f\left(z\right)=z+{\sum }_{l=2}^{\infty }{a}_l{z}^l,$$a_1=1$ is analytic, univalent, and convex in $\mathbb{E}$, we have

$$\sum _{l=1}^{\infty }{b}_l{a}_l{z}^l\prec f\left(z\right)\left(z\in \mathbb{E}\right).$$

Lemma 1

([37]). The sequence ${\left\{b_l\right\}}_{l=1}^{\infty }$ is subordinating factor sequence if and only if

$$\Re \left(1+2\sum _{l=1}^{\infty }{b}_l{z}^l\right)>0\left(z\in \mathbb{E}\right).$$

Let ${\mathcal{A}}_{p,q}^{*}(n,j,\beta )$ denoted the class of functions $f\left(z\right)\in {\mathcal{A}}_p\left(j\right)$ whose coefficients satisfy the condition (6).

Theorem 2.

Let $f\left(z\right)$$\in {\mathcal{A}}_{p,q}^{*}(n,j,\beta )$, $g\left(z\right)\in K$, and

$$\epsilon =\frac{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}{2\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}},$$

then

$$\left(\epsilon z^{1-p}f\left(z\right)\right)\ast g\left(z\right)\prec g\left(z\right)(z\in \mathbb{E}),$$

(7)

and

$$\Re \left(\frac{f\left(z\right)}{z^{p-1}}\right)>\frac{-1}{2\epsilon }.$$

(8)

The constant

$$\frac{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}{2\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}},$$

is the best estimate.

Proof. 

Let $f\left(z\right)$$\in {\mathcal{A}}_{p,q}^{*}(n,j,\beta )$, and let $g\left(z\right)=z+{\sum }_{l=2}^{\infty }{c}_l{z}^l$ belong to the subclass $K.$ Then

$$\left(\epsilon z^{1-p}f\left(z\right)\right)\ast g\left(z\right)=\sum _{l=1}^{\infty }{b}_l{c}_l{z}^l(z\in \mathbb{E}),$$

where

$$b_l=\left\{\begin{array}{cc}\epsilon & (l=1),\\ 0& (2\le l\le j),\\ \epsilon a_{p+l-1}& (l\ge j+1).\end{array}\right.$$

Hence, by using Definition 2, the subordination result (7) will be true, if ${\left\{b_l\right\}}_{l=1}^{\infty }$ is the subordinating factor sequence. Since

$$\mathrm{\Psi }\left(l\right)={\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q+\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|\right),$$

(9)

is an increasing function of $l(l\ge j+1),$ we have

$$\begin{array}{ccc}& & \Re \left\{1+2\sum _{l=1}^{\infty }{b}_l{z}^l\right\}=\Re \left\{1+2\epsilon z+2\sum _{l=j+1}^{\infty }{b}_l{z}^l\right\}\hfill \\ & =& \Re \left\{1+\frac{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}{\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}}z\right.\hfill \\ & & +\frac{1}{\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}}\hfill \\ & & \left.\times \sum _{l=j+p}^{\infty }{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)a_l{z}^{l+1-p}\right\}\hfill \\ & \ge & \Re \left\{1-\frac{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}{\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}}r\right.\hfill \\ & & \left.-\sum _{l=j+p}^{\infty }\frac{{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q+\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)}\left|a_l\right|r^{j+1}\right\}\hfill \end{array}$$

Thus, by using Theorem 1, and Lemma 1 we deduce that

$$\begin{array}{ccc}& & \Re \left\{1+2\sum _{l=1}^{\infty }{b}_l{z}^k\right\}\hfill \\ & \ge & 1-\frac{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)r}{\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}}\hfill \\ & & -\frac{\left(\beta -{\left[p\right]}_q\right)}{\left\{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left(q^p{\left[j\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)+\left(\beta -{\left[p\right]}_q\right)\right\}}r\hfill \\ & >& 0.\hfill \end{array}$$

This proves the subordination result (7). Letting $g\left(z\right)=\frac{z}{1-z}={\sum }_{l=1}^{\infty }{z}^l$$(z\in \mathbb{E})$ in (7), we easily get the result (8). □

Theorem 3.

