Strong Sandwich-Type Results for Fractional Integral of the Extended q-Analogue of Multiplier Transformation

Abstract

In this research, we obtained several strong differential subordinations and strong differential superordinations, which gave sandwich-type results for the fractional integral of the extended q-analogue of multiplier transformation.

1. Introduction

Fractional calculus is being used nowadays as part of a growing variety of scientific domains, wherein researchers are developing and discovering novel applications. Additionally, q-calculus is used in physics, engineering, and a number of mathematical fields. In addition, combining fractional and q-calculus fields in geometric function theory have revealed a variety of noteworthy findings that Srivastava emphasized in a review paper that elucidates the relevance of the advancements and encourages more research on the topic [1].

In 1910, Jackson used the q-calculus in Mathematical Analysis to define the concepts of q-derivative [2] and q-integral [3]. Research incorporating quantum calculus features into geometric function theory studies began the development after Srivastava provided a general framework for these types of investigations in a chapter of a book released in 1989 [4]. The foundations laid by Srivastava have generated continued research in this context that have led to the development of numerous concepts among which q-analogue operators play an important part. Noteworthy operators generating numerous important results in geometric function theory can be listed as the q-analogue of the Sălăgean differential operator [5] for which new applications can be seen in [6,7,8]; the q-analogue of the Ruscheweyh differential operator developed by Răducanu and Kanas [9] and used in the research by Mohammed and Darus [10] and by Mahmood and Sokół [11]; the q-analogue of multiplier transformation [5,12]; and Bazilevič functions [13].

Romaguera and Antonino [14] employed the concept of strong differential subordination for the first time in their analysis of the strong differential subordination of Briot–Bouquet. It appeared as an extension of Mocanu and Miller’s classical concept of differential subordination [15,16].

The notion emerged in 2009 [17], laying the groundwork for the field of strong differential subordination. The researchers in this theory expanded the concepts from the well-known theory of differential subordination [18]. The strong differential superordination has a dual notion introduced in 2009 [19] that is based on the pattern established for classical field of differential superordination [20].

The next period showed the development of both theories. In [21] were given methods for determining the best subordinant for the strong differential superordination, and in [22] were studied special cases of strong differential superordinations and subordinations. By linking various operators to the research, such as the Liu–Srivastava operator [23], the Sălăgean differential operator [24], the Ruscheweyh operator [25], multiplier transformation [26,27], combinations of the Sălăgean and Ruscheweyh operators [28], the Komatu integral operator [29,30], or differential operators [31,32], wherein strong differential subordinations could be further obtained. Recently released papers [33,34,35,36] show that the topic is current and in the present.

The operator studied in this paper is defined by the Riemann–Liouville fractional integral applied to the q-multiplier transformation following a popular line of study, where a number of operators have been defined and explored utilizing fractional integrals such as the Riemann–Liouville or Atagana–Băleanu [37,38,39,40].

The endeavor starts by reviewing some of the standard terminology and symbols applied in geometric function theory.

Consider $\mathcal{H}(\Delta \times \overline{\Delta })$ the class of analytic functions in $\Delta \times \overline{\Delta },$ where $\Delta =\{t\in \mathbb{C}:|t|<1\}$ and $\overline{\Delta }=\{t\in \mathbb{C}:|t|\le 1\}$.

Special subclasses of $\mathcal{H}(\Delta \times \overline{\Delta })$ are defined in [41] regarding to the strong differential subordination and strong differential superordination theories:

$${\mathcal{A}}_{n\tau }^{\ast }=\{g(t,\tau )=t+a_{n+1}\left(\tau \right)t^{n+1}+\cdots \in \mathcal{H}(\Delta \times \overline{\Delta })\},$$

with ${\mathcal{A}}_{1\tau }^{\ast }={\mathcal{A}}_{\tau }^{\ast }$ and $a_k\left(\tau \right)$ holomorphic functions in $\overline{\Delta },$ $k\ge n+1$, $n\in \mathbb{N}$, and

$${\mathcal{H}}^{\ast }[a,n,\tau ]=\{g(t,\tau )=a+a_n\left(\tau \right)t^n+a_{n+1}\left(\tau \right)t^{n+1}+\cdots \in \mathcal{H}(\Delta \times \overline{\Delta })\},$$

with $a_k\left(\tau \right)$ holomorphic functions in $\overline{\Delta },$ $k\ge n$, $a\in \mathbb{C}$, and $n\in \mathbb{N}.$

The notion of strong differential subordination used in [14] and developed in [17,41] is defined below.

