Quantum–Fractal–Fractional Operator in a Complex Domain

Abstract

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions.

1. Introduction

The fractal–fractional operators extend traditional operations like differentiation and integration to non-integer or fractional dimensions. Fractals, turbulence, and other complex systems are examples of events that can exhibit self-similarity or scale invariant and can be described and analyzed using these operators. The fractional derivative, which expands the classical derivative to non-integer orders, is an illustration of a fractal–fractional operator. The functioning of complex systems that display long-range dependence and memory effects can be described using the fractional derivative. The fractional Laplacian, which extends the classical Laplacian to non-integer dimensions, is another illustration of a fractal–fractional operator. The conduct of complex systems that display non-local interactions and power law decay is described by the fractional Laplacian. Fractal–fractional operators are a crucial tool for comprehending and modeling complicated processes and have applications across a variety of fields, including physics, engineering, economics, and ecology. Atangana was the first researcher that offered a thorough analysis encompassing definitions of there operators covering differential and integral of various understandings in [1].

Fractal–fractional operators, which combine ideas from fractals with fractional calculus, have attracted a lot of attention lately because of their capacity to represent complex structures with variation in space and memory. There are now new definitions of fractional operators that take into account fractal properties like self-similarity or localized scaling. Riemann–Liouville-like and Caputo-like operators on fractal domains are two examples. In order to represent systems having fractional memory effects on fractal geometries, hybrid operators involving fractal–fractional kernels are being created. It might be difficult for practitioners to evaluate fractal–fractional systems because they frequently lack geometric or physical intuition. Linking fractional ordering and fractal dimensions to visible effects is challenging.

The analysis of fractional derivatives and integrals of analytical functions is known as fractional calculus of analytic functions. The Cauchy integral formulation and complex analysis are used to define fractional derivatives and integrals of analytical functions. For example, the fractional derivative of an analytic function $\upsilon$ of order $\nu$ is defined by

$$D^{\nu }\upsilon \left(\xi \right)={\displaystyle \frac{1}{\Gamma (n-\nu )}}{\int }_{\left(\gamma \right)}\upsilon \left(\eta \right){(\xi -\eta )}^{(n-\nu -1)}d\eta ,$$

where $\gamma$ is a contour in the complex plane, n is the smallest integer that is greater than $\nu$, and $\Gamma$ is the gamma function. Similarly, the fractional integral of an analytic function $\upsilon$ of order $\nu$ is defined by

$$J^{\nu }\upsilon \left(\xi \right)={\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_{\left(\gamma \right)}\upsilon \left(\eta \right){(\eta -\xi )}^{(\nu -1)}d\eta ,$$

where $\gamma$ is a contour in the complex plane and $\Gamma$ is the gamma function. Numerous researchers in physics, engineering, and finance use fractional calculus of analytic functions. It can be applied, for instance, to simulate fractional-order control systems, nonlinear actions of materials, and strange diffusion in permeable environments (see [2,3,4,5,6]).

The area of fractional calculus that is concerned with the calculus of local fractional derivatives and integrals is also referred to as local fractional calculus, occasionally referred to as fractional calculus with local derivatives. In this method, local fractional derivatives and integrals are used to define the fractional derivative and integral operators. Numerous branches of the sciences and engineering, such as fluid mechanics, image processing, and signal processing, as well as others, use the study of local fractional calculus of analytic functions. It offers a potent tool for the development and evaluation of intricate systems with fractal characteristics (see [7,8,9,10]). In the evaluation of complex systems where concentrated conduct is significant, such as fractional propagation, wave propagation, and chaos, the assumption is particularly useful (see [11,12,13,14]).

Quantum calculus (Jackson calculus [15]) is a branch of mathematics that extends traditional calculus to include non-commutative operations on functions. In traditional calculus, the derivative of a function is defined as the limit of the difference quotient as the change in the input approaches zero [16,17,18]. In contrast, a non-commutative operator that takes into account the possibility that the data inputs may not commute with one another operator replaces the difference division in quantum calculus. In the investigation of quantum physics, where non-commutative operators control the movement of particles and systems, this kind of calculus is crucial. It is possible to simulate a range of stock prices that may deviate from the standard commutative property norms in other domains, such as economics (see [19,20]). Quantum calculus can be approached in various ways, such as through operator calculus, non-commutative calculus, and q-calculus (see [21,22]). The explanations and criteria for derivatives, integrals, and other mathematical operations differ according to these methods. There are numerous unanswered questions and difficulties in the study of quantum calculus, which is still a hot area of inquiry.

With the help of the quantum calculus derivative, we expand the fractal–fractional operators into the complex plane in this attempt at developing quantum–fractal–fractional operators (QFFOs). We develop an entire different subclass of analytical functions in the unit disk, utilizing our recently invented operator. We concentrate on the prerequisites for attaining a bounded turning QFFO. Examples incorporating unique functions, such as Bessel and the generalized hypergeometric functions, are taken into consideration.

This paper is organized as follows: Section 2 presents the definitions of the concepts that will be used. Moreover, it contains the definitions of the proposed quantum–fractal–fractional operators, with some properties. Section 3 deals with the main outcomes that are obtained. These results include differential subordination inequalities and some geometric properties. Finally, Section 4 provides a conclusion of our results and future directions.

2. Concepts

In this section, we outline the key concepts relevant to the study of quantum–fractal–fractional operators (QFFOs) in the complex domain.

2.1. Fractal–Fractional Operators

Definition 1 

([23]). Let $\upsilon \left(\xi \right)$ be analytic in the open unit disk ($\mathcal{U}:=\{\xi \in \mathbb{C}:|\xi |<1\}$) and a real number $0<\sigma \le 1.$ The following are regarded as the Riemann–Liouville and Caputo differentiation operators of any order σ:

$${\mathcal{RL}}_{{\Lambda }_{\xi }^{\sigma }\upsilon \left(\xi \right)}:={\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\displaystyle \frac{d}{d\xi }}{\int }_0^{\xi }{\displaystyle \frac{\upsilon \left(\tau \right)}{{(\xi -\tau )}^{\sigma }}}\mathrm{d}Ø,$$

and

$${\mathcal{C}}_{{\Lambda }_{\xi }^{\sigma }\upsilon \left(\xi \right)}:={\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\int }_0^{\xi }{\displaystyle \frac{{\upsilon }^{\prime }\left(\tau \right)}{{(\xi -\tau )}^{\sigma }}}\mathrm{d}Ø,$$

individually. In accordance with this, the fractional integral is shown as follows:

$$J_{\xi }^{\sigma }\upsilon \left(\xi \right):={\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{\int }_0^{\xi }\upsilon \left(\tau \right){(\xi -\tau )}^{\sigma -1}\mathrm{d}Ø,\sigma >0.$$

Definition 2. 

