Further characterizations of the $m$-weak group inverse of a complex matrix

1.   Introduction

Fractional analysis can be regarded as a generalization of classical analysis. Fractional analysis has been investigated and discussed by several mathematicians and they have explored the fractional derivatives and integrals in variant ways with numerous notations. Although the expressions of these generalized definitions can be transformed into each other, but have variant physical meanings. It is well known that the first fractional integral operator is the Riemann-Liouville fractional integral operator. Recently, fractional calculus has become one of the prestigious fields in research because of its inherent applications in different fields like numerical physical science ^[1], fluid mechanics ^[2], biological modeling ^[3,4], and many more. Fractional calculus owes its starting point to whether or not the importance of the derivative of an integer order could be generalized to a fractional order which is not an integer. Analysis of fractional calculus is the generalization of classical analysis, it has been developed rapidly with the interesting convexity theory attributed to Jensen ^[5]. Its fertile applications in functional analysis, stochastic theory, finite element method, and optimization theory have made it a fascinating topic for research. Therefore, it acts as an incorporative subject among quantum theory, orthogonal polynomials, combinatorics, etc. These are the principal urges that led to the deep research and improvement of the theory of inequality in literature (see ^[6,7,8]).

Let $0 < \gamma_{1}\leq \gamma_{2}\leq...\leq \gamma_{\mathcal{P}}$ and $\varphi = \big(\varphi_{1}, \varphi_{2}, ..., \varphi_{\mathcal{P}}\big)$ be non-negative weights such that $\sum_{k = 1}^{\mathcal{P}}\ \varphi_{k} = 1$. The Jensen's inequality (see ^[5]) in the literature, reads as:

If the convexity of $\Phi$ holds true on the interval $\big[\, \wp_1, \wp_2\big]$, then

for all $\gamma_{k}\in [\, \mathrm{d}, \mathrm{D}],$ $\varphi _{k}\in \left[ 0, 1 \right]\, {\rm and} \left(k = 1, 2, ..., \mathcal{P}\right).$ This inequality helps to obtain new bounds for useful distances in information theory (see ^[9]).

Mercer ^[10] in the year 2003, investigated a generalized form of Jensen's inequality, which is famously known as Jensen-Mercer (J-M) inequality:

If $\Phi$ is a convex function on $\left[\, \mathrm{d}, \mathrm{D}\right]$, then

holds true for all $\gamma_{k}\in \left[\, \mathrm{d}, \mathrm{D}\right],$ $\varphi _{k}\in \left[ 0, 1 \right]$ and $\left(k = 1, 2, ..., \mathcal{P}\right).$

Some generalizations on J-M operator inequalities are proposed by Pečarić et al. in ^[11]. Later, Niezgoda in ^[12] proved several generalizations of Mercer's type inequalities to higher dimensions. Kian ^[13] contemplated the idea of Jensen inequality for superquadratic functions. Moreover, reverse Mercer's type operator inequalities and the aforesaid inequalities for super-quadratic functions have been proposed in ^[14,15]. Also, the integral and integral mean version have been established in ^[16]. Cortez et al. (see Remark 3.4 ^[17]) presented the concept of $s$-convex variant of J-M inequality in a generalized form. For some recent generalizations on J-M type inequalities, we direct readers to go through ^[18,19,20] and the references cited therein.

Hudzik et al. ^[21], introduced the notion of $s$-convex functions in the first and second sense and also discussed some of their notable features.

Definition 1.1. Let $s\in (0, 1]$ and $\Phi$ be a real valued function on an interval $\mathcal {I} = [0, \infty)$. Then $\phi$ is said to be $s$-convex in the first sense if

holds true for all $\gamma_1, \gamma_2 \in \mathcal{I},$ $\varphi_1, \varphi_2\geq 0$ with $\varphi_1^{s}+\varphi_2^{s} = 1$.

Definition 1.2. Let $s\in (0, 1]$ and $\Phi$ be a real valued function on an interval $\mathcal {I} = [0, \infty)$. Then $\phi$ is said to be $s$-convex in the second sense if

holds true for all $\gamma_1, \gamma_2 \in \mathcal{I},$ $\varphi_1, \varphi_2\geq 0$ with $\varphi_1+\varphi_2 = 1$.