Definition 1 

([17]). The analytic function $h_1\left(t,\tau \right)$ is strongly subordinate to the analytic function $h_2\left(t,\tau \right)$, which is denoted $h_1\left(t,\tau \right)\prec \prec h_2\left(t,\tau \right),$ if there exists an analytic function f in Δ, with $f\left(0\right)=0,$ $\left|f\left(t\right)\right|<1$, $t\in \Delta ,$ and $h_1\left(t,\tau \right)=h_2\left(f\left(t\right),\tau \right),$ where $\tau \in \overline{\Delta }$.

Remark 1 

([17]). (i) When $h_2\left(t,\tau \right)$ is univalent in $\Delta ,$ ∀ $\tau \in \overline{\Delta },$ Definition 1 is equivalent, with $h_1\left(\Delta \times \overline{\Delta }\right)\subset h_2\left(\Delta \times \overline{\Delta }\right)$ and $h_1\left(0,\tau \right)=h_2\left(0,\tau \right),$ $\tau \in \overline{\Delta }.$

(ii) For the special case where $h_1\left(t,\tau \right)=h_1\left(t\right)$ and $h_2\left(t,\tau \right)=h_2\left(t\right),$ the strong differential subordination becomes the differential subordination.

To investigate strong differential subordinations, we need the following lemma:

Lemma 1 

([42]). Consider the univalent function w in $\Delta \times \overline{\Delta }$ and the analytic functions f and g in a domain $D\supset w\left(\Delta \times \overline{\Delta }\right)$ such that $g\left(z\right)\ne 0$ for $z\in w\left(\Delta \times \overline{\Delta }\right)$. Define the functions $F\left(t,\tau \right)=t{w}_t^{\prime }\left(t,\tau \right)g\left(w\left(t,\tau \right)\right)$ and $G\left(t,\tau \right)=f\left(w\left(t,\tau \right)\right)+F\left(t,\tau \right)$. Then, we have the following conditions:

(1) F is starlike univalent in $\Delta \times \overline{\Delta },$

(2) $Re\left({\displaystyle \frac{t{G}_t^{\prime }\left(t,\tau \right)}{F\left(t,\tau \right)}}\right)>0$ for $\left(t,\tau \right)\in \Delta \times \overline{\Delta }$,

(3) The analytic function u, having the properties $u\left(0,\tau \right)=w\left(0,\tau \right)$ and $u\left(\Delta \times \overline{\Delta }\right)\subseteq D,$ is a solution of the strong differential subordination.

$$f\left(u\left(t,\tau \right)\right)+t{u}_t^{\prime }\left(t,\tau \right)g\left(u\left(t,\tau \right)\right)\prec \prec f\left(w\left(t,\tau \right)\right)+t{w}_t^{\prime }\left(t,\tau \right)g\left(w\left(t,\tau \right)\right),$$

so the strong differential subordination holds as

$$u\left(t,\tau \right)\prec \prec w\left(t,\tau \right),\left(t,\tau \right)\in \Delta \times \overline{\Delta },$$

and w is the best dominant.

The strong differential superordination is defined below.

Definition 2 

([19]). The analytic function $h_1\left(t,\tau \right)$ is strongly superordinate to the analytic function $h_2\left(t,\tau \right)$, which is denoted $h_2\left(t,\tau \right)\prec \prec h_1\left(t,\tau \right),$ if there exists an analytic function f in Δ, with $f\left(0\right)=0,$ $\left|f\left(t\right)\right|<1,$ $t\in \Delta ,$ and $h_2\left(t,\tau \right)=h_1\left(f\left(t\right),\tau \right),$ $\tau \in \overline{\Delta }$.