Assume that $\upsilon \left(\xi \right)$ is analytic in $\mathcal{U}.$ If υ is a fractal analytic function on $\mathcal{U}$ of order $\varsigma \in (0,1]$, then the differential FFOs of order $\sigma \in (0,1]$ in the realizing of Riemann–Liouville and Caputo operators are as follows, individually:

$${\mathcal{RL}}_{{\Lambda }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}:={\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\displaystyle \frac{d}{d{\xi }^{\varsigma }}}{\int }_0^{\xi }{\displaystyle \frac{\upsilon \left(\tau \right)}{{(\xi -\tau )}^{\sigma }}}\mathrm{d}Ø,$$

and

$${\mathcal{C}}_{{\Lambda }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}:={\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\int }_0^{\xi }{\displaystyle \frac{d}{d{\tau }^{\varsigma }}}{\displaystyle \frac{\upsilon \left(\tau \right)}{{(\xi -\tau )}^{\sigma }}}\mathrm{d}Ø,$$

where

$${\displaystyle \frac{d\upsilon \left(\xi \right)}{d{\xi }^{\nu }}}=\underset{\xi \to \tau }{lim}{\displaystyle \frac{\upsilon \left(\xi \right)-\upsilon \left(\tau \right)}{{\xi }^{\varsigma }-{\tau }^{\varsigma }}}.$$

As a consequence, the integral of an FFO of order $\sigma >0$ is formulated as follows:

$$J_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right):={\displaystyle \frac{\varsigma }{\Gamma \left(\sigma \right)}}{\int }_0^{\xi }{\tau }^{\sigma -1}\upsilon \left(\tau \right){(\xi -\tau )}^{\sigma -1}\mathrm{d}Ø,\sigma >0.$$

Remember that Atangana [1] proposed a collection of fractal–fractional integral operators of the power kernel and exponential kernel. The prolonged complex FFO of the power kernel that matches the specifications of the traditional fractal–fractional differential operators is selected in this effort. Furthermore, we use instances to show how FFOs behave in certain particular functions. Normalized FFOs are then displayed in $\mathcal{U}.$ We are able to investigate the operators geometrically thanks to FFOs’ normalization.

Application 1. 

• Let $\upsilon \left(\xi \right)={\xi }^n.$ Then,

  $${\mathcal{RL}}_{{\Delta }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}=\left({\displaystyle \frac{\Gamma (n+1)}{\Gamma (n+2-\sigma )}}\right){(n+1-\sigma )}^{\left(\varsigma \right)}{\xi }^{n+1-\sigma -\varsigma },$$

  where $\left(\varsigma \right)$ designs the factorial powers, as follows:

  $${\displaystyle \begin{array}{cc}\hfill L^{\left(n\right)}& =\stackrel{k\mathit{factors}}{\overbrace{L(L+1)(L+2)\cdots (L+k-1)}}\hfill \\ & =\prod _{m=1}^k(L+m-1)=\prod _{m=0}^{k-1}(L+m).\hfill \end{array}}$$

  $$={\displaystyle \frac{\Gamma (L+n)}{\Gamma \left(L\right)}},$$

  while

  $${\mathcal{C}}_{{\Delta }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}=\left({\displaystyle \frac{\Gamma (n+1-\varsigma )}{\Gamma (n+2-\sigma -\varsigma )}}\right){\xi }^{n+1-\sigma -\varsigma },$$

  and

  $$J_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)=\left({\displaystyle \frac{\varsigma \Gamma (n+\sigma )}{\Gamma (n+2\sigma )}}\right){\xi }^{2\sigma +n-1},\sigma +n>0.$$
• Consider that $\upsilon \left(\xi \right)=sin\left(\xi \right)$. Then,

  $$J_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)=\varsigma \pi 2^{-2\sigma -1}\Gamma (\sigma +1){\xi }^{2\sigma }{}_2\breve{F}_3\left({\displaystyle \frac{(\sigma +1)}{2}},{\displaystyle \frac{(\sigma +2)}{2}};{\displaystyle \frac{3}{2}},\sigma +{\displaystyle \frac{1}{2}},\sigma +1;-{\displaystyle \frac{{\xi }^2}{4}}\right),$$

  where ₂${\breve{F}}_3$ refers to the regularized modified hypergeometric function.
• While, for $\upsilon \left(\xi \right)=sinh\left(\xi \right),$ we have

  $$J_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)=\varsigma \pi 2^{-2\sigma -1}\Gamma (\sigma +1){\xi }^{2\sigma }{}_2\breve{F}_3\left({\displaystyle \frac{(\sigma +1)}{2}},{\displaystyle \frac{(\sigma +2)}{2}};{\displaystyle \frac{3}{2}},\sigma +{\displaystyle \frac{1}{2}},\sigma +1;{\displaystyle \frac{{\xi }^2}{4}}\right).$$
• Assume that $\upsilon \left(\xi \right)=cos\left(\xi \right)$. Then,

  $$J_{\zeta }^{\sigma ,\varsigma }\upsilon \left(\xi \right)=\varsigma \sqrt{\pi }{\xi }^{\sigma -1/2}cos\left({\displaystyle \frac{\xi }{2}}\right)B_{\sigma -1/2}\left({\displaystyle \frac{\xi }{2}}\right),$$

  where B presents the Bessel function.
• Let $\upsilon \left(\xi \right)=log\left(\xi \right)$. Then,

  $$J_{\zeta }^{\sigma ,\varsigma }\upsilon \left(\xi \right)={\displaystyle \frac{\varsigma \sqrt{\pi }{2}^{1-2\sigma }{\xi }^{2\sigma -1}({\psi }^{\left(0\right)}\left(\sigma \right)-{\psi }^{\left(0\right)}\left(2\sigma \right)+log\left(\xi \right))}{\Gamma (\sigma +1/2)}},$$

  where $\psi$ indicates the digamma function.
• Let $\upsilon \left(\xi \right)=log\left(\xi \right),\varsigma =1$. Then,

  $$\begin{array}{cc}\hfill {\mathcal{C}}_{{\Delta }_{\zeta }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}& ={\displaystyle \frac{1}{\Gamma (1-\sigma )}}\left({(\xi -\tau )}^{-\sigma }\right(\sigma (\tau -\xi ){}_{2}F_1(1,1-\sigma ,2-\sigma ,1-\tau /\xi )\hfill \\ \hfill & +(\sigma -1)\xi {(}_2{F}_1(1,-\sigma ,1-\sigma ,1-\tau /\xi )+\sigma log\left(\tau \right)-1)\left)\right)/\left((\sigma -1)\sigma \xi \right).\hfill \end{array}$$

2.2. The Normalized Case

The normalized case is then used to define FFOs. We give the normalized class of analytic functions in this work, dented by $\mathcal{A}$ of the type

$$\upsilon \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{a}_n{\xi }^n,\xi \in \mathcal{U}.$$

In order to normalize FFOs, we have the following result:

Proposition 1. 

Let $\upsilon \in \mathcal{A}.$ Then, the normalized FFOs are as follows:

$$\begin{array}{cc}\hfill {\mathcal{RL}}_{{\Lambda }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}& :={\displaystyle \frac{\Gamma (3-\sigma )}{{(2-\sigma )}^{\left(\varsigma \right)}}}{\xi }^{\sigma +\varsigma -1}\left({\mathcal{RL}}_{{\Delta }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}\right)\hfill \\ \hfill & =\xi +\sum _{n=2}^{\infty }{a}_n\left({\displaystyle \frac{\Gamma (3-\sigma )}{{(2-\sigma )}^{\left(\varsigma \right)}}}{\displaystyle \frac{\Gamma (n+1){(n+1-\sigma )}^{\left(\varsigma \right)}}{\Gamma (n+2-\sigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +\sum _{n=2}^{\infty }{a}_n{\lambda }_n{\xi }^n;\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{C}}_{{\Lambda }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}& :={\displaystyle \frac{\Gamma (3-\sigma -\varsigma )}{\Gamma (2-\varsigma )}}{\xi }^{\sigma +\varsigma -1}\left({\mathcal{C}}_{{\Delta }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}\right)\hfill \\ \hfill & =\xi +\sum _{n=2}^{\infty }{a}_n\left({\displaystyle \frac{\Gamma (3-\sigma -\varsigma )}{\Gamma (2-\varsigma )}}{\displaystyle \frac{\Gamma (n+1-\varsigma )}{\Gamma (n+2-\sigma -\varsigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +\sum _{n=2}^{\infty }{a}_n{\gamma }_n{\xi }^n;\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)& :={\displaystyle \frac{\Gamma (1+2\sigma )}{\varsigma \Gamma (1+\sigma )}}{\xi }^{1-2\sigma }{J}_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)\hfill \\ \hfill & =\xi +\sum _{n=2}^{\infty }{a}_n\left({\displaystyle \frac{\Gamma (1+2\sigma )}{\Gamma (1+\sigma )}}{\displaystyle \frac{\Gamma (n+\sigma )}{\Gamma (n+2\sigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +\sum _{n=2}^{\infty }{a}_n{\tau }_n{\xi }^n.\hfill \end{array}$$

Proof. 