The class of $s$-convex function in the first and second sense are denoted by $(\Phi \in \ K_{s}^{1})$ and $(\Phi \in \ K_{s}^{2})$ respectively. Moreover, they also studied that the class of $s$-convex function in the second sense is more stronger than the $s$-convexity in first sense for $s \in (0, 1)$. Several features of $s$-convex function in both senses are discussed taking some examples into consideration. It is interesting to see that if $s \in (0, 1)$ and $\Phi\in \ K_{s}^{2},$ then $\Phi$ is non negative.

For more detail on $s$-convex function (see ^{[21,22,23,24]}) and references cited therein. Chen ^[23] gave a nice relationship between convex functions and $s$-convex functions as:

Lemma 1.1. (see ^[23]) Suppose $\Phi:[\, \mathrm{d}, \mathrm{D}]\rightarrow \Re$ is a convex function. Then

(i) If $\Phi$ is non-negative, then it is $s$-convex for $s \in(0, 1].$

(ii) If $\Phi$ is non-positive, then it is $s$-convex for $s \in[1, \infty).$

Butt et al. ^[25] investigated the J-M inequality for $s$-convex functions in Breckner sense $\Big(s > 0\Big)$ as follows:

Theorem 1.1. Let $\varphi_1, \varphi_2, ..., \varphi_\mathcal{P}$ be positive probability distributions and $\underset{\jmath = 1}{\overset{\mathcal{P}}{\sum \limits}}\varphi_\jmath^s\leq1.$ If $\Phi:[\mathrm{d}, \mathrm{D}]\subset (0, \infty)\to \Re$ is $s$-Breckner sense convex function, then the following inequality is valid

for any increasing sequence $\{\gamma_\mathcal{\jmath}\}_{\jmath = 1}^{\mathcal{P}} \in [\, \mathrm{d}, \mathrm{D}].$

The following conditions are necessary:

(i) By considering $s = 1$ in (1.3), we get the J-M inequality (1.1).

(ii) By considering $s = 1$ and $\varphi_1 = \mathrm{d}$ and $\varphi_2 = \mathrm{D},$ in (1.3) it gives the definition of the classical convex function.

The following inequality named as Simpson's inequality, is given in the literature as (see ^[26,27,28]).

where $\Phi:[\mathrm{d}, \mathrm{D}]\rightarrow \Re$ is a four times continuously differentiable mapping on $(\mathrm{d}, \mathrm{D})$ and $\left\vert \left\vert \Phi^{4}\right\vert \right\vert _{\infty } = \sup_{x\in \left(\mathrm{d}, \mathrm{D}\right) }\left\vert \Phi^{4}\left(x\right) \right\vert < \infty.$

Several authors have concentrated on Simpson-type inequalities for different classes of mappings. Precisely, a few mathematicians have dealt with Simpson-type inequalities for convex mappings, since convexity theory is an efficient and stable process for taking care of an extraordinary number of issues that emerge in various branches in pure and applied sciences. For some recent improvements on Simpson type inequality via different convexities and fractional operators see references ^{[29,30,31,32]} and cited therein.

Let $\Phi \in \mathcal L[\mathrm{d}, \mathrm{D}].$ Then the left and right Riemann-Liouville (R-L) fractional integrals of order $\delta > 0$ with $\mathrm{d}\geq0$ are defined as follows:

and

For further details one may (see ^[33,34]).

In ^[35], the idea of the J-M inequality has been used by Kian and Moslehian and the following Hermite-Hadamard-Mercer inequality was demonstrated:

Recently, the notions of fractional operators have drawn the attention of several researchers. Fractional calculus helps mathematicians solve many real-life problems in a convincing manner than classical calculus. Mainly, there are two types of nonlocal fractional derivatives, the classical Riemann-Liouville and Caputo derivatives with singular kernels and the others with non-singular kernels, which have been introduced recently, such as the Caputo-Fabrizio and Atangana-Baleanu derivatives. However, the fractional derivative operators with non-singular kernels are effective to solve the non-locality of real-world problems in an appropriate manner. The introduction of new notions of fractional operators is a journey to succeed momentum to the fractional calculus and to gain the most efficient operators to the discussion. Now, we recall the notion of the Caputo-Fabrizio fractional operator:

Definition 1.3. Let $p\in[1, \infty)$ and $[\omega_{1}, \omega_{2}]$ be an open subset of $\Re$, the Sobolev space $\mathcal{H}^{p}(\omega_{1}, \omega_{2})$ is defined by

Definition 1.4. ^[36] Let $\Phi\in \mathcal H^{1}(0, \kappa)$, $\kappa > \omega$, $\delta\in[0, 1]$ then, the definition of the new Caputo fractional derivative is given as:

where $\mathcal M(\delta)$ is normalization function with $\mathcal{M}(0) = \mathcal{M}(1) = 1$.