Remark 2 

([19]). (i) When $h_1\left(t,\tau \right)$ is univalent in $\Delta ,$ where ∀ $\tau \in \overline{\Delta },$ Definition 2 is equivalent with $h_2\left(\Delta \times \overline{\Delta }\right)\subset h_1\left(\Delta \times \overline{\Delta }\right)$ and $h_2\left(0,\tau \right)=h_1\left(0,\tau \right),$ $\tau \in \overline{\Delta }.$

(ii) For the special cases $h_1\left(t,\tau \right)=h_1\left(t\right)$ and $h_2\left(t,\tau \right)=h_2\left(t\right),$ the strong differential superordination become the differential superordination.

Definition 3 

([43]). Denote by $Q^{\ast }=\{f\in \mathcal{H}(\Delta \times \overline{\Delta }):f$ the injective $\overline{\Delta }\times \overline{\Delta }\setminus E\left(f,\tau \right)$, $f_t^{\prime }\left(z,\tau \right)\ne 0,$ $z\in \partial \Delta \times \overline{\Delta }\setminus E\left(f,\tau \right)\}$, with $E\left(f,\tau \right)=\{z\in \partial \Delta :\underset{t\to z}{lim}f\left(t,\tau \right)=\infty \}$ and $Q^{\ast }\left(a\right)$ being the subclass of $Q^{\ast }$ and with $f\left(0,\tau \right)=a$.

To investigate strong differential superordinations, we need the following lemma:

Lemma 2 

([42]). Consider the convex univalent function w in $\Delta \times \overline{\Delta }$ and the analytic functions f and g in a domain $D\supset w\left(\Delta \times \overline{\Delta }\right).$ The, we have the following conditions:

(1) Function $F\left(t,\tau \right)=t{w}_t^{\prime }\left(t,\tau \right)g\left(w\left(t,\tau \right)\right)$ is starlike univalent in $\Delta \times \overline{\Delta },$

(2) $Re\left({\displaystyle \frac{f_t^{\prime }\left(w\left(t,\tau \right)\right)}{g\left(w\left(t,\tau \right)\right)}}\right)>0$ for $\left(t,\tau \right)\in \Delta \times \overline{\Delta }$,

(3) The function $f\left(u\left(t,\tau \right)\right)+t{u}_t^{\prime }\left(t\right)g\left(u\left(t,\tau \right)\right)$ is univalent in $\Delta \times \overline{\Delta },$

(4) The function $u\left(t,\tau \right)\in {\mathcal{H}}^{\ast }\left[w\left(0,\tau \right),1,\tau \right] \cap Q^{\ast }$, with $u\left(\Delta \times \overline{\Delta }\right)\subseteq D,$ satisfies the strong differential superordination.

$$f\left(w\left(t,\tau \right)\right)+t{w}_t^{\prime }\left(t,\tau \right)g\left(w\left(t,\tau \right)\right)\prec \prec f\left(u\left(t,\tau \right)\right)+t{u}_t^{\prime }\left(t,\tau \right)g\left(u\left(t,\tau \right)\right),$$

so the strong differential superordination holds as

$$w\left(t,\tau \right)\prec \prec u\left(t,\tau \right),\left(t,\tau \right)\in \Delta \times \overline{\Delta },$$

and w is the best subordinant.

We remind the definition of Riemann–Liouville fractional integral [44,45] applied to a function $f\in {\mathcal{A}}_{\zeta }^{\ast }$.

Definition 4 

([44,45]). The fractional integral of order α ($\alpha >0$) applied to an analytic function f is defined by

$$D_t^{-\alpha }f\left(t,\tau \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{f\left(y,\tau \right)}{{\left(t-y\right)}^{1-\alpha }}}dy,$$

with condition $log\left(t-y\right)$ to be real when $\left(t-y\right)>0.$

The q-analogue of multiplier transformation is defined below.

Definition 5 

([5]). The q-analogue of multiplier transformation is defined by

$${\mathcal{I}}_q^{m,l}f\left(t\right)=t+\sum _{k=2}^{\infty }{\left({\displaystyle \frac{{\left[l+k\right]}_q}{{\left[l+1\right]}_q}}\right)}^m{a}_k{t}^k,$$

where $q\in \left(0,1\right)$, $m,l\in \mathbb{R}$, $l>-1$, and $f\left(t\right)=t+{\sum }_{k=2}^{\infty }{a}_k{t}^k\in \mathcal{A}$, $t\in \Delta .$

2. Main Results

A new operator defined by applying the Riemann–Liouville fractional integral to the extended q-analogue of multiplier transformation is introduced in the following:

Definition 6. 