A direct application of 1. □

These operators have a frequent role in a variety of applications, including control systems, signal processing, and image processing. They have been shown to have stronger mathematical properties than traditional operators and to be useful for simulating complicated systems.

2.3. Quantum Operators

Definition 3. 

The Jackson derivative might be presented in the following manner, employing the difference operator:

$$\begin{array}{c}\hfill \left({⋎}_q\right)\upsilon \left(\xi \right)={\displaystyle \frac{\upsilon \left(\xi \right)-\upsilon \left(q\xi \right)}{\xi (1-q)}},q\in (0,1),\end{array}$$

(1)

where

$${⋎}_q\left({\xi }^k\right)=\left({\displaystyle \frac{1-q^k}{1-q}}\right){\xi }^{k-1},k\in \mathbb{R}.$$

Maclaurin’s series formulation additionally takes into account the sum of the numbers

$$\begin{array}{c}\hfill \left({⋎}_q\upsilon \right)\left(\xi \right)=\sum _{n=0}^{\infty }{a}_n{\left[n\right]}_q{\xi }^{n-1},\end{array}$$

(2)

where ${\left[m\right]}_q:={\displaystyle \frac{1-q^m}{1-q}}.$

Note that

$${⋎}_{q}C=0,\underset{q\to 1^{-}}{lim}\left({⋎}_q\upsilon \right)\left(\xi \right)={\upsilon }^{\prime }\left(\xi \right),$$

where C is a constant.

Suppose that $\rho \in \mathbb{C}$. Then, the q-shifted factorials are formulated in the next equality [15] as follows:

$$\begin{array}{c}\hfill {(\rho ;q)}_{\varrho }=\prod _{i=0}^{\varrho -1}(1-q^i\rho ),\varrho \in \mathbb{N},{(\rho ;q)}_0=1.\end{array}$$

(3)

By (3), the q-shifted gamma function is formulated as follows (see Table 1 for examples):

$$\begin{array}{c}\hfill {(q^{\rho };q)}_{\varrho }={\displaystyle \frac{{\Gamma }_q(\rho +\varrho ){(1-q)}^{\varrho }}{{\Gamma }_q\left(\rho \right)}},{\Gamma }_q\left(\rho \right)={\displaystyle \frac{{(q;q)}_{\infty }{(1-q)}^{1-\rho }}{{(q^{\rho };q)}_{\infty }}}={(1-q)}^{1-\rho }\prod _{n=0}^{\infty }\left({\displaystyle \frac{1-q^{1+n}}{1-q^{\rho +n}}}\right),\end{array}$$

(4)

where

$${\Gamma }_q(\rho +1)={\displaystyle \frac{{\Gamma }_q\left(\rho \right)(1-q^{\rho })}{1-q}},q\in (0,1)$$

and

$$\begin{array}{c}\hfill {(\rho ;q)}_{\infty }=\prod _{i=0}^{\infty }(1-q^i\rho ).\end{array}$$

(5)

Table 1. Evaluation of ${\Gamma }_q\left(\rho \right)$ in Equation (4) for different values of the quantum number $q$.

Quantum Number (q)	${\mathbf{\Gamma }}_{\mathit{q}}$ (2.1)	${\mathbf{\Gamma }}_{\mathit{q}}$ (2.5)	${\mathbf{\Gamma }}_{\mathit{q}}$ (2.9)
0.001	1.0000995556762426	1.0004994054811371	1.000899855923606
0.1	1.0082659793606632	1.046066572448892	1.088784698676843
0.2	1.0147511896471197	1.0868086641261354	1.17558530615081
0.3	1.0202506978830732	1.1239476292023016	1.2607967455441123
0.4	1.0250805873649542	1.1583598546562799	1.344668018051892
0.5	1.029414446594846	1.1905936250275284	1.427376969260213
0.6	1.0333611545829302	1.221025326199809	1.5090582738591247
0.7	1.036994746225153	1.2499290452711966	1.5898179667143768
0.8	1.0403812870012996	1.2775335316152772	1.6697747269289396
0.9	1.0488444513789157	1.3128314103324294	1.763713044723999
0.95	1.1357798101736767	1.4733614062423337	2.058688968144806

We use the quantum gamma function to extend the FFOs as follows:

Proposition 2. 

Let $\upsilon \in \mathcal{A}.$ Then, the quantum normalized FFOs (QFFOs) are given by

$$\begin{array}{cc}\hfill \underset{q}{\mathcal{R}}{\mathcal{L}}_{{\Lambda }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)}& :=\xi +\sum _{n=2}^{\infty }{a}_n\left({\displaystyle \frac{{\Gamma }_q(3-\sigma )}{{(2-\sigma )}_q^{\left(\varsigma \right)}}}{\displaystyle \frac{{\Gamma }_q(n+1){(n+1-\sigma )}_q^{\left(\varsigma \right)}}{{\Gamma }_q(n+2-\sigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +\sum _{n=2}^{\infty }{a}_n{\left({\lambda }_n\right)}_q{\xi }^n;\hfill \end{array}$$

$$\begin{array}{cc}\hfill \underset{q}{\mathcal{C}}{\Lambda }_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)& :=\xi +\sum _{n=2}^{\infty }{a}_n\left({\displaystyle \frac{{\Gamma }_q(3-\sigma -\varsigma )}{{\Gamma }_q(2-\varsigma )}}{\displaystyle \frac{{\Gamma }_q(n+1-\varsigma )}{{\Gamma }_q(n+2-\sigma -\varsigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +\sum _{n=2}^{\infty }{a}_n{\left({\gamma }_n\right)}_q{\xi }^n;\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {}_q\mathcal{J}_{\xi }^{\sigma ,\varsigma }\upsilon \left(\xi \right)& :=\xi +\sum _{n=2}^{\infty }{a}_n\left({\displaystyle \frac{{\Gamma }_q(1+2\sigma )}{{\Gamma }_q(1+\sigma )}}{\displaystyle \frac{{\Gamma }_q(n+\sigma )}{{\Gamma }_q(n+2\sigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +\sum _{n=2}^{\infty }{a}_n{\left({\tau }_n\right)}_q{\xi }^n.\hfill \end{array}$$

In the sequel, we shall denote all the above QFFOs’ type by

$$Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right):=\xi +\sum _{n=2}^{\infty }{a}_n{\left({\kappa }_n\right)}_q{\xi }^n,$$

where ${\left({\kappa }_n\right)}_q$ indicates one of the coefficients ${\left({\lambda }_n\right)}_q,{\left({\gamma }_n\right)}_q$ or ${\left({\tau }_n\right)}_q.$

Remark 1. 

• Special functions are frequently used in quantum mechanics to characterize the weight of pathways in fractal geometries when calculating path integrals over fractional or fractal domains. As an illustration, consider the following: solving radial equations on fractal geometries frequently results in modified Bessel functions in fractional quantum systems. Hypergeometric functions: show up when evaluating Green’s functions or path integrals for quantum–fractal–fractional operators.
• The mathematical structures underlying quantum–fractal–fractional operators and their solutions give rise to the relationship between these operators and special functions. When evaluating differential equations with fractional derivatives or equations stated on fractal domains, special functions frequently arise naturally. These functions offer computational tools, analytical expressions, and information on the characteristics of the solutions. The generalized hypergeometric function, for instance, is a generalization of the Mittag–Leffler function and is helpful in characterizing more intricate systems that use fractal–fractional operators or multi-term fractional derivatives.