Moreover, the corresponding Caputo-Fabrizio fractional integral operator is given as:

Definition 1.5. ^[37] Let $\Phi\in \mathcal H^{1}(0, \kappa)$, $\kappa > \omega$, $\delta \in [0, 1]$. Then, the definition of Caputo-Fabrizio fractional integrals are given as:

and

where $\mathcal M (\delta)$ is normalization function with $\mathcal{M}(0) = \mathcal{M}(1) = 1$.

Recently, Atangana & Baleanu introduced a new fractional operator containing the Mittag-Leffler function in the kernel, that solves the problem of retrieving the original function (a clear advantage over the Caputo-Fabrizio operator). Mittag-Leffler's function is more suitable than a power law in modeling nature and physical phenomena. This attribute has made the A-B operator more effective and helpful. As a result, many researchers have shown keen interest in utilizing this operator. Atangana & Baleanu introduced the derivative operator both in Caputo and Riemann-Liouville sense:

Definition 1.6. ^[38] Let $\kappa > \omega$, $\delta \in \lbrack 0, 1]$ and $\Phi\in \mathcal H^{1}(\omega, \kappa)$. The new fractional derivative is given as:

where $E_{\delta}$ is the Mittag-Leffler function and is defined as $E_{\delta}(-t^{\delta}) = \sum_{k = 0}^{\infty}\frac{(-t)^{\delta k}}{\Gamma(\delta k +1)}$.

Definition 1.7. ^[38] Let $\Phi\in \mathcal H^{1}(\omega, \kappa)$, $\omega > \kappa$, $\delta \in [0, 1]$. The new fractional derivative is given as:

where $E_{\delta}$ is the Mittag-Leffler function.

However, in the same paper the authors also presented corresponding A-B fractional integral operator as:

Definition 1.8. ^[38] The fractional integral operator with non-local kernel of a function $\Phi\in \mathcal H^{1}(\omega, \kappa)$ is defined as:

where $\kappa > \omega, \delta\in[0, 1].$

In ^[39], the right hand side of A-B fractional integral operator is presented as follows:

Here, $\Gamma (\delta)$ is the Gamma function. The positivity of the normalization function $\mathcal M(\delta)$ implies that the fractional A-B integral of a positive function is positive. It is worth noticing that the case when the order $\delta\rightarrow 1$ yields the classical integral and the case when $\delta\rightarrow 0$ provides the initial function.

Sarikaya et al. ^[33] used Riemann-Liouville fractional integrals to generalise the Hermite-Hadamard inequality. Iscan ^[40] generalised the conclusions reached by Sarikaya et al. ^[33] to Hermite-Hadamard-Fejér type inequalities. Chen ^[41] employed the methods of Sarikaya et al. ^[33] and used the product of two convex functions to produce fractional Hermite-Hadamard type integral inequalities. Recently, Sun proved different variants of the Hermite-Hadamard inequality via generalized local fractional integral operators with Mittag-Leffler kernel for $h$-convex functions ^[42], $s$-preinvex functions ^[43] and generalized preinvex fuctions ^[44]. Fernandez and Mohammed ^[45] presented a nice relation between Riemann-Liouville fractional integrals and A-B fractional integral operators and also proved new type of Hermite-Hadamard inequalities for convex functions. Ogulmus et al. ^[46] established Hermite-Hadamard-Mercer type inequalities involving Riemann-Liouville fractional operator. Sahoo et al. ^[47] presented some mid-point versions of Hermite-Hadamard inequalities via Caputo-Fabrizio fractional operator. Latif et al. ^[48] worked on Hermite-Hadamard-Fejér type fractional inequalities relating to a convex harmonic function and a positive symmetric increasing function.