Let $q,m,l,\alpha$ be real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha >0.$ The fractional integral applied to the extended q-analogue of multiplier transformation is defined by

$$D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{{\mathbb{I}}_q^{m,l}f\left(y,\tau \right)}{{\left(t-y\right)}^{1-\alpha }}}dy=$$

(1)

$${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{y}{{\left(t-y\right)}^{1-\alpha }}}dy+\sum _{k=2}^{\infty }{\left({\displaystyle \frac{{\left[l+k\right]}_q}{{\left[l+1\right]}_q}}\right)}^m{a}_k\left(\tau \right){\int }_0^t{\displaystyle \frac{y^k}{{\left(t-y\right)}^{1-\alpha }}}dy.$$

After a simple calculation, it takes the following form:

$$D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)={\displaystyle \frac{1}{\Gamma \left(\alpha +2\right)}}{t}^{\alpha +1}+\sum _{k=2}^{\infty }{\left({\displaystyle \frac{{\left[l+k\right]}_q}{{\left[l+1\right]}_q}}\right)}^m{\displaystyle \frac{\Gamma \left(k+1\right)}{\Gamma \left(k+\alpha +1\right)}}{a}_k\left(\tau \right)t^{k+\alpha },$$

(2)

when $f(t,\tau )=t+{\sum }_{k=2}^{\infty }{a}_k\left(\tau \right)t^k\in {\mathcal{A}}_{\tau }^{\ast }$. We note that $D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)\in \mathcal{H}\left[0,\alpha +1,\tau \right].$

The strong differential subordination result obtained using the operator given by (2) is the next theorem.

Theorem 1. 

Consider ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\in \mathcal{H}\left(\Delta \times \overline{\Delta }\right)$ and a univalent function $w\left(t,\tau \right)$ in $U\times \overline{U}$ with the property $w\left(t,\tau \right)\ne 0$, ∀$t\in \Delta \setminus \left\{0\right\},$$\tau \in \overline{\Delta }$, $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0$ Assuming that the function ${\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}$ is starlike univalent in $\Delta \times \overline{\Delta }$ and that

$$Re\left(1+{\displaystyle \frac{b}{d}}w\left(t,\tau \right)+{\displaystyle \frac{2c}{d}}{\left(w\left(t,\tau \right)\right)}^2-{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}+{\displaystyle \frac{t{w}_{t^2}^{\prime \prime }\left(t,\tau \right)}{w_t^{\prime }\left(t,\tau \right)}}\right)>0,$$

(3)

for $a,b,c,d\in \mathbb{C}$, where $d\ne 0$, $t\in \Delta \setminus \left\{0\right\},$ $\tau \in \overline{\Delta },$ we denote

$$H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right):=a+b{\left[{\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right]}^n+$$

(4)

$$c{\left[{\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right]}^{2n}+dn\left[{\displaystyle \frac{t{\left(D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)\right)}_t^{\prime }}{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}}-1\right].$$

If w is a solution of the strong subordination

$$H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)\prec \prec a+bw\left(t,\tau \right)+c{\left(w\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}},$$

(5)

then w is the best dominant of the strong subordination

$${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec w\left(t,\tau \right),\left(t,\tau \right)\in \Delta \times \overline{\Delta }.$$

(6)

Proof. 

Considering the function $u\left(t,\tau \right)={\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n$, $t\in \Delta \setminus \left\{0\right\},$ $\tau \in \overline{\Delta }$, differentiating it with respect to t, we obtain $u_t^{\prime }\left(t,\tau \right)=n{\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^{n-1}\left[{\displaystyle \frac{{\left(D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)\right)}_t^{\prime }}{t}}-{\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t^2}}\right]$ = $n{\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^{n-1}{\displaystyle \frac{{\left(D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)\right)}_t^{\prime }}{t}}-{\displaystyle \frac{n}{t}}u\left(t,\tau \right),$ and this yields ${\displaystyle \frac{t{u}_t^{\prime }\left(t,\tau \right)}{u\left(t,\tau \right)}}=n\left[{\displaystyle \frac{t{\left(D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)\right)}_t^{\prime }}{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}}-1\right].$

Define the analytic functions $f\left(z\right)=a+bz+c{z}^2$ and $g\left(z\right)={\displaystyle \frac{d}{z}}$, with $g\left(z\right)\ne 0,$ $z\in \mathbb{C}\setminus \left\{0\right\}.$

Define also the functions $F\left(t,\tau \right)=t{w}_t^{\prime }\left(t,\tau \right)g\left(w\left(t,\tau \right)\right)=d{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}$ and $G\left(t,\tau \right)=f\left(w\left(t,\tau \right)\right)+F\left(t,\tau \right)=a+bw\left(t,\tau \right)+c{\left(w\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}$.