Application 2. 

In this part, we present the acting of QFFOs on the following univalent convex function:

$$f\left(\xi \right)={\displaystyle \frac{\xi }{1-\xi }}=\xi +\sum _{n=2}^{\infty }{\xi }^n,\xi \in \mathcal{U}.$$

We aim to approximate the above sum by q-hypergeometric function. Thus, we have

$$\begin{array}{cc}\hfill \underset{q}{\mathcal{R}}{\mathcal{L}}_{{\Lambda }_{\xi }^{\sigma ,\varsigma }f\left(\xi \right)}& =\xi +\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(3-\sigma )}{{(2-\sigma )}_q^{\left(\varsigma \right)}}}{\displaystyle \frac{{\Gamma }_q(n+1){(n+1-\sigma )}_q^{\left(\varsigma \right)}}{{\Gamma }_q(n+2-\sigma )}}\right){\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(3-\sigma )}{{(2-\sigma )}_q^{\left(\varsigma \right)}}}\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(n+1){(n+1-\sigma )}_q^{\left(\varsigma \right)}}{{\Gamma }_q(n+2-\sigma )}}\right){\xi }^n\hfill \\ \hfill & :=\xi +C\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(n+1){(n+1-\sigma )}_q^{\left(\varsigma \right)}}{{\Gamma }_q(n+2-\sigma )}}\right){\xi }^n\hfill \\ \hfill & =\xi +C{{[}_2{\varphi }_1\left(\xi \right)]}_q(1,q^{\varsigma },q;\xi )\hfill \end{array}$$

where

$${(n+1-\sigma )}_q^{\left(\varsigma \right)}=\prod _{j=0}^{\varsigma -1}{\displaystyle \frac{(n+1-\sigma -j)}{(1-q^j)}}={\displaystyle \frac{{(n+1-\sigma )}_q!}{{\left(\varsigma \right)}_q!{(n+1-\sigma -\varsigma )}_q!}}$$

and

$${{[}_2{\varphi }_1\left(\xi \right)]}_q(a,b,c;\xi )=\sum _{n=0}^{\infty }{\displaystyle \frac{{(a;q)}_n{(b;q)}_n}{{(c;q)}_n}}{\xi }^n,$$

such that

$${(a;q)}_n=(1-a)(1-aq)(1-a{q}^2)....(1-a{q}^{n-1})n\ge 1.$$

For the fractal–fractional values $\sigma =\varsigma =0.5$ and $q=0.5,$ we obtain the function

$$\begin{array}{cc}\hfill \underset{0.5}{\mathcal{RL}}{\Lambda }_{\xi }^{0.5,0.5}f\left(\xi \right)& =\xi (\cdots +0.304955960839043{\xi }^9+0.320203758880996{\xi }^8\hfill \\ \hfill & +0.337992856596606{\xi }^7+0.359117410133894{\xi }^6\hfill \\ \hfill & +0.384768653714887{\xi }^5+0.416832708191127{\xi }^4+0.45851597901024{\xi }^3\hfill \\ \hfill & +0.51583047638652{\xi }^2+0.60180222245094\xi +1)\hfill \end{array}$$

While for $\sigma =\varsigma =0.5$ and $q=0.9,$

$$\begin{array}{cc}\hfill \underset{0.9}{\mathcal{RL}}{\Lambda }_{\xi }^{0.5,0.5}f\left(\xi \right)& =\xi (\cdots +0.790057469942021{\xi }^9+0.797958044641441{\xi }^8+0.806824245137457{\xi }^7\hfill \\ \hfill & +0.816909548201675{\xi }^6+0.828579684604556{\xi }^5+0.842389346014632{\xi }^4+0.859237132934925{\xi }^3\hfill \\ \hfill & +0.880718061258298{\xi }^2+0.910075329966907\xi +1).\hfill \end{array}$$

Now, we consider the following data $\sigma =0.3, \varsigma =0.4$ and $q=0.5$:

$$\begin{array}{cc}\hfill \underset{0.5}{\mathcal{RL}}{\Lambda }_{\xi }^{0.3,0.4}f\left(\xi \right)& =\xi (...+0.291061349491525{\xi }^9+0.311435643955932{\xi }^8+0.335658416263616{\xi }^7\hfill \\ \hfill & +0.365028527686682{\xi }^6+0.40153138045535{\xi }^5+0.44837670817514{\xi }^4\hfill \\ \hfill & +0.51114944731966{\xi }^3+0.6006006006006{\xi }^2+0.740740740740741\xi +1)\hfill \end{array}$$

The above conclusion implies that

$$\underset{q}{\mathcal{R}}{}^{\mathcal{L}}\Lambda _{\xi }^{\sigma ,\varsigma }f\left(\xi \right)={\displaystyle \frac{\xi }{1-\mu \xi }},\mu >0.$$

All the coefficients of $\underset{q}{\mathcal{R}}{}^{\mathcal{L}}\Lambda _{\xi }^{\sigma ,\varsigma }f\left(\xi \right)$ are majored by the coefficients of $f\left(\xi \right)=\xi /(1-\xi ).$

$$\begin{array}{cc}\hfill {}_q{}^{\mathcal{C}}\Lambda _{\xi }^{\sigma ,\varsigma }f\left(\xi \right)& =\xi +\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(3-\sigma -\varsigma )}{{\Gamma }_q(2-\varsigma )}}{\displaystyle \frac{{\Gamma }_q(n+1-\varsigma )}{{\Gamma }_q(n+2-\sigma -\varsigma )}}\right){\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(3-\sigma -\varsigma )}{{\Gamma }_q(2-\varsigma )}}\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(n+1-\varsigma )}{{\Gamma }_q(n+2-\sigma -\varsigma )}}\right){\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(3-\sigma -\varsigma )}{{\Gamma }_q(2-\varsigma )}}\sum _{n=2}^{\infty }{\displaystyle \frac{q^{\sigma +\varsigma -1}}{1-q^{n+1-\varsigma }}}{\xi }^n\hfill \\ \hfill & \approx \xi +{\displaystyle \frac{{\Gamma }_q(3-\sigma -\varsigma )}{{\Gamma }_q(2-\varsigma )}}{q}^{\sigma +\varsigma -1}\sum _{n=2}^{\infty }{\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(3-\sigma -\varsigma )}{{\Gamma }_q(2-\varsigma )}}{\displaystyle \frac{{(q^{1-\varsigma };q)}_2}{{(q^{2-\sigma -\varsigma };q)}_2}}{\xi }^2{}_1\varphi _1\left(\begin{array}{c}{q}^{3-\varsigma }\\ q^{4-\sigma -\varsigma }& ;& q,\xi \end{array}\right).\hfill \end{array}$$

For example,

$$\begin{array}{cc}\hfill & {}_{0.5}{}^{\mathcal{C}}\Lambda _{\xi }^{0.3,0.4}f\left(\xi \right)=\xi +0.629058640329445{\xi }^2{}_1\varphi _1\left(\begin{array}{c}0.164938488846612\\ 0.101531549544529& ;& 0.5,\xi \end{array}\right)\hfill \\ \hfill & {}_{0.5}{}^{\mathcal{C}}\Lambda _{\xi }^{0.5,0.5}f\left(\xi \right)=\xi +0.569725647106914{\xi }^2{}_1\varphi _1\left(\begin{array}{c}0.176776695296637\\ 0.125& ;& 0.5,\xi \end{array}\right)\hfill \\ \hfill & {}_{0.9}{}^{\mathcal{C}}\Lambda _{\xi }^{0.5,0.5}f\left(\xi \right)=\xi +0.445515790985195{\xi }^2{}_1\varphi _1\left(\begin{array}{c}0.768433471420916\\ 0.729& ;& 0.5,\xi \end{array}\right).\hfill \end{array}$$