The main motivation of this study is to obtain an integral identity employing A-B fractional integral operator, which has a unique place among fractional integral operators, and to generate some new Simpson-Mercer's like inequalities for $s$-convex functions based on this identity. Considering many special cases of the main findings, the study also includes the applications of the results.

2.   Main results

In this section, we study several Simpson-Mercer's like inequalities via the A-B fractional integral operator for differentiable functions on $(\mathrm{d}, \mathrm{D})$. For this, we establish a new A-B integral identity that will serve as an auxiliary result to produce subsequent results for improvements. Throughout the paper, let us assume that $\mathcal P_{1}$: = $\left(\frac{1+\kappa}{2}\right)^s+\left(\frac{1-\kappa}{2}\right)^s\le 1$, for $\kappa\in[0, 1].$

Lemma 2.1. Suppose the mapping $\Phi:\mathcal I = [\mathrm{d}, \mathrm{D}] \rightarrow \Re$ is differentiable on $(\mathrm{d}, \mathrm{D})$ with $\mathrm{D} > \mathrm{d}$. If $\Phi^{{ \prime }} \in \mathcal L[\mathrm{d}, \mathrm{D}],$ then for all $\omega_1, \omega_2 \in [\mathrm{d}, \mathrm{D}]$ and $\delta > 0,$ the following A-B fractional identity of Simpson-Mercer type holds true:

for $\kappa \in[0, 1]$.

Proof. Let us assume that

Implies that

Integrating by parts, we get

By substituting the variables, we get

Similarly,

By using the values of ${\mathcal{J}}_{1}$ and ${\mathcal{J}}_{2}$ in (2.2), we get (2.1).

Corollary 2.1. If we consider $\delta = 1$ in (2.1), then we have the following Simpson-Mercer type equality

Remark 2.1. If we set $\omega_1 = \mathrm{d}$ and $\omega_2 = \mathrm{D}$ in Corollary 2.1, we obtain Lemma 1 in ^[27].

Theorem 2.1. Taking all the conditions as defined in Lemma 2.1 and assumption $\mathcal P_{1}$ into consideration and let $|\Phi^{{\prime}}|$ be $s$-convex function on $[\mathrm{d}, \mathrm{D}], \;$ for $s > 0$. Then, for all $\delta > 0,$ the following Simpson-Mercer type inequality for A-B fractional integral holds true for $\kappa\in [0, 1]$:

where

and

Proof. Employing Lemma 2.1 with the J-M inequality and the $s$-convexity of $|\Phi^{{\prime}}|,$ we have

which completes the proof.

Remark 2.2. If we consider $\omega_1 = \mathrm{d}$ and $\omega_2 = \mathrm{D}$ and $\delta = 1$ in Theorem 2.1, we obtain Theorem 7 in ^[27].

Corollary 2.2. Taking the same assumptions as defined in Theorem 2.1 with $s = 1$ into consideration, we get the following Simpson-Mercer type inequality

Corollary 2.3. If we consider $\delta = 1$ in Theorem 2.1, then we have the following Simpson-Mercer type inequality

In the results to follow hereon, we will give the new upper bounds for the right hand side of the Simpson-Mercer type inequality via $s$-convexity involving A-B fractional integrals operator.

Theorem 2.2. Taking all the conditions as defined in Lemma 2.1 and assumption $\mathcal P_{1}$ into consideration, if $|\Phi^{{\prime}}|^q$ is $s$-convex function on $[\mathrm{d}, \mathrm{D}], \;$ for $s > 0$, then for all $\delta > 0,$ the following inequality for A-B fractional integral holds true for $\kappa\in [0, 1]$:

where $\frac{1}{p}+\frac{1}{q} = 1 \ and\ q > 1$.

Proof. Employing Lemma 2.1 and the Hölder's inequality, we have

Since $|\Phi^{{\prime}}|^q$ is $s$-convex on $[ \mathrm{d}, \mathrm{D}],$ with the J-M inequality, we obtain

and

By using the Eqs (2.8) and (2.9) with (2.7), we get the (2.6). This concludes the proof.

Remark 2.3. If we set $\omega_1 = \mathrm{d},$ $\omega_2 = \mathrm{D}$ and $\delta = 1$ in Theorem 2.2, we obtain Theorem 8 in ^[27].