Let $f\left(z\right)$ be in the class ${\mathcal{A}}_{p,q}^{*}(n,j,\beta ),$ defined by (1). Then for $\left|z\right|=r<1$, we have

$$\begin{array}{ccc}& & \left(\frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}-\frac{2\left(\beta -{\left[p\right]}_q\right)\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}{r}^j}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}\right)r^{p-m}\hfill \\ & \le & \left|{\partial }_q^m\left(f\left(z\right)\right)\right|\le \hfill \\ & & \left(\frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}+\frac{2\left(\beta -{\left[p\right]}_q\right)\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}{r}^j}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}\right)r^{p-m}.\hfill \end{array}$$

The result is sharp for the function $f\left(z\right)$ given by

$$f\left(z\right)=z^p+\frac{2\left(\beta -{\left[p\right]}_q\right)\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}{z}^{p+j}.$$

(10)

Proof. 

Since $\mathrm{\Psi }\left(l\right)$ given by (9) is an increasing function of $l(l\ge j+1),$ Theorem 1 gives

$$\begin{array}{ccc}& & {\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)\sum _{l=p+j}^{\infty }\left|a_l\right|\hfill \\ & \le & \sum _{l=p+j}^{\infty }{\left(\frac{{\left[l\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[l\right]}_q-{\left[p\right]}_q+\left|{\left[l\right]}_q+{\left[p\right]}_q-2\beta \right|\right)\left|a_l\right|\le 2\left(\beta -{\left[p\right]}_q\right).\hfill \end{array}$$

That is

$$\sum _{l=p+j}^{\infty }\left|a_l\right|\le \frac{2\left(\beta -{\left[p\right]}_q\right)}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}.$$

(11)

The $m^{th}$q-derivative of the functions $f\left(z\right)\in {\mathcal{A}}_p\left(j\right)$ is given by

$${\partial }_q^m\left(f\left(z\right)\right)=\frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}{z}^{p-m}+\sum _{l=p+j}^{\infty }\frac{{\left[l\right]}_q!}{{\left[l-m\right]}_q!}{a}_l{z}^{l-m},$$

then we have

$$\begin{array}{ccc}\hfill \left|{\partial }_q^m\left(f\left(z\right)\right)\right|& \ge & \frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}{\left|z\right|}^{p-m}-\sum _{l=p+j}^{\infty }\frac{{\left[l\right]}_q!}{{\left[l-m\right]}_q!}\left|a_l\right|{\left|z\right|}^{l-m}\hfill \\ & \ge & \frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}{r}^{p-m}-r^{p+j-m}\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}\sum _{l=p+j}^{\infty }\left|a_l\right|\hfill \end{array}$$

$$\ge \left(\frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}-\frac{2\left(\beta -{\left[p\right]}_q\right)\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}{r}^j}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}\right)r^{p-m},$$

and

$$\begin{array}{ccc}\hfill \left|{\partial }_q^m\left(f\left(z\right)\right)\right|& \le & \frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}{\left|z\right|}^{p-m}+\sum _{l=p+j}^{\infty }\frac{{\left[l\right]}_q!}{{\left[l-m\right]}_q!}\left|a_l\right|{\left|z\right|}^{l-m}\hfill \\ & \le & \frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}{r}^{p-m}+\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}{r}^{p+j-m}\sum _{l=p+j}^{\infty }\left|a_l\right|\hfill \end{array}$$

$$\le \left(\frac{{\left[p\right]}_q!}{{\left[p-m\right]}_q!}+\frac{2\left(\beta -{\left[p\right]}_q\right)\frac{{\left[p+j\right]}_q!}{{\left[p+j-m\right]}_q!}{r}^j}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left({\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right)}\right)r^{p-m}.$$

□

Putting $m=0$ in Theorem 3 we have the following corollary

Corollary 2.

Let $f\left(z\right)$ defined by (1) be in the class ${\mathcal{A}}_{p,q}^{*}(n,j,\beta )$.Then for $\left|z\right|=r<1$, we have

$$\left|f\left(z\right)\right|\ge \left(1-\frac{2\left(\beta -{\left[p\right]}_q\right)r^j}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}\right)r^p,$$

and

$$\left|f\left(z\right)\right|\le \left(1+\frac{2\left(\beta -{\left[p\right]}_q\right)r^j}{{\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}\right)r^p.$$

This result is sharp.