As a consequence, the coefficients of the operator ${}_q{}^{\mathcal{C}}\Lambda _{\xi }^{\sigma ,\varsigma }f\left(\xi \right)$ are majored by the coefficients of $\xi /(1-\xi ).$ Therefore,

$${}_q{}^{\mathcal{C}}\Lambda _{\xi }^{\sigma ,\varsigma }f\left(\xi \right)={\displaystyle \frac{\xi }{1-t\xi }},t>0.$$

Finally, we analyze the integral operator

$$\begin{array}{cc}\hfill {}_q\mathcal{J}_{\xi }^{\sigma ,\varsigma }f\left(\xi \right)& =\xi +\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(1+2\sigma )}{{\Gamma }_q(1+\sigma )}}{\displaystyle \frac{{\Gamma }_q(n+\sigma )}{{\Gamma }_q(n+2\sigma )}}\right){\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(1+2\sigma )}{{\Gamma }_q(1+\sigma )}}\sum _{n=2}^{\infty }\left({\displaystyle \frac{{\Gamma }_q(n+\sigma )}{{\Gamma }_q(n+2\sigma )}}\right){\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(1+2\sigma )}{{\Gamma }_q(1+\sigma )}}\sum _{n=2}^{\infty }\left({\displaystyle \frac{{(q^{1+\sigma +n};q)}_{\infty }}{{(q^{1+2\sigma +n};q)}_{\infty }}}\right){\xi }^n\hfill \\ \hfill & =\xi +{\displaystyle \frac{{\Gamma }_q(1+2\sigma )}{{\Gamma }_q(1+\sigma )}}{\xi }^2{}_2\varphi _1(q^{\sigma },q^{\sigma };q^{2\sigma };q;\xi ).\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {}_{0.5}\mathcal{J}_{\xi }^{0.5,\varsigma }f\left(\xi \right)=\xi +1.128{\xi }^2{}_2\varphi _1(0.5^{0.5},0.5^{0.5};0.5^{2\ast 0.5};0.5;\xi )\hfill \\ \hfill & {}_{0.9}\mathcal{J}_{\xi }^{0.5,\varsigma }f\left(\xi \right)=\xi +1.07963736672053{\xi }^2{}_2\varphi _1(0.9^{0.5},0.9^{0.5};0.9^{2\ast 0.5};0.9;\xi )\hfill \\ \hfill & {}_{0.9}\mathcal{J}_{\xi }^{0.5,\varsigma }f\left(\xi \right)=\xi +1.07963736672053{\xi }^2{}_2\varphi _1({0.5}^{0.5},{0.5}^{0.5};0.5^{2\ast 0.5};0.5;\xi ).\hfill \end{array}$$

2.4. Lemmas

For a more detailed investigation, we need the following well-known results:

Lemma 1 

([24], p. 19). For ${\xi }_0\in \mathcal{U}$ and ${\rho }_0=\left|{\xi }_0\right|$, the function $u\left(\xi \right)=u_n{\xi }^n+u_{n+1}{\xi }^{n+1}+\cdots$ is smooth on ${\mathcal{U}}_{{\rho }_0}\cup \left\{{\rho }_0\right\}$, where $u\not\equiv 0$ and $n\ge 1$. If $|u\left({\xi }_0\right)|=max\{|u\left(\xi \right)|:\xi \in {\mathcal{U}}_{{\rho }_0}\},$ then there exists an $m\ge n$, such that

• $\Re \left({\displaystyle \frac{{\xi }_0{u}^{\prime }\left({\xi }_0\right)}{u\left({\xi }_0\right)}}\right)=m$, and
• ${\displaystyle \Re \left(\frac{{\xi }_0{u}^{″}\left({\xi }_0\right)}{u^{\prime }\left({\xi }_0\right)}+1\right)\ge m}$.

Lemma 2 

([25], p. 217, Theorem 4). Let $f\left(\xi \right)=\xi +a_2{\xi }^2+....$ be analytic in $\mathcal{U}.$ If $|f^{″}\left(\xi \right)|<\sqrt{6}\left|f^{\prime }\left(\xi \right)\right|,\xi \in \mathcal{U}$, then $f\in \mathcal{A}.$

Lemma 3 

([24], p. 73, Theorem 3.1c). Let $p\left(\xi \right)=1+c_1\xi +c_2{\xi }^2+...\in \mathcal{P}$ be analytic in $\mathcal{U};$ then,

$$p\left(\xi \right)+\lambda \xi p^{\prime }\left(\xi \right)\prec {\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{\alpha }\Rightarrow p\left(\xi \right)\prec {\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{\beta },\beta \le 3-\alpha .$$

Some geometric properties are discovered in the next two propositions.

Proposition 3. 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ satisfies the inequality

$$\sum _{n=2}^{\infty }n(n-(1-\sqrt{6}))|{\left({\kappa }_n\right)}_q\left|\right|a_n|<\sqrt{6},$$

then $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is univalent in $\mathcal{U}.$

Proof. 

A calculation produces

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right|& =\left|{\displaystyle \frac{{\sum }_{n=2}^{\infty }n(n-1){\left({\kappa }_n\right)}_q{a}_n{\xi }^{n-1}}{1+{\sum }_{n=2}^{\infty }n{\left({\kappa }_n\right)}_q{a}_n{\xi }^{n-1}}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{{\sum }_{n=2}^{\infty }n(n-1)|{\left({\kappa }_n\right)}_q\left|\right|a_n|}{1-{\sum }_{n=2}^{\infty }n|{\left({\kappa }_n\right)}_q\left|\right|a_n|}}.\hfill \end{array}$$

To show that $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is univalent, we note that from the condition of the outcome

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right|& <\left|{\displaystyle \frac{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right|\hfill \\ \hfill & <\sqrt{6}.\hfill \end{array}$$

The operator $\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]$ then has univalency, according to Lemma 2. □

We will examine certain essential elements of an uncommon instance of QFFOs, where $\upsilon$ is the Koebe function because of the sold properties of the Koebe function as a univalent function.

Proposition 4. 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ achieves the relation

$${\displaystyle \frac{{\xi }^2{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{‴}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}-{\left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right)}^2+{\displaystyle \frac{\left(2\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}\right)}{\left({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\right)}}+1$$

$$\prec \alpha +(1-\alpha ){\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{3/2},\alpha \in [0,1),$$

then $\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]$ is convex of order $\alpha .$

Proof. 