We will check the conditions from Lemma 1. It is evident that $F\left(t,\tau \right)$ is starlike univalent.

Differentiating the function G with respect to t, we obtain $G_t^{\prime }\left(t,\tau \right)=d+w_t^{\prime }\left(t,\tau \right)+2cw\left(t,\tau \right)w_t^{\prime }\left(t,\tau \right)+d{\displaystyle \frac{\left(w_t^{\prime }\left(t,\tau \right)+t{w}_{t^2}^{\prime \prime }\left(t,\tau \right)\right)w\left(t,\tau \right)-t{\left(w_t^{\prime }\left(t,\tau \right)\right)}^2}{{\left(w\left(t,\tau \right)\right)}^2}}$ and ${\displaystyle \frac{t{G}_t^{\prime }\left(t,\tau \right)}{F\left(t,\tau \right)}}={\displaystyle \frac{t{G}_t^{\prime }\left(t,\tau \right)}{d\frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}}=$ $1+{\displaystyle \frac{b}{d}}w\left(t,\tau \right)+{\displaystyle \frac{2c}{d}}{\left(w\left(t,\tau \right)\right)}^2-{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}+{\displaystyle \frac{t{w}_{t^2}^{\prime \prime }\left(t,\tau \right)}{w_t^{\prime }\left(t,\tau \right)}}.$

The second condition Re$\left({\displaystyle \frac{t{G}_t^{\prime }\left(t,\tau \right)}{F\left(t,\tau \right)}}\right)=$ Re$(1+{\displaystyle \frac{b}{d}}w\left(t,\tau \right)+{\displaystyle \frac{2c}{d}}{\left(w\left(t,\tau \right)\right)}^2-{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}$ + ${\displaystyle \frac{t{w}_{t^2}^{\prime \prime }\left(t,\tau \right)}{w_t^{\prime }\left(t,\tau \right)}})>0$ is true from the relation (3).

We obtain the function $a+bu\left(t,\tau \right)+c{\left(u\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{u}_t^{\prime }\left(t,\tau \right)}{u\left(t,\tau \right)}}=a+b{\left[{\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right]}^n$ + $c{\left[{\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right]}^{2n}+$ $d\alpha \left[{\displaystyle \frac{t{\left(D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)\right)}_t^{\prime }}{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}}-1\right]=H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ from the function from relation (4).

Strong differential subordination (5) can be written in the following form: $a+bu\left(t,\tau \right)+c{\left(u\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{u}_t^{\prime }\left(t,\tau \right)}{u\left(t,\tau \right)}}\prec \prec a+bw\left(t,\tau \right)+c{\left(w\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}.$

With the conditions from Lemma 1 being fulfilled, we obtain $u\left(t,\tau \right)\prec \prec w\left(t,\tau \right)$, $\left(t,\tau \right)\in \Delta \times \overline{\Delta },$ which iswritten as ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec w\left(t,\tau \right)$, with w the being best dominant. □

Corollary 1. 

Assuming that relation (3) takes place for $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0$, if the strong differential subordination

$$H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)\prec \prec a+b{\displaystyle \frac{Mt+\tau }{Nt+\tau }}+c{\left({\displaystyle \frac{Mt+\tau }{Nt+\tau }}\right)}^2+d{\displaystyle \frac{\left(M-N\right)t\tau }{\left(Mt+\tau \right)\left(Nt+\tau \right)}},$$

is verified for $a,b,c,d\in \mathbb{C}$, $d\ne 0,$ $-1\le N<M\le 1$, and the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ is defined by relation (4), then ${\displaystyle \frac{Mt+\tau }{Nt+\tau }}$ is the best dominant for the strong differential subordination

$${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec {\displaystyle \frac{Mt+\tau }{Nt+\tau }},\left(t,\tau \right)\in \Delta \times \Delta .$$

Corollary 2. 