4. Application of $\mathbf{q}$-Fractional Calculus Operators

Let the function $f\left(z\right)$ be defined by (1). Then the q-Bernardi integral operator ${\mathcal{J}}_{c,p}^q$ is given by

$${\mathcal{J}}_{c,p}^{q}f\left(z\right)=\frac{{\left[c+p\right]}_q}{z^c}{\int }_0^z{t}^{c-1}f\left(z\right)d_{q}t=z^p+{\displaystyle \sum _{l=p+j}^{\infty }}\frac{{\left[c+p\right]}_q}{{\left[c+l\right]}_q}{a}_l{z}^l\left(c>-p\right),$$

(12)

this operator introduced by El-Qadeem and Mamon [23] (see also [17,38]). For $f\left(z\right)\in {\mathcal{A}}_p\left(j\right),$ we define the following q-fractional calculus operators given by Purohit and Raina [39,40].

Definition 3.

The fractional q-integral operator of order $m(m>0)$ is defined, for a function $f,$ by

$${\mathrm{\Omega }}_{q,z}^{-m}f\left(z\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(m\right)}{\int }_0^z{\left(z-qt\right)}_{m-1}f\left(t\right)d_{q}t,$$

where f is analytic in a simply-connected region of the z-plane containing the origin and the function ${\left(z-qt\right)}_{-m}$ is single-valued when $\left|arg(-\frac{-t{q}^m}{z})\right|<\pi ,\left|\frac{t{q}^m}{z})\right|<1$ and $\left|argz\right|<\pi$.

Definition 4.

The fractional q-derivative operator of order m is defined, for a function $f,$ by

$${\mathrm{\Omega }}_{q,z}^m\left(f\left(z\right)\right)=\frac{1}{{\mathrm{\Gamma }}_q(1-m)}{\partial }_q{\int }_0^z{\left(z-qt\right)}_{-m}f\left(t\right)d_{q}t(1>m\ge 0),$$

where f suitably constrained and removing the multiplicity of ${\left(z-qt\right)}_{-m}$ as in Definition 3 above.

Remark 3.

From Definitions 3 and 4, we see that

$$\begin{array}{ccc}\hfill {\mathrm{\Omega }}_{q,z}^m{z}^{\gamma }& =& \frac{{\mathrm{\Gamma }}_q(\gamma +1)}{{\mathrm{\Gamma }}_q(\gamma -m+1)}{z}^{\gamma -m}(m\ge 0,\gamma >-1),\hfill \\ \hfill {\mathrm{\Omega }}_{q,z}^{-m}{z}^{\gamma }& =& \frac{{\mathrm{\Gamma }}_q(\gamma +1)}{{\mathrm{\Gamma }}_q(\gamma +m+1)}{z}^{\gamma +m}(m>0,\gamma >-1).\hfill \end{array}$$

This gives that, for $f\left(z\right)\in {\mathcal{A}}_p\left(j\right),$

$${\mathrm{\Omega }}_{q,z}^{-m}f\left(z\right)=\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{z}^{p+m}+{\displaystyle \sum _{l=p+j}^{\infty }}\frac{{\mathrm{\Gamma }}_q\left(l+1\right)}{{\mathrm{\Gamma }}_q\left(l+1+m\right)}{a}_l{z}^{l+m},$$

(13)

and

$${\mathrm{\Omega }}_{q,z}^{m}f\left(z\right))=\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}{z}^{p-m}+{\displaystyle \sum _{l=p+j}^{\infty }}\frac{{\mathrm{\Gamma }}_q\left(l+1\right)}{{\mathrm{\Gamma }}_q\left(l+1-m\right)}{a}_l{z}^{l-m}.$$

(14)

Using the formulas (13), (14), and (12), we have

$${\mathrm{\Omega }}_{q,z}^{-m}\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)=\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{z}^{p+m}+{\displaystyle \sum _{l=p+j}^{\infty }}\frac{{\left[c+p\right]}_q}{{\left[c+l\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(l+1\right)}{{\mathrm{\Gamma }}_q\left(l+1+m\right)}{a}_l{z}^{l+m},$$

(15)

$${\mathrm{\Omega }}_{q,z}^m\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)=\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}{z}^{p-m}+{\displaystyle \sum _{l=p+j}^{\infty }}\frac{{\left[c+p\right]}_q}{{\left[c+l\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(l+1\right)}{{\mathrm{\Gamma }}_q\left(l+1-m\right)}{a}_l{z}^{l-m},$$

(16)

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{-m}f\left(z\right)\right)& =& \frac{{\left[c+p\right]}_q}{{\left[c+p+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{z}^{p+m}\hfill \\ & & +\sum _{l=p+j}^{\infty }\frac{{\left[c+p\right]}_q}{{\left[c+l+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(l+1\right)}{{\mathrm{\Gamma }}_q\left(l+1+m\right)}{a}_l{z}^{l+m},\hfill \end{array}$$