Define two functions as follows:

$$\Phi \left(\xi \right)=1+\left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right),\Psi \left(\xi \right)={\displaystyle \frac{\Phi \left(\xi \right)-\alpha }{1-\alpha }}.$$

Clearly, $\Psi \left(0\right)=1$ and

$$\xi {\Psi }^{\prime }\left(\xi \right)={\displaystyle \frac{\xi {\Phi }^{\prime }\left(\xi \right)}{1-\alpha }}.$$

Then, in view of Lemma 3, with $\lambda =\beta =1,$ we obtain

$$\Psi \left(\xi \right)+\xi {\Psi }^{\prime }\left(\xi \right)\prec {\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{3/2},$$

where $\Psi \left(\xi \right)\prec {\displaystyle \frac{1+\xi }{1-\xi }}.$ However,

$$\Phi \left(\xi \right)+\xi {\Phi }^{\prime }\left(\xi \right)={\displaystyle \frac{{\xi }^2{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{‴}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}-{\left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right)}^2+{\displaystyle \frac{\left(2\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}\right)}{\left({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\right)}}+1$$

$$\prec \alpha +(1-\alpha ){\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{3/2},\xi \in [0,1).$$

Then,

$$\Phi \left(\xi \right)\prec {\displaystyle \frac{1+(1-2\alpha )\xi }{1-\xi }},$$

which is equivalent to $\Re (\Phi (\xi \left)\right)>\alpha ;$ hence, $\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]$ is convex of order $\alpha .$ □

3. Outcomes

A bounded turning function of a complex variable is a function that satisfies a certain geometric condition related to the behavior of the function near its critical points. Specifically, let $\upsilon \left(\xi \right)$ be a complex-valued function of a complex variable $\xi$, and let ${\xi }_0$ be a critical point of $\upsilon \left(\xi \right)$, i.e., a point where the derivative ${\upsilon }^{\prime }\left({\xi }_0\right)$ is zero or does not exist. We say that $\upsilon \left(\xi \right)$ has a bounded turning at ${\xi }_0$ if there exists a disk $\mathcal{U}$ centered at ${\xi }_0$ such that for all $\xi$ in $\mathcal{U}$ except possibly ${\xi }_0$, the argument of $\upsilon \left(\xi \right)-\upsilon \left({\xi }_0\right)$ is constant, i.e., the difference $\upsilon \left(\xi \right)-\upsilon \left({\xi }_0\right)$ lies on a ray emanating from $\upsilon \left({\xi }_0\right)$. Geometrically, this means that near ${\xi }_0$, the graph of $\upsilon \left(\xi \right)$ turns around $\upsilon \left({\xi }_0\right)$ without crossing itself. Functions that satisfy this condition are sometimes called “univalent” or “sense-preserving” near ${\xi }_0$, since they preserve the sense of rotation around $\upsilon \left({\xi }_0\right)$. Bounded turning functions have a number of interesting properties, including a connection to conformal mapping and the theory of quasicrystals. They also arise naturally in the study of geometric function theory, a branch of complex analysis concerned with the properties of analytic functions. In this section, we aim to present a set of conditions on the QFFO to be in the class of bounded turning functions.

Theorem 1. 

Let

$$\begin{array}{c}\hfill \Re \left(1+{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right)<1+{\displaystyle \frac{c}{2}},\left|\xi \right|<1,0<c\le 1.\end{array}$$

(6)

(i) 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)/\xi \ne 0,$ then $\Re {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }>0,\left|\xi \right|<1$.

(ii) 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)/\xi \ne 0,$ then

$${\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\prec (1-\xi )/(1-k\xi ),\left|\xi \right|<1,k=1/(1+c).$$

(iii) 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)/\xi \ne 0$, then $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function.

Proof. 

The proof of (i).

We aim to show that ${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\ne 0$ for all $\left|\xi \right|<1$ and $\xi \ne 0$ (because ${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(0\right)\right]}^{\prime }=1$ in Proposition 2). If it occurs at ${\xi }_1$, $0<|{\xi }_1|<1$ and

$${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }={(\xi -{\xi }_1)}^{m_1}\psi \left(\xi \right),$$

where $m_1\ge 1$ and $\psi$ is smooth in $\mathcal{K}$ such that $\psi \left({\xi }_1\right)\ne 0,$ then

$$1+{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}=1+{\displaystyle \frac{m_1\xi }{\xi -{\xi }_1}}+{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{\psi \left(\xi \right)}}.$$

Therefore, $1+{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\to \infty$ when $\xi \to {\xi }_1$, which is a contradiction (6).

Let

$$\begin{array}{c}\hfill {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }={\left(1-u\left(\xi \right)\right)}^{{\displaystyle \frac{c}{n-1}}},\end{array}$$

(7)

where $0<c/(n-1)\le 1$. Since ${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\ne 0$ for all $\left|\xi \right|<1$, $u\left(\xi \right)=u_{n-1}{\xi }^{n-1}+\cdots$ is analytic in the unit disk with $u\left(0\right)=0$. Moreover, we get

$$\begin{array}{cc}\hfill \left(1+{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}\right)=& 1-{\displaystyle \frac{c\xi u^{\prime }\left(\xi \right)}{(n-1)(1-u(\xi \left)\right)}}.\hfill \end{array}$$

Assume that ${max}_{\left|\xi \right|\le |{\xi }_0|}\left|u\left(\xi \right)\right|=|u\left({\xi }_0\right)|=1.$ According to Lemma 1, we get

$$\begin{array}{c}\hfill {\xi }_0{u}^{\prime }\left({\xi }_0\right)=ku\left({\xi }_0\right),(k\ge n-1,u\left({\xi }_0\right)=e^{i\vartheta },\vartheta \in \mathbb{R}).\end{array}$$

Hence, we obtain

$$\begin{array}{cc}\hfill \Re \left(1+{\displaystyle \frac{{\xi }_0{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{\prime }}}\right)& =1-{\displaystyle \frac{c}{n-1}}\Re \left({\displaystyle \frac{k{e}^{i\vartheta }}{1-e^{i\vartheta }}}\right).\hfill \end{array}$$

Therefore, the following fact is obtained:

$$\begin{array}{cc}\hfill \Re \left(1+{\displaystyle \frac{{\xi }_0{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{\prime }}}\right)& =1+{\displaystyle \frac{kc}{2}}\ge 1+{\displaystyle \frac{c}{2}},\hfill \end{array}$$

which is a contradiction to (6). That is, $\left|u\right(\xi \left)\right|<1,\left|\xi \right|<1$, and by (7), we receive

$$\begin{array}{cc}\hfill |arg({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\left)\right|& =c|arg((1-u(\xi \left)\right)|\hfill \\ \hfill & <{\displaystyle \frac{\pi c}{2(n-1)}}\hfill \\ \hfill & \le {\displaystyle \frac{\pi }{2}},\hfill \end{array}$$

such that $0<c\le 1$, which implies that $\Re {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }>0,\left|\xi \right|<1.$

The proof of (ii).

Formulate a function $ϵ$ as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}}={\displaystyle \frac{1-ϵ\left(\xi \right)}{1-kϵ\left(\xi \right)}},\end{array}$$

(8)

where $k=1/(1+c).$ Then, $ϵ\left(\xi \right)=c_1\xi +\cdots$ is analytic in the open unit disk. By putting ${max}_{\left|\xi \right|\le |{\xi }_0|}\left|ϵ\left(\xi \right)\right|=|ϵ\left({\xi }_0\right)|=1.$ On the authority of Lemma 1, we obtain

$$\begin{array}{c}\hfill {\xi }_0{ϵ}^{\prime }\left({\xi }_0\right)=kϵ\left({\xi }_0\right);(k\ge 1;ϵ\left({\xi }_0\right)=e^{i\vartheta };\vartheta \in \mathbb{R}).\end{array}$$

