A study on the fractal-fractional tobacco smoking model

Hasib Khan; Jehad Alzabut; Anwar Shah; Sina Etemad; Shahram Rezapour; Choonkil Park; Hasib Khan; Jehad Alzabut; Anwar Shah; Sina Etemad; Shahram Rezapour; Choonkil Park

doi:10.3934/math.2022767

AIMS Mathematics

2022, Volume 7, Issue 8: 13887-13909. doi: 10.3934/math.2022767

Previous Article Next Article

Research article

A study on the fractal-fractional tobacco smoking model

1.
Department of Mathematics, Shaheed Benazir Bhutto Uniersity, Sheringal, Dir Upper, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
3.
Department of Industrial Engineering, OSTİM Technical University, 06374 Ankara, Türkiye
4.
Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhtunkhwa, Pakistan
5.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
7.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received: 30 March 2022 Revised: 01 May 2022 Accepted: 08 May 2022 Published: 25 May 2022
MSC : 26A33, 34A08, 35R11

In this article, we consider a fractal-fractional tobacco mathematical model with generalized kernels of Mittag-Leffler functions for qualitative and numerical studies. From qualitative point of view, our study includes; existence criteria, uniqueness of solution and Hyers-Ulam stability. For the numerical aspect, we utilize Lagrange's interpolation polynomial and obtain a numerical scheme which is further illustrated simulations. Lastly, a comparative analysis is presented for different fractal and fractional orders. The numerical results are divided into four figures based on different fractal and fractional orders. We have found that the fractional and fractal orders have a significant impact on the dynamical behaviour of the model.

Keywords:

Citation: Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park. A study on the fractal-fractional tobacco smoking model[J]. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767

Related Papers:

[1]	Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak . Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546
[2]	Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334
[3]	Miguel Vivas-Cortez, Muhammad Uzair Awan, Muhammad Zakria Javed, Artion Kashuri, Muhammad Aslam Noor, Khalida Inayat Noor . Some new generalized $\kappa$ –fractional Hermite–Hadamard–Mercer type integral inequalities and their applications. AIMS Mathematics, 2022, 7(2): 3203-3220. doi: 10.3934/math.2022177
[4]	Jia-Bao Liu, Saad Ihsan Butt, Jamshed Nasir, Adnan Aslam, Asfand Fahad, Jarunee Soontharanon . Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator. AIMS Mathematics, 2022, 7(2): 2123-2141. doi: 10.3934/math.2022121
[5]	Yanping Yang, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function. AIMS Mathematics, 2021, 6(11): 12260-12278. doi: 10.3934/math.2021710
[6]	Yamin Sayyari, Mana Donganont, Mehdi Dehghanian, Morteza Afshar Jahanshahi . Strongly convex functions and extensions of related inequalities with applications to entropy. AIMS Mathematics, 2024, 9(5): 10997-11006. doi: 10.3934/math.2024538
[7]	Jamshed Nasir, Saber Mansour, Shahid Qaisar, Hassen Aydi . Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel. AIMS Mathematics, 2023, 8(5): 10001-10020. doi: 10.3934/math.2023506
[8]	Tahir Ullah Khan, Muhammad Adil Khan . Hermite-Hadamard inequality for new generalized conformable fractional operators. AIMS Mathematics, 2021, 6(1): 23-38. doi: 10.3934/math.2021002
[9]	Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Kottakkaran Sooppy Nisar, Dumitru Baleanu . Some generalized fractional integral inequalities with nonsingular function as a kernel. AIMS Mathematics, 2021, 6(4): 3352-3377. doi: 10.3934/math.2021201
[10]	Paul Bosch, Héctor J. Carmenate, José M. Rodríguez, José M. Sigarreta . Generalized inequalities involving fractional operators of the Riemann-Liouville type. AIMS Mathematics, 2022, 7(1): 1470-1485. doi: 10.3934/math.2022087

Abstract

1. Introduction

For a convex function $\sigma:\mathrm{I}\subseteq \bf{R}\to \bf{R}$ on $\mathrm{I}$ with $\nu_{1}, \nu_{2}\in \mathrm{I}$ and $\nu_{1} < \nu_{2}$ , the Hermite-Hadamard inequality is defined by ^[1]:

$\begin{equation} \sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \leq \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma(\eta)d\eta \leq \frac{\sigma(\nu_{1})+\sigma(\nu_{2})}{2}. \end{equation}$

(1.1)

The Hermite-Hadamard integral inequality (1.1) is one of the most famous and commonly used inequalities. The recently published papers ^[2,3,4] are focused on extending and generalizing the convexity and Hermite-Hadamard inequality.