Assuming that relation (3) takes place for $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0$, if the strong differential subordination

$$H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)\prec \prec a+b{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^p+c{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{2p}+d{\displaystyle \frac{2pt\tau }{{\tau }^2-t^2}}$$

is verified for $a,b,c,d\in \mathbb{C}$, $0<p\le 1$, and $d\ne 0,$ and the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ is defined by relation (4), then ${\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^p$ is the best dominant for the strong differential subordination

$${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec {\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^p,\left(t,\tau \right)\in \Delta \times \overline{\Delta }.$$

The strong differential superordination result obtained using the operator given by (2) is the next theorem.

Theorem 2. 

Consider that the analytic and univalent function w in $\Delta \times \overline{\Delta }$, with the properties $w\left(t,\tau \right)\ne 0$ and ${\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}$, is starlike univalent. Suppose that

$$Re\left({\displaystyle \frac{2c}{d}}{\left(w\left(t,\tau \right)\right)}^2+{\displaystyle \frac{b}{d}}w\left(t,\tau \right)\right)>0,forb,c,d\in \mathbb{C},d\ne 0.$$

(7)

If ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\in \mathcal{H}\left[w\left(0,\tau \right),\left(\alpha -1\right)n,\tau \right] \cap Q^{\ast }$, the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ defined by the relation (4), is univalent in $\Delta \times \overline{\Delta }$, then the strong differential subordination

$$a+bw\left(t,\tau \right)+c{\left(w\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}\prec \prec H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$$

(8)

is endowed for $a,b,c,d\in \mathbb{C}$, $d\ne 0$, $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0;$ then, w is the best subordinant for the following strong differential superordination

$$w\left(t,\tau \right)\prec \prec {\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n,\left(t,\tau \right)\in \Delta \times \overline{\Delta }.$$

(9)

Proof. 

Considering again the function $u\left(t,\tau \right)={\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n$, $\left(t,\tau \right)\in \left(\Delta \setminus \left\{0\right\}\right)\times \overline{\Delta }$, and the analytic functions $f\left(z\right)=a+bz+c{z}^2$ and $g\left(z\right)={\displaystyle \frac{d}{z}},$ with $g\left(z\right)\ne 0,$ $z\in \mathbb{C}\setminus \left\{0\right\}$, we verify the conditions from Lemma 2.

Taking into account that ${\displaystyle \frac{f_t^{\prime }\left(w\left(t,\tau \right)\right)}{g\left(w\left(t,\tau \right)\right)}}={\displaystyle \frac{w_t^{\prime }\left(t,\tau \right)\left[b+2cw\left(t,\tau \right)\right]w\left(t,\tau \right)}{d}}$, it follows that Re$\left({\displaystyle \frac{f_t^{\prime }\left(w\left(t,\tau \right)\right)}{g\left(w\left(t,\tau \right)\right)}}\right)$ = Re$\left({\displaystyle \frac{2c}{d}}{\left(w\left(t,\tau \right)\right)}^2+{\displaystyle \frac{b}{d}}w\left(t,\tau \right)\right)>0$, for $b,c,d\in \mathbb{C},$ $d\ne 0$ by relation (7).

The strong differential superordination of (8) can be written as

$$a+bw\left(t,\tau \right)+c{\left(w\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{w}_t^{\prime }\left(t,\tau \right)}{w\left(t,\tau \right)}}\prec \prec a+bu\left(t,\tau \right)+c{\left(u\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{u}_t^{\prime }\left(t,\tau \right)}{u\left(t,\tau \right)}}.$$

With the conditions from Lemma 2 being fulfilled, we obtain

$$w\left(t,\tau \right)\prec \prec u\left(t,\tau \right)={\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n,\left(t,\tau \right)\in \Delta \times \overline{\Delta },$$

and w is the best subordinant. □

Corollary 3. 