As maintained by (8), a logarithmic differentiation yields

$$\begin{array}{cc}\hfill \Re \left(1+{\displaystyle \frac{{\xi }_0{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{\prime }}}\right)& =\Re \left(a{\displaystyle \frac{{\xi }_0{ϵ}^{\prime }\left({\xi }_0\right)}{1-kϵ\left({\xi }_0\right)}}-{\displaystyle \frac{{\xi }_0{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{\prime }}{1-ϵ\left({\xi }_0\right)}}+{\displaystyle \frac{1-ϵ\left({\xi }_0\right)}{1-kϵ\left({\xi }_0\right)}}\right)\hfill \\ \hfill & =\Re \left(k{\displaystyle \frac{k{e}^{i\vartheta }}{1-k{e}^{i\vartheta }}}-{\displaystyle \frac{k{e}^{i\vartheta }}{1-e^{i\vartheta }}}+{\displaystyle \frac{1-e^{i\vartheta }}{1-k{e}^{i\vartheta }}}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{1+k}{2}}\right)g\left(\tau \right),\tau =cos\vartheta ,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill g\left(\tau \right)={\displaystyle \frac{k(1-k)+2(1-\tau )}{1-2k\tau +k^2}},\tau \in [-1,1].\end{array}$$

But $0<c\le 1$ and $k=1/(1+c)<1$ and $2\ge 1+c.$ Thus, we obtain $2k-1\ge 0$. Hence,

$$\begin{array}{cc}\hfill g^{\prime }\left(\tau \right)& =2(1-k)\left({\displaystyle \frac{(k+1)k-1}{{(1-2k\tau +k^2)}^2}}\right)\hfill \\ \hfill & \ge 2(1-k)\left({\displaystyle \frac{2k-1}{{(1-2k\tau +k^2)}^2}}\right)\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

which yields that g is a non-decreasing function and

$$g\left(\tau \right)\ge g(-1)={\displaystyle \frac{(n-1)(1-k)+4}{{(1+k)}^2}}.$$

Therefore, we get

$$\begin{array}{cc}\hfill \Re \left(1+{\displaystyle \frac{{\xi }_0{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{″}}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_0\right)\right]}^{\prime }}}\right)& \ge {\displaystyle \frac{(1-k)+4}{2(1+k)}}\hfill \\ \hfill & \ge 1+{\displaystyle \frac{c}{2}},\hfill \end{array}$$

which contradicts (6). As claimed by $\left|ϵ\right(\xi \left)\right|<1,\left|\xi \right|<1$, the argument of the theorem is present.

The proof of (iii).

Let

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\right)}^2{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }-1=W\left(\xi \right).\end{array}$$

(9)

Then, $W\left(\xi \right)=d_1\xi +\cdots$ is analytic in the open unit disk. A logarithmic differentiation gives

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime \prime }}{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}}={\displaystyle \frac{\xi W^{\prime }\left(\xi \right)}{1+W\left(\xi \right)}}+2{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}-1.\end{array}$$

(10)

According to (10), we obtain

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{\xi W^{\prime }\left(\xi \right)}{1+W\left(\xi \right)}}+2{\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}-1\right)<1+{\displaystyle \frac{c}{2}}.\end{array}$$

(11)

Thus, we receive

$$\begin{array}{c}\hfill {\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\prec {\displaystyle \frac{1-\xi }{1-k\xi }},\end{array}$$

where $k=1/(1+c),$ which brings $\Re \left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\right)>0.$ Therefore, (11) presents

$$\begin{array}{cc}\hfill \Re \left({\displaystyle \frac{\xi W^{\prime }\left(\xi \right)}{1+W\left(\xi \right)}}-1\right)& <1+{\displaystyle \frac{c}{2}}-2\Re \left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\right)\hfill \\ \hfill & <1+{\displaystyle \frac{c}{2}}.\hfill \end{array}$$

(12)

For $|{\xi }_0|<1$ with $\left|W\right({\xi }_0\left)\right|=1$. In virtue of Lemma 1, we get

$$\begin{array}{c}\hfill {\xi }_0{W}^{\prime }\left({\xi }_0\right)=kW\left({\xi }_0\right);(k\ge 1;W\left({\xi }_0\right)=e^{i\vartheta };\vartheta \in \mathbb{R}).\end{array}$$

Thus,

$$\begin{array}{cc}\hfill \Re \left({\displaystyle \frac{{\xi }_0{W}^{\prime }\left({\xi }_0\right)}{1+W\left({\xi }_0\right)}}-1\right)& =\Re \left({\displaystyle \frac{k{e}^{i\vartheta }}{1+e^{i\vartheta }}}\right)-1\hfill \\ \hfill & ={\displaystyle \frac{k}{2}}-1.\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{{\xi }_0{W}^{\prime }\left({\xi }_0\right)}{W\left({\xi }_0\right)}}-1\right)\ge 1+{\displaystyle \frac{c}{2}},\end{array}$$

where $0<c\le 1$, which contradicts the assumption. That is, $\left|W\right(\xi \left)\right|<1,\left|\xi \right|<1$. In addition, (9) shows that

$$\begin{array}{c}\hfill \left|{\left({\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\right)}^2{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }-1\right|<1,\left|\xi \right|<1,\end{array}$$

which concludes the desired assertion. □

The following result shows the upper bound of $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right).$ The proof is a direct application of Lemma 3.

Theorem 2. 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ satisfies the inequality

$${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }+\lambda \xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}\prec {\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{{\gamma }_1},\lambda >0,$$

then

$${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\prec {\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{{\gamma }_2},$$

where ${\gamma }_1={\gamma }_2+(2/\pi )ta{n}^{-1}\left(x{\gamma }_2\right).$

Proof. 

Since $\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]\in \mathcal{A}$, ${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\in \mathcal{P}.$ Let

$$p\left(\xi \right):={\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime },\xi \in \mathcal{U}.$$

Consequently, we obtain

$$\begin{array}{cc}\hfill p\left(\xi \right)+\lambda \xi p^{\prime }\left(\xi \right)& ={\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }+\lambda \xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}\hfill \\ \hfill & \prec {\left({\displaystyle \frac{1+\xi }{1-\xi }}\right)}^{{\gamma }_1}.\hfill \end{array}$$

Then, by the condition on ${\gamma }_1$ and ${\gamma }_2$, Lemma 3 implies the desired result. □

Theorem 3. 

If $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ has the symmetrical inequality

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)-Q_q^{\sigma ,\varsigma }\upsilon (-\xi )}}\right)>0,\end{array}$$

(13)

then $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function.

Proof. 

Since $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ satisfies the equations $Q_q^{\sigma ,\varsigma }\upsilon \left(0\right)=0$ and ${\left[Q_q^{\sigma ,\varsigma }\upsilon \left(0\right)\right]}^{\prime }=1$, it is a normalized function. By substituting $-\xi$ by $\xi$ in (13), this yields

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon (-\xi )\right]}^{\prime }}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)-Q_q^{\sigma ,\varsigma }\upsilon (-\xi )}}\right)>0.\end{array}$$

(14)

Adding the last two inequalities, this gives

$$\begin{array}{c}\hfill \Re \left({\displaystyle \frac{\xi \left({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }-{\left[Q_q^{\sigma ,\varsigma }\upsilon (-\xi )\right]}^{\prime }\right)}{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)-Q_q^{\sigma ,\varsigma }\upsilon (-\xi )}}\right)>0.\end{array}$$

(15)

This leads to $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)-Q_q^{\sigma ,\varsigma }\upsilon (-\xi )$ being a bounded turning function. Thus, according to the Kaplan theorem [26], one can observe that $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function. □

Also, the following result shows different conditions of a bounded turning function.

Theorem 4. 

Suppose that $\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]$ has the following positive real part:

$$\begin{array}{c}\hfill \Re \left({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }+L\left(\xi \right){\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime \prime }\right)>0\end{array}$$

(16)

whenever $\Re \left(L\right(\xi \left)\right)\ge 0$. Then, $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function.

Proof. 