The situation of the fractional calculus (integrals and derivatives) has won vast popularity and significance throughout the previous five decades or so, due generally to its demonstrated applications in numerous seemingly numerous and great fields of science and engineering ^[5,6,7].

Now, we recall the definitions of Riemann-Liouville fractional integrals.

Definition 1.1 (^[5,6,7]). Let $\sigma\in L_{1}[\nu_{1}, \nu_{2}].$ The Riemann-Liouville integrals $J_{\nu_{1}+}^{\vartheta }\sigma$ and $J_{\nu_{2}-}^{\vartheta }\sigma$ of order $\vartheta > 0$ with $\nu_{1}\geq 0$ are defined by

$\begin{equation} J_{\nu_{1}+}^{\vartheta }\sigma(x) = \frac{1}{\Gamma (\vartheta )}\int_{\nu_{1}}^{x}(x-\eta)^{\vartheta -1}\sigma(\eta)d\eta,\; \ \ \ \nu_{1} < x \end{equation}$

(1.2)

and

$\begin{equation} J_{\nu_{2}-}^{\vartheta }\sigma(x) = \frac{1}{\Gamma (\vartheta )}\int_{x}^{\nu_{2}}(\eta-x)^{\vartheta -1}\sigma(\eta)d\eta,\; \ \ x < \nu_{2}, \end{equation}$

(1.3)

respectively. Here $\Gamma (\vartheta)$ is the well-known Gamma function and $J_{\nu_{1}+}^{0}\sigma(x) = J_{\nu_{2}-}^{0}\sigma(x) = \sigma(x).$

With a huge application of fractional integration and Hermite-Hadamard inequality, many researchers in the field of fractional calculus extended their research to the Hermite-Hadamard inequality, including fractional integration rather than ordinary integration; for example see ^{[8,9,10,11,12,13,14,15,16,17,18,19,20,21]}.

In this paper, we consider the integral inequality of Hermite-Hadamard-Mercer type that relies on the Hermite-Hadamard and Jensen-Mercer inequalities. For this purpose, we recall the Jensen-Mercer inequality: Let $0 < x_{1}\leq x_{2}\leq \cdots\leq x_{n}$ and $\mu = (\mu _{1}, \mu_{2}, \ldots, \mu _{n})$ nonnegative weights such that $\sum_{i = 1}^{n}\mu_{i} = 1$ . Then, the Jensen inequality ^[22,23] is as follows, for a convex function $\sigma$ on the interval $[\nu_{1}, \nu_{2}]$ , we have

$\begin{equation} \sigma\Bigg(\sum\limits_{i = 1}^{n}\mu _{i}x_{i}\Bigg)\leq \sum\limits_{i = 1}^{n}\mu _{i}\sigma(x_{i}), \end{equation}$

(1.4)

where for all $x_{i}\in \lbrack \nu_{1}, \nu_{2}]$ and $\mu_{i}\in \lbrack 0, 1]$ , $(i = \overline{1, n})$ .

Theorem 1.1 (^[2,23]). If $\sigma$ is convex function on $[\nu_{1}, \nu_{2}]$ , then

$\begin{equation} \sigma\left(\nu_{1}+\nu_{2}-\sum\limits^{n}_{i = 1}\mu_{i}x_{i}\right)\leq \sigma(\nu_{1})+\sigma(\nu_{2})-\sum\limits^{n}_{i = 1}\mu_{i}\sigma(x_{i}), \end{equation}$

(1.5)

for each $x_{i}\in[\nu_{1}, \nu_{2}]$ and $\mu_{i}\in[0, 1]$ , $(i = \overline{1, n})$ with $\sum^{n}_{i = 1}\mu_{i} = 1$ . For some results related with Jensen-Mercer inequality, see ^[24,25,26].

In view of above indices, we establish new integral inequalities of Hermite-Hadamard-Mercer type for convex functions via the Riemann-Liouville fractional integrals in the current project. Particularly, we see that our results can cover the previous researches.