Corollary 2.4. Under the same assumptions as defined in Theorem 2.2 with $s = 1,$ we get the following A-B fractional inequality of Simpson-Mercer type:

Corollary 2.5. Under the same assumptions as defined in Theorem 2.2 with $\delta = 1,$ we get the following Simpson-Mercer type inequality

Theorem 2.3. Taking all the conditions as defined in Lemma 2.1 and assumption $\mathcal P_{1}$ into consideration, if $|\Phi^{{\prime}}|^q$ is $s$-convex function on $[\mathrm{d}, \mathrm{D}],$ for $s > 0$ and $q > 1,$ then for all $\delta > 0,$ the following inequality for A-B fractional integral holds true for $\kappa\in[0, 1]$$:$

where $\frac{1}{p}+\frac{1}{q} = 1 \ and\ q > 1$.

Proof. Employing Lemma 2.1 and the Hölder inequality, we have

Since $|\Phi^{{\prime}}|^q$ is $s$-convex on $[\mathrm{d}, \mathrm{D} ],$ with the J-M inequality, we obtain

and

Upon combining the Eqs (2.13) and (2.14) in (2.12) and using the tool of calculus, we get (2.11). This completes the proof.

Remark 2.4. If we set $\omega_1 = \mathrm{d},$ $\omega_2 = \mathrm{D}$ and $\delta = 1$ in Theorem 2.3, we obtain Theorem 9 in ^[27].

Corollary 2.6. Under the same assumptions as defined in Theorem 2.3 with $s = 1,$ we get the following A-B fractional inequality of Simpson-Mercer type:

Corollary 2.7. Under the same assumptions as defined in Theorem 2.3 with $\delta = 1,$ we get the following Simpson-Mercer type inequality:

Theorem 2.4. Taking all the conditions as defined in Lemma 2.1 and assumption $\mathcal P_{1}$ into consideration, if $|\Phi^{{\prime}}|^q$ is $s$-convex function on $[\mathrm{d}, \mathrm{D}],$ for $s > 0,$ $\ q\geq1,$ then for all $\delta > 0,$ the following Simpson- Mercer type inequality for A-B fractional integral {holds true} for $\kappa\in[0, 1]$$:$

where

and

Proof. Employing Lemma 2.1 and well-known Power mean inequality, we have

Since $|\Phi^{{\prime}}|^q$ is $s$-convex on $[\mathrm{d}, \mathrm{D}],$ with the J-M inequality, we obtain

and

By using the Eqs (2.20) and (2.21) in (2.19), we get

which completes the proof.

Remark 2.5. If we consider $\omega_1 = \mathrm{d},$ $\omega_2 = \mathrm{D}$ and $\delta = 1$ in Theorem 2.4, we obtain Theorem 10 in ^[27].

Corollary 2.8. Under the same assumptions as defined in Theorem 2.4 with $s = 1,$ we get the following A-B fractional inequality of Simpson-Mercer type:

3.   Applications

3.1.   Modified Bessel functions

Example 3.1. Let the function $\mathcal{J}_{\varrho}:{\Re}\to [1, \infty)$ be defined ^[54] as

Here, we consider the modified Bessel function of first kind given in

The first and second order derivative are given as

If we use, $\Phi(u) = \mathcal{J}^{\prime}_{\varrho}(u)$ and the above functions in Corollary 2.3, we have

Example 3.2. If we use, $\Phi(u) = \mathcal{J}^{\prime}_{\varrho}(u)$ and the above functions in Corollary 2.5, we have

3.2.   Special means

Consider the following two special means for $0 < \mathrm{d} < \mathrm{D}.$

The arithmetic mean:

The generalized logarithmic-mean:

Proposition 3.1. Suppose that $\mathrm{d}, \mathrm{D} \in \Re$, $s \in (0, 1],$ $0 < \mathrm{d} < \mathrm{D},$ $0\notin[\mathrm{d}, \mathrm{D}]$. Then,

Proof. The proof follows an immediate consequences from Corollary 2.3 by considering $\Phi(u) = u^s.$

Proposition 3.2. Suppose that $\mathrm{d}, \mathrm{D} \in \Re$, $s \in (0, 1],$ $0 < \mathrm{d} < \mathrm{D},$ $0\notin[\mathrm{d}, \mathrm{D}]$. Then for all $q > 1,$ the following inequality holds true:

Proof. The proof follows an immediate consequence from Corollary 5 by considering $\Phi(u) = {u^{-s}}.$