Define an admissible function $\chi :{\mathbb{C}}^2\to \mathbb{C}$ as follows:

$$\begin{array}{c}\hfill \chi (f,g)=f\left(\xi \right)+L\left(\xi \right)g\left(\xi \right).\end{array}$$

By (16) with

$$f\left(\xi \right):={\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime },g\left(\xi \right):=\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″},$$

we get

$$\Re \left(\chi \left({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime },\xi {\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}\right)\right)>0.$$

In virtue of ([27], Theorem 5), we have

$$\Re \left({\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{\prime }\right)>0,$$

which implies that $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function. □

Theorem 5. 

Let $\Omega \left(\xi \right)$ be a bounded turning function with

$$inf\left({\displaystyle \frac{\Omega \left({\xi }_1\right)-\Omega \left({\xi }_2\right)}{{\xi }_1-{\xi }_2}}\right)>0,|{\xi }_1|<1,|{\xi }_2|<1.$$

If

$$\left|{\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}-{\displaystyle \frac{\xi }{\Omega \left(\xi \right)}}\right|\le {\displaystyle \frac{2inf\left({\displaystyle \frac{\Omega \left({\xi }_1\right)-\Omega \left({\xi }_2\right)}{{\xi }_1-{\xi }_2}}\right)}{{\left[{sup}_{\left|\xi \right|<1}\left(\Omega \left(\xi \right)\right)\right]}^2}},$$

then $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function.

Proof. 

Let

$$Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{s}_n{\xi }^n$$

and

$$\Omega \left(\xi \right)=\xi +\sum _{n=2}^{\infty }{\omega }_n{\xi }^n.$$

Define the function $H,$ as follows:

$$H\left(\xi \right)={\left({\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}-{\displaystyle \frac{\xi }{\Omega \left(\xi \right)}}\right)}^{″}.$$

Integrating yields

$${\left({\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}-{\displaystyle \frac{\xi }{\Omega \left(\xi \right)}}\right)}^{\prime }={\omega }_2-s_2+{\int }_0^{\xi }H\left(t\right)dt.$$

As a conclusion, we get

$${\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}-{\displaystyle \frac{\xi }{\Omega \left(\xi \right)}}=({\omega }_2-s_2)\xi +{\int }_0^{\xi }dy{\int }_0^{y}H\left(t\right)dt.$$

Hence, we have

$$Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)={\displaystyle \frac{\Omega \left(\xi \right)}{1+({\omega }_2-s_2)\Omega \left(\xi \right)+\Omega \left(\xi \right)\left(h\left(\xi \right)/\xi \right)}},$$

where

$$h\left(\xi \right)={\int }_0^{\xi }dy{\int }_0^{y}H\left(t\right)dt.$$

Therefore, we obtain

$${\left({\displaystyle \frac{h\left(\xi \right)}{\xi }}\right)}^{\prime }={\displaystyle \frac{1}{{\xi }^2}}{\int }_0^{\xi }\tau h^{″}\left(\tau \right)d\tau ={\displaystyle \frac{1}{{\xi }^2}}{\int }_0^{\xi }tH\left(t\right)dt.$$

According to the condition, we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{h\left({\xi }_2\right)}{{\xi }_2}}-{\displaystyle \frac{h\left({\xi }_1\right)}{{\xi }_1}}\right|& =\left|{\int }_{{\xi }_1}^{{\xi }_2}{\left({\displaystyle \frac{h\left(\xi \right)}{\xi }}\right)}^{\prime }d\xi \right|\hfill \\ \hfill & \le \left({\displaystyle \frac{2inf\left({\displaystyle \frac{\Omega \left({\xi }_1\right)-\Omega \left({\xi }_2\right)}{{\xi }_1-{\xi }_2}}\right)}{{\left[{sup}_{\left|\xi \right|<1}\left(\Omega \left(\xi \right)\right)\right]}^2}}\right)\left({\displaystyle \frac{|{\xi }_2-{\xi }_1|}{2}}\right),\hfill \end{array}$$

where ${\xi }_1\ne {\xi }_2.$ Next, we aim to show that

$$Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_1\right)\ne Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_2\right),{\xi }_1\ne {\xi }_2$$

or

$$\left|Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_1\right)-Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_2\right)\right|>0,{\xi }_1\ne {\xi }_2.$$

$$\begin{array}{cc}\hfill & \left|Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_1\right)-Q_q^{\sigma ,\varsigma }\upsilon \left({\xi }_2\right)]\right|\hfill \\ \hfill & ={\displaystyle \frac{\left|\Omega \left({\xi }_1\right)-\Omega \left({\xi }_2\right)+\Omega \left({\xi }_2\right)\Omega \left({\xi }_1\right)\left({\displaystyle \frac{h\left({\xi }_2\right)}{{\xi }_2}}-{\displaystyle \frac{h\left({\xi }_1\right)}{{\xi }_1}}\right)\right|}{\left|1+({\omega }_2-s_2)\Omega \left({\xi }_1\right)+\Omega \left({\xi }_1\right)\left({\displaystyle \frac{h\left({\xi }_1\right)}{{\xi }_1}}\right)\right|\left|1+({\omega }_2-s_2)\Omega \left({\xi }_2\right)+\Omega \left({\xi }_2\right)\left({\displaystyle \frac{h\left({\xi }_2\right)}{{\xi }_2}}\right)\right|}}\hfill \\ \hfill & >{\displaystyle \frac{|\Omega \left({\xi }_1\right)-\Omega \left({\xi }_2\right)|-inf\left({\displaystyle \frac{\Omega \left({\xi }_1\right)-\Omega \left({\xi }_2\right)}{{\xi }_1-{\xi }_2}}\right)\left({\xi }_2-{\xi }_1\right)}{\left|1+({\omega }_2-s_2)\Omega \left({\xi }_1\right)+\Omega \left({\xi }_1\right)\left({\displaystyle \frac{h\left({\xi }_1\right)}{{\xi }_1}}\right)\right|\left|1+({\omega }_2-s_2)\Omega \left({\xi }_2\right)+\Omega \left({\xi }_2\right)\left({\displaystyle \frac{h\left({\xi }_2\right)}{{\xi }_2}}\right)\right|}}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Thus, $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function. □

Corollary 1. 

Let

$$\left|{\displaystyle \frac{\xi }{{\left[Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)\right]}^{″}}}\right|\le 2.$$

Then, $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function.

Proof. 

Assuming $\Omega \left(\xi \right)=\xi$ in Theorem 5, we obtain the outcome, where it is precise for

$$Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)={\displaystyle \frac{\xi }{{(1+\xi )}^{2+\alpha }}},$$

where

$$\left|{\left({\displaystyle \frac{\xi }{Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)}}\right)}^{″}\right|=(2+\alpha )(1+\alpha ){(1+\xi )}^{\alpha },\alpha >0.$$

□

In view of Corollary 1, we obtain the following result:

Corollary 2. 

If

$$Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)={\displaystyle \frac{\xi }{1+{\sum }_{n=1}^{\infty }{c}_n{\xi }^n}},$$

where

$$\sum _{n=2}^{\infty }n(n-1)\left|c_n\right|\le 2,$$

then $Q_q^{\sigma ,\varsigma }\upsilon \left(\xi \right)$ is a bounded turning function.

4. Conclusions

We created QFFOs in a complex domain, acting on the class of analytic functions. Additionally, we aimed to understand the geometric actions of these QFFOs; as a result, these QFFOs were normalized. We also determined the QFFOs’ upper bounds in terms of the normalized special function. Under a set of conditions, the proposed operators are univalent and convex in the open unit disk. Finally, we present a set of conditions to obtain the bounded turning functions’ properties. Our methods are given from the field of the geometric function theory, specifically the differential subordination theory. For future work, one can use the suggested operators to develop different classes of analytic functions, such as meromorphic and harmonic functions. Moreover, QFFOs can be selected to obtain univalent solutions of quantum–fractal–fractional differential equations.