2. Main results

Theorem 2.1. For a convex function $\sigma:[\nu_{1}, \nu_{2}]\subseteq \bf{R} \to\bf{R}$ with $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

$\begin{multline} \sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\leq \frac{2^{\vartheta -1} \Gamma\left(\vartheta +1\right)}{\left( y-x\right)^{\vartheta}} \Biggl[ J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \\ +\; J_{\left(\nu_{1}+\nu_{2}-x\right)-}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\Biggr] \leq \sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right)-\frac{\sigma\left(x\right)+\sigma\left(y\right)}{2}. \end{multline}$

(2.1)

Proof. By using the convexity of $\sigma$ , we have

$\begin{equation} \sigma\left( \nu_{1}+\nu_{2}-\frac{u+v}{2}\right) \leq \frac{1}{2}\left[ \sigma\left( \nu_{1}+\nu_{2}-u\right) +\sigma\left( \nu_{1}+\nu_{2}-v\right) \right], \end{equation}$

(2.2)

and above with $u = \frac{1-\eta}{2}x+\frac{1+\eta}{2}y$ , $v = \frac{1+\eta}{2}x+\frac{1-\eta}{2}y$ , where $x, y\in[\nu_{1}, \nu_{2}]$ and $\eta\in [0, 1]$ , leads to

$\begin{multline} \sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\leq\frac{1}{2} \Biggl[\sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right) \\ +\sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\Biggr]. \end{multline}$

(2.3)

Multiplying both sides of (2.3) by $\eta^{\vartheta -1}$ and then integrating with respect to $\eta$ over $[0, 1]$ , we get

$\begin{align*} \frac{1}{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) &\leq \frac{1}{2}\left[\int_{0}^{1}\eta^{\vartheta-1} \sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)d\eta\right. \\ &+\left. \int_{0}^{1}\eta^{\vartheta -1}\sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)d\eta\right] \\ & = \frac{1}{2}\left[ \frac{2^{\vartheta }}{\left( y-x\right) ^{\vartheta }}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-\frac{x+y}{2}} \left( \left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)-w\right) ^{\vartheta -1}\sigma\left( w\right) dw\right. \\ &\left. +\frac{2^{\vartheta}}{\left(y-x\right)^{\vartheta}}\int_{\nu_{1}+\nu_{2}-\frac{x+y}{2}}^{\nu_{1}+\nu_{2}-x} \left( w-\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right) ^{\vartheta-1}\sigma\left( w\right) dw\right] \\ & = \frac{2^{\vartheta-1}\Gamma \left(\vartheta\right)}{\left(y-x\right)^{\vartheta}} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left(\nu_{1}+\nu_{2}-x\right)-}^{\vartheta}\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right], \end{align*}$

and thus the proof of first inequality in (2.1) is completed.

On the other hand, we have by using the Jensen-Mercer inequality:

$\begin{eqnarray} \sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) &\leq &\sigma\left( \nu_{1}\right) +\sigma\left( \nu_{2}\right) -\left( \frac{1-\eta}{2}\sigma\left( x\right) +\frac{1+\eta}{2}\sigma\left( y\right) \right) \end{eqnarray}$

(2.4)

$\begin{eqnarray} \sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right) \right) &\leq &\sigma\left( \nu_{1}\right) +\sigma\left( \nu_{2}\right) -\left( \frac{1+\eta}{2}\sigma\left( x\right) +\frac{1-\eta}{2}\sigma\left( y\right) \right). \end{eqnarray}$

(2.5)

Adding inequalities (2.4) and (2.5) to get

$\begin{multline} \sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) +\sigma\left(\nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right) \right) \\ \leq 2\left[\sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right)\right]-\left[\sigma\left(x\right)+\sigma\left(y\right)\right]. \end{multline}$

(2.6)

Multiplying both sides of (2.6) by $\eta^{\vartheta -1}$ and then integrating with respect to $\eta$ over $[0, 1]$ to obtain

$\begin{multline*} \int_{0}^{1}\eta^{\vartheta -1}\sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right)d\eta +\int_{0}^{1}\eta^{\vartheta-1}\sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)d\eta \\ \leq \frac{2}{\vartheta}\left[\sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right)\right] -\frac{1}{\vartheta }\left[ \sigma\left( x\right) +\sigma\left( y\right) \right]. \end{multline*}$

By making use of change of variables and then multiplying by $\frac{\vartheta }{2}$ , we get the second inequality in (2.1).