Assuming that relation (7) takes place and that ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\in \mathcal{H}[w\left(0,\tau \right),$ $\left(\alpha -1\right)n$, $\tau ] \cap Q^{\ast }$ for $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0$, if the strong differential superordination

$$a+b{\displaystyle \frac{Mt+\tau }{Nt+\tau }}+c{\left({\displaystyle \frac{Mt+\tau }{Nt+\tau }}\right)}^2+d{\displaystyle \frac{\left(M-N\right)t\tau }{\left(Mt+\tau \right)\left(Nt+\tau \right)}}\prec \prec H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right),$$

is fulfilled for $a,b,c,d\in \mathbb{C}$, $d\ne 0,$ $-1\le N<M\le 1,$, and the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ is defined by relation (4), then ${\displaystyle \frac{Mt+\tau }{Nt+\tau }}$ is the best subordinant for the strong differential superordination

$${\displaystyle \frac{Mt+\tau }{Nt+\tau }}\prec \prec {\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n,\left(t,\tau \right)\in \Delta \times \overline{\Delta }.$$

Corollary 4. 

Assuming that relation (7) takes place and that ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\in \mathcal{H}[w\left(0,\tau \right)$, $\left(\alpha -1\right)n$, $\tau ] \cap Q^{\ast }$ for $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0$, if the strong differential superordination

$$a+b{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^p+c{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{2p}+d{\displaystyle \frac{2pt\tau }{{\tau }^2-t^2}}\prec \prec H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right),$$

is fulfilled for $a,b,c,d\in \mathbb{C}$, $d\ne 0,$ $0<p\le 1$, and the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ is defined by relation (4), then ${\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^p$ is the best subordinant for the strong differential superordination

$${\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^p\prec \prec {\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n,\left(t,\tau \right)\in \Delta \times \overline{\Delta }.$$

Looking at Theorems 1 and 2 together, they generate a strong sandwich-type result.

Theorem 3. 

Consider the analytic and univalent functions $w_1$ and $w_2$ in $\Delta \times \overline{\Delta }$ with the properties $w_1\left(t,\tau \right)\ne 0,$ $w_2\left(t,\tau \right)\ne 0$, ∀ $\left(t,\tau \right)\in \Delta \times \overline{\Delta }$. Assuming that the functions ${\displaystyle \frac{t{\left(w_1\right)}_t^{\prime }\left(t,\tau \right)}{w_1\left(t,\tau \right)}},$${\displaystyle \frac{t{\left(w_2\right)}_t^{\prime }\left(t,\tau \right)}{w_2\left(t,\tau \right)}}$ are starlike univalent in $\Delta \times \overline{\Delta }$, that $w_1$ satisfies relation (3), and that $w_2$ satisfies relation (7) if ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\in \mathcal{H}\left[w\left(0,\tau \right),\left(\alpha -1\right)n,\tau \right] \cap Q^{\ast },$ the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ defined in (4) is univalent in $\Delta \times \overline{\Delta }$, and the sandwich-type result

$$a+b{w}_1\left(t,\tau \right)+c{\left(w_1\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{\left(w_1\right)}_t^{\prime }\left(t,\tau \right)}{w_1\left(t,\tau \right)}}\prec \prec H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$$

$$\prec \prec a+b{w}_2\left(t,\tau \right)+c{\left(w_2\left(t,\tau \right)\right)}^2+d{\displaystyle \frac{t{\left(w_2\right)}_t^{\prime }\left(t,\tau \right)}{w_2\left(t,\tau \right)}},$$

is endowed for $a,b,c,d\in \mathbb{C}$, $d\ne 0,$ $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1$, and $\alpha ,n>0;$ then, $w_1$ and $w_2$ are, respectively, the best subordinant and the best dominant for the following sandwich-type result

$$w_1\left(t,\tau \right)\prec \prec {\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec w_2\left(t,\tau \right),\left(t,\tau \right)\in \Delta \times \overline{\Delta }.$$

Corollary 5. 