Proposition 3.3. Suppose that $\mathrm{d}, \mathrm{D} \in \Re$, $s \in (0, 1],$ $0 < \mathrm{d} < \mathrm{D},$ $0\notin[\mathrm{d}, \mathrm{D}]$. Then,

Proof. The proof follows an immediate consequences from Corollary 2.7 by considering $\Phi(u) = u^s.$

3.3.   Q-digamma function

The $q$-digamma(psi) function $\varrho_\rho,$ is the $\rho$-analogue of the digamma function $\varrho$ (see ^[54,55]) given as:

For $\rho > 1$ and $\gamma > 0,$ $\rho$-digamma function $\varrho_\rho$ can be given as:

Proposition 3.4. Suppose $\mathrm{d}, \mathrm{D}, \rho$ be real numbers such that $0 < \mathrm{d} < \mathrm{D}$, $0 < \rho < 1,$ $s \in (0, 1]$ and $\frac{1}{p}+\frac{1}{q} = 1, q > 1$. Then the following inequality holds true:

Proof. By employing the definition of $\rho$-digamma function $\varrho _{\rho}(\gamma),$ it is easy to notice that $\rho$ -digamma function $\Phi(\gamma) = \varrho _{\rho}(\gamma)$ is completely monotonic on $(0, \infty).$ This ensures that the function $\Phi^{\prime}(\gamma) = \varrho^{\prime} _{\rho}(\gamma)$ is $s$-convex on $(0, \infty)$ for each $\rho \in (0, 1)$. Now by applying Corollary 2.7, we get the desired result.

Proposition 3.5. Suppose $\mathrm{d}, \mathrm{D}, q, \rho$ be real numbers such that $0 < \mathrm{d} < \mathrm{D},$ $\frac{1}{p}+\frac{1}{q} = 1, q > 1$ and $0 < \rho < 1,$ $s \in (0, 1].$ Then the following inequality holds true:

Proof. By employing the definition of $\rho$-digamma function $\varrho _{\rho}(\gamma),$ it is easy to notice that $\rho$ -digamma function $\Phi(\gamma) = \varrho _{\rho}(\gamma)$ is completely monotonic on $(0, \infty).$ This ensures that the function $\Phi^{\prime}(\gamma) = \varrho^{\prime} _{\rho}(\gamma)$ is $s$-convex on $(0, \infty)$ for each $\rho \in (0, 1)$. Now by applying Corollary 2.5, we get the desired result.

Proposition 3.6. Suppose $\mathrm{d}, \mathrm{D}, \rho$ be real numbers such that $0 < \mathrm{d} < \mathrm{D}, \,$ and $0 < \rho < 1,$ $s \in (0, 1].$ Then the following inequality holds true:

Proof. By employing the definition of $\rho$-digamma function $\varrho _{\rho}(\gamma),$ it is easy to notice that $\rho$ -digamma function $\Phi(\gamma) = \varrho _{\rho}(\gamma)$ is completely monotonic on $(0, \infty).$ This ensures that the function $\Phi^{\prime}(\gamma) = \varrho^{\prime} _{\rho}(\gamma)$ is $s$-convex on $(0, \infty)$ for each $\rho \in (0, 1)$. Now by applying Corollary 2.3, we get the desired result.

4.   Conclusions

One can see that the recent developments in the field of inequalities are related to finding generalized versions and new bounds of various well-known inequalities employing different fractional integral operators. Researchers use new methodologies, new applications, and new operators to add originality to this subject. In this article, the incorporation of the Simpson-Mercer inequalities and the Atangana-Baleanu (A-B) fractional integral operator is studied for differentiable mappings, whose derivatives in absolute value at certain powers are $s$-convex in the second sense. Some novel cases of the established results are discussed as well. Moreover, the study also have applications to special means, modified Bessel functions, and $q$-digamma functions. An interesting scope is to check whether the methodology of this paper can be utilized to refine the Simpson-Mercer inequality via concepts such as interval analysis, quantum calculus, and coordinates. In future readers can work on modified A-B fractional operators as given in manucripts (see ^[49,50,51]) and modified Caputo-Fabrizio fractional operators (see ^[52,53]).

Acknowledgements

The authors received financial support from Taif University Researches Supporting Project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare no conflicts of interest.