Remark 2.1. If we choose $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ in Theorem 2.1, then the inequality (2.1) reduces to (1.1).

Corollary 2.1. Theorem 2.1 with

● $\vartheta = 1$ becomes [24, Theorem 2.1].

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} \sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)\leq\frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(\nu_{2}-\nu_{1}\right)^{\vartheta}} \left[J_{\nu_{1}+}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)+\; J_{\nu_{2}-}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)\right] \leq \frac{\sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right) }{2}, \end{align*}$

which is obtained by Mohammed and Brevik in ^[10].

The following Lemma linked with the left inequality of (2.1) is useful to obtain our next results.

Lemma 2.1. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R} \to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $\sigma'\in L[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left( y-x\right)^{\vartheta}} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right)-}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right] \\ -\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) = \frac{\left(y-x\right)}{4}\int_{0}^{1}\eta^{\vartheta} \Bigl[\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right) \\ -\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\Bigr]d\eta. \end{multline}$

(2.7)

Proof. From right hand side of (2.7), we set

$\begin{align} \varpi_{1}-\varpi_{2}&:= \int_{0}^{1}\eta^{\vartheta}\left[\sigma'\left(\nu_{1}+\nu_{2} -\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right) -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\right]d\eta \\ & = \int_{0}^{1}\eta^{\vartheta }\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)d\eta \\ &-\int_{0}^{1}\eta^{\vartheta}\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right) d\eta. \end{align}$

(2.8)

By integrating by parts with $w = \nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)$ , we can deduce:

$\begin{align*} \varpi_{1}& = -\frac{2}{\left( y-x\right) }\sigma\left(\nu_{1}+\nu_{2}-y\right)+\frac{2\vartheta }{\left(y-x\right)} \int_{0}^{1}\eta^{\vartheta -1}\sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) d\eta \\ & = -\frac{2}{\left(y-x\right)}\sigma\left(\nu_{1}+\nu_{2}-y\right)+\frac{2^{\vartheta+1}\vartheta}{\left(y-x\right)^{\vartheta+1}} \int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-\frac{x+y}{2}}\sigma\left(\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)-w\right)^{\vartheta -1} \sigma\left(w\right) dw \\ & = -\frac{2}{\left(y-x\right)}\sigma\left( \nu_{1}+\nu_{2}-y\right) +\frac{2^{\vartheta+1}\Gamma\left(\vartheta+1\right)}{\left( y-x\right) ^{\vartheta +1}} J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right). \end{align*}$

Similarly, we can deduce:

$\begin{equation*} \varpi_{2} = \frac{2}{y-x}\sigma\left(\nu_{1}+\nu_{2}-x\right) -\frac{2^{\vartheta+1}\Gamma\left(\vartheta+1\right)}{\left( y-x\right) ^{\vartheta +1}} J_{\left( \nu_{1}+\nu_{2}-x\right)-}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right). \end{equation*}$

By substituting $\varpi_{1}$ and $\varpi_{2}$ in (2.8) and then multiplying by $\frac{\left(y-x\right)}{4}$ , we obtain required identity (2.7).

Corollary 2.2. Lemma 2.1 with

● $\vartheta = 1$ becomes:

$\begin{multline*} \frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma\left(w\right) dw-\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) = \frac{\left( y-x\right) }{4}\int_{0}^{1}\eta\left[ \sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) \right. \\ \left. -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right) \right] d\eta. \end{multline*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) = \frac{\left( \nu_{2}-\nu_{1}\right) }{4}\int_{0}^{1}\eta \left[ \sigma'\left(\nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}\nu_{1}+\frac{1+\eta}{2}\nu_{2}\right) \right) \right. \\ \left. -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}\nu_{1}+\frac{1-\eta}{2}\nu_{2}\right)\right) \right] d\eta. \end{multline*}$