Assuming that relations (3) and (7) take place and that ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\in \mathcal{H}[w\left(0,\tau \right)$, $\left(\alpha -1\right)n,\tau ] \cap Q^{\ast }$ for $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1,$ and $\alpha ,n>0$, if the sandwich-type result

$$a+b{\displaystyle \frac{M_{1}t+\tau }{N_{1}t+\tau }}+c{\left({\displaystyle \frac{M_{1}t+\tau }{N_{1}t+\tau }}\right)}^2+d{\displaystyle \frac{\left(M_1-N_1\right)t\tau }{\left(M_{1}t+\tau \right)\left(N_{1}t+\tau \right)}}\prec \prec H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$$

$$\prec \prec a+b{\displaystyle \frac{M_{2}t+\tau }{N_{2}t+\tau }}+c{\left({\displaystyle \frac{M_{2}t+\tau }{N_{2}t+\tau }}\right)}^2+d{\displaystyle \frac{\left(M_2-N_2\right)t\tau }{\left(M_{2}t+\tau \right)\left(N_{2}t+\tau \right)}},$$

is fulfilled for $a,b,c,d\in \mathbb{C}$, $d\ne 0,$ $-1\le N_2<N_1<M_1<M_2\le 1$, and the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ is defined by relation (4), then ${\displaystyle \frac{M_{1}t+\tau }{N_{1}t+\tau }}$ and ${\displaystyle \frac{M_{2}t+\tau }{N_{2}t+\tau }}$ are, respectively, the best subordinant and the best dominant for the following sandwich-type result

$${\displaystyle \frac{M_{1}t+\tau }{N_{1}t+\tau }}\prec \prec {\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec {\displaystyle \frac{M_{2}t+\tau }{N_{2}t+\tau }}.$$

Corollary 6. 

Assuming that relations (3) and (7) take place and that ${\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n$ $\in \mathcal{H}[w\left(0,\tau \right)$, $\left(\alpha -1\right)n,\tau ] \cap Q^{\ast }$ for $q,m,l,\alpha$ real numbers, $q\in \left(0,1\right)$, $l>-1,$ and $\alpha ,n>0$, if the sandwich-type result

$$a+b{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{p_1}+c{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{2p_1}+d{\displaystyle \frac{2p_{1}t\tau }{{\tau }^2-t^2}}\prec \prec H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$$

$$\prec \prec a+b{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{p_2}+c{\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{2p_2}+d{\displaystyle \frac{2p_{2}t\tau }{{\tau }^2-t^2}},$$

is fulfilled for $a,b,c,d\in \mathbb{C}$, $d\ne 0,$ $0<p_1<p_2\le 1$, and the function $H_{\alpha }^{m,l,q}\left(n,a,b,c,d;t,\tau \right)$ is defined by relation (4), then ${\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{p_1}$ and ${\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{p_2}$ are, respectively, the best subordinant and the best dominant for the following sandwich-type result

$${\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{p_1}\prec \prec {\left({\displaystyle \frac{D_t^{-\alpha }{\mathbb{I}}_q^{m,l}f\left(t,\tau \right)}{t}}\right)}^n\prec \prec {\left({\displaystyle \frac{\tau +t}{\tau -t}}\right)}^{p_2}.$$

3. Conclusions

Motivated by the inspiring outcomes of studies pertaining to geometric function theory that incorporate aspects of quantum calculus and fractional calculus, the theories of strong differential subordination and its dual, strong differential superordination, embed such aspects in an attempt of this work to revive a study started in [46], but which has not been pursued up to this point. The novel aspects of this research’s conclusion consist of the definition of the Riemann–Liouville fractional integral to the extended q-analogue of multiplier transformation, stated in Definition 6 and provided in relations (1) and (2), and in the way it is applied to derive new strong differential subordination results and dual new strong differential superordinations. In each theorem established, the best dominants and best subordinants are provided. When functions distinguished by their geometric properties are substituted as best dominants or best subordinants in the theorems, significant corollaries are derived. The new results of the research concerning the two dual theories of strong differential subordination and strong differential superordination considered in this paper are connected by sandwich-type theorems and corollaries.

The aim of the work is to suggest a new direction for the study of strong differential subordination and its dual, strong differential superordination that integrate quantum calculus and fractional calculus. By applying the concepts discussed in this paper to other operators defined with them, further intriguing operators could be obtained.

For future research, using the Riemann–Liouville fractional integral to the extended q-analogue of multiplier transformation introduced in this paper, we can define the q-subclasses of univalent functions and study some geometric properties like coefficient estimates, closure theorems, distortion theorems, neighborhoods, and the radii of starlikeness, with the convexity and close-to-convexity of functions belonging to the defined subclasses.