(17)

and

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{m}f\left(z\right)\right)& =& \frac{{\left[c+p\right]}_q}{{\left[c+p-m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}{z}^{p-m}\hfill \\ & & +\sum _{l=p+j}^{\infty }\frac{{\left[c+p\right]}_q}{{\left[c+l-m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(l+1\right)}{{\mathrm{\Gamma }}_q\left(l+1-m\right)}{a}_l{z}^{l-m}.\hfill \end{array}$$

(18)

Here, we investigate the distortion properties of functions in the class ${\mathcal{A}}_{p,q}^{*}(n,j,\beta )$ involving the operators ${\mathcal{J}}_{q,z}^q,{\mathrm{\Omega }}_{q,z}^{-m}$, and ${\mathrm{\Omega }}_{q,z}^m.$

Theorem 4.

Let $f\left(z\right)$ be in the class ${\mathcal{A}}_{p,q}(n,j,\beta ),$ defined by (1). Then for $\left|z\right|=r<1$, we have

$$\left|{\mathrm{\Omega }}_{q,z}^{-m}\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)\right|\ge \left\{\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}-{\mathrm{{\rm Y}}}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m},$$

(19)

$$\left|{\mathrm{\Omega }}_{q,z}^{-m}\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)\right|\le \left\{\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}+{\mathrm{{\rm Y}}}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m},$$

(20)

where

$${\mathrm{{\rm Y}}}_{p,q}^{c,j,m}\left(n,\beta \right)=\frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right)(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}},$$

$(0\le m<1,c>-p,p\in \mathbb{N}).$ Each of the assertions are sharp for f given by (10).

Proof. 

By using (11) and (15), we have

$$\begin{array}{ccc}\hfill \left|{\mathrm{\Omega }}_{q,z}^{-m}\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)\right|& \ge & \frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{\left|z\right|}^{p+m}-\sum _{l=p+j}^{\infty }\frac{{\left[c+p\right]}_q}{{\left[c+l\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(k+1\right)}{{\mathrm{\Gamma }}_q\left(l+1+m\right)}\left|a_l\right|{\left|z\right|}^{l+m}\hfill \\ & \ge & \frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{\left|z\right|}^{p+m}-\frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+j+1\right)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right)}{\left|z\right|}^{p+j+m}\sum _{l=p+j}^{\infty }\left|a_l\right|\hfill \\ & \ge & \left\{\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}-{\mathrm{{\rm Y}}}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m},\hfill \end{array}$$

where

$${\mathrm{{\rm Y}}}_{p,q}^{c,j,m}\left(n,\beta \right)=\frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right)(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}.$$

Similarly, using (15) and (11) we have

$$\left|{\mathrm{\Omega }}_{q,z}^{-m}\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)\right|\le \left\{\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}+{\mathrm{{\rm Y}}}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m}.$$

Thus, the proof of the theorem is completed. □

Theorem 5.

Let $f\left(z\right)$ be in the class ${\mathcal{A}}_{p,q}(n,j,\beta ),$ defined by (1). Then for $\left|z\right|=r<1$, we have

$$\left|{\mathrm{\Omega }}_{q,z}^m\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)\right|\ge \left\{\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}-{\mathrm{\Theta }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p-m},$$

(21)

$$\left|{\mathrm{\Omega }}_{q,z}^m\left({\mathcal{J}}_{c,p}^{q}f\left(z\right)\right)\right|\le \left\{\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}+{\mathrm{\Theta }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p-m},$$

(22)

where

$$\begin{array}{ccc}& & {\mathrm{\Theta }}_{p,q}^{c,j,m}\left(n,\beta \right)\hfill \\ & =& \frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right)(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1-m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}.\hfill \end{array}$$

According to (10), each assertion is sharp.

Proof. 

Using (11) and (16), the assertions (21) and (22) of Theorem 5 can now be proved similarly to Theorem 4. □

Theorem 6.

Let $f\left(z\right)$ be in the class ${\mathcal{A}}_{p,q}(n,j,\beta ),$ defined by(1). Then for $\left|z\right|=r<1$, we have

$$\left|{\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{-m}f\left(z\right)\right)\right|\ge \left\{\frac{{\left[c+p\right]}_q}{{\left[c+p+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}-{\mathrm{\Lambda }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m}$$

$$\left|{\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{-m}f\left(z\right)\right)\right|\le \left\{\frac{{\left[c+p\right]}_q}{{\left[c+p+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}+{\mathrm{\Lambda }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m},$$

where

$${\mathrm{\Lambda }}_{p,q}^{c,j,m}\left(n,\beta \right)=\frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right)(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}},$$

$(m>0,c>-p,p\in \mathbb{N}).$ The result is sharp for the function f given by (10).

Proof. 

We only prove the first inequality. The argument for the second inequality is similar and hence omitted. Using (17) and (11), we have

$$\begin{array}{ccc}& & \left|{\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{-m}f\left(z\right)\right)\right|\hfill \\ & \ge & \frac{{\left[c+p\right]}_q}{{\left[c+p+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{\left|z\right|}^{p+m}-\frac{{\left[c+p\right]}_q}{{\left[c+p+j+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+j+1\right)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right)}{\left|z\right|}^{p+j+m}\sum _{l=p+j}^{\infty }\left|a_l\right|\hfill \\ & \ge & \frac{{\left[c+p\right]}_q}{{\left[c+p+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}{\left|z\right|}^{p+m}\hfill \\ & & -\frac{{\left[c+p\right]}_q}{{\left[c+p+j+m\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right){\left|z\right|}^{p+j+m}(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}\hfill \\ & =& \left\{\frac{{\left[c+p\right]}_q}{{\left[c+p+m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1+m\right)}-{\mathrm{\Lambda }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p+m},\hfill \end{array}$$

where

$${\mathrm{\Lambda }}_{p,q}^{c,j,m}\left(n,\beta \right)=\frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right)(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}}.$$

□

Theorem 7.

Let $f\left(z\right)$ be in the class ${\mathcal{A}}_{p,q}(n,j,\beta ),$ defined by (1). Then for $\left|z\right|=r<1$, we have

$$\left|{\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{m}f\left(z\right)\right)\right|\ge \left\{\frac{{\left[c+p\right]}_q}{{\left[c+p-m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}-{\mathrm{\Phi }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p-m},$$

$$\left|{\mathcal{J}}_{c,p}^q\left({\mathrm{\Omega }}_{q,z}^{m}f\left(z\right)\right)\right|\le \left\{\frac{{\left[c+p\right]}_q}{{\left[c+p-m\right]}_q}\frac{{\mathrm{\Gamma }}_q\left(p+1\right)}{{\mathrm{\Gamma }}_q\left(p+1-m\right)}+{\mathrm{\Phi }}_{p,q}^{c,j,m}\left(n,\beta \right){\left|z\right|}^j\right\}{\left|z\right|}^{p-m},$$

where

$$\begin{array}{ccc}& & {\mathrm{\Phi }}_{p,q}^{c,j,m}\left(n,\beta \right)\hfill \\ & =& \frac{{\left[c+p\right]}_q}{{\left[c+p+j\right]}_q}\frac{2{\mathrm{\Gamma }}_q\left(p+j+1\right)(\beta -{\left[p\right]}_q)}{{\mathrm{\Gamma }}_q\left(p+j+1+m\right){\left(\frac{{\left[p+j\right]}_q}{{\left[p\right]}_q}\right)}^n\left\{{\left[p+j\right]}_q-{\left[p\right]}_q+\left|{\left[p+j\right]}_q+{\left[p\right]}_q-2\beta \right|\right\}},\hfill \end{array}$$

$(0\le m<1,c>-p,p\in \mathbb{N}).$ Each of the assertions are sharp for f given by (10).

Proof. 

As the same manner in proving Theorem 6, we can easily deduce the proof of this theorem. □

5. Conclusions

Quantum calculus is classical calculus without limits. The field of q-calculus has recently attracted researchers’ attention. Its application in various branches of mathematics and physics is responsible for this extraordinary interest. Jackson [1,2] was one of the first to define the q-analog to derivative the integral operators and provide some applications for them. Numerous subclasses of normalized analytic functions in the open symmetric unit disc associated with q-derivatives have already been investigated in geometric function theory. Using the Sălăgean q-difference operator, we introduce a new class of analytic p-valent functions in the open symmetric unit disc. Several subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems are also presented. The paper also generalizes some known results. For future work, we can study some new classes of analytic p-valent functions in the open symmetric unit disc in the same way.