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left( \nu_{2}-\nu_{1}\right)^{\vartheta }} \left[ J_{\nu_{1}+}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) +\; J_{\nu_{2}-}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right]-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \\ = \frac{\left( \nu_{2}-\nu_{1}\right) }{4}\int_{0}^{1}\eta^{\vartheta } \left[\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}\nu_{1}+\frac{1+\eta}{2}\nu_{2}\right) \right) \right. \left. -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}\nu_{1}+\frac{1-\eta}{2}\nu_{2}\right) \right) \right] d\eta. \end{multline*}$

Theorem 2.2. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R}\to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $|\sigma'|$ is convex on $[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \Biggl| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \Biggl[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \Biggr] \\ -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\Biggr| \leq \frac{\left( y-x\right) }{2\left( 1+\vartheta \right) }\left[ \left| \sigma'\left( \nu_{1}\right) \right| +\left| \sigma'\left(\nu_{2}\right) \right| -\frac{\left| \sigma'\left( x\right) \right| +\left| \sigma'\left( y\right) \right| }{2}\right]. \end{multline}$

(2.9)

Proof. By taking modulus of identity (2.7), we get

$\begin{multline*} \Biggl|\frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(y-x\right)^{\vartheta}} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] \\ -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \Biggr|\leq \frac{\left( y-x\right) }{4} \left[\int_{0}^{1}\eta^{\vartheta }\left|\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) \right| d\eta\right. \\ \left. +\int_{0}^{1}\eta^{\vartheta }\left| \sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right) \right) \right| d\eta\right]. \end{multline*}$

Then, by applying the convexity of $|\sigma'|$ and the Jensen-Mercer inequality on above inequality, we get

$\begin{align*} &\Biggl|\frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(y-x\right)^{\vartheta}} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] \\ &-\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \Biggr|\leq \frac{\left( y-x\right) }{4}\left[ \int_{0}^{1}\eta^{\vartheta }\left[ \left| \sigma'\left( \nu_{1}\right) \right| +\left| \sigma'\left( \nu_{2}\right) \right| -\left( \frac{1+\eta}{2}\left| \sigma^{\prime }\left( x\right) \right| +\frac{1-\eta}{2}\right) \left| \sigma'\left( y\right) \right| \right] d\eta\right. \\ &\left. +\int_{0}^{1}\eta^{\vartheta }\left[ \left| \sigma'\left(\nu_{1}\right) \right| +\left| \sigma'\left( \nu_{2}\right) \right| -\left(\frac{1-\eta}{2}\left|\sigma'\left(x\right) \right| +\frac{1+\eta}{2}\right)\left| \sigma'\left( y\right)\right| \right] d\eta\right] \\ & = \frac{\left(y-x\right)}{2\left(1+\vartheta \right)}\left[ \left|\sigma'\left( \nu_{1}\right)\right| +\left|\sigma'(\nu_{2})\right|-\frac{\left|\sigma'\left(x\right)\right|+\left|\sigma'\left(y\right)\right|}{2}\right], \end{align*}$

which completes the proof of Theorem 2.2.

Corollary 2.3. Theorem 2.2 with

● $\vartheta = 1$ becomes:

$\begin{align*} \left|\frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma(w)dw-\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right| \leq \frac{\left(y-x\right)}{4}\left[\left| \sigma'\left(\nu_{1}\right)\right|+\left|\sigma'\left(\nu_{2}\right)\right| -\frac{\left|\sigma'\left(x\right)\right|+\left|\sigma'\left(y\right)\right|}{2}\right]. \end{align*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes [27, Theorem 2.2].

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} \left\vert \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right\vert \leq \frac{\left( \nu_{2}-\nu_{1}\right) }{4}\left[ \frac{\left\vert \sigma^{\prime }\left( \nu_{1}\right) \right\vert +\left\vert \sigma^{\prime }\left( \nu_{2}\right) \right\vert }{2}\right] . \end{align*}$

Theorem 2.3. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R}\to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $\left| \sigma'\right|^{q}, q > 1$ is convex on $[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left( y-x\right) ^{\vartheta }}\left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}- \frac{x+y}{2}\right) \right. \right. \\ \left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{4\sqrt[p]{\vartheta p+1}} \left[ \left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left| \sigma'\left(\nu_{2}\right)\right| ^{q} -\left(\frac{\left|\sigma'(x) \right|^{q}+3\left|\sigma'(y)\right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left|\sigma'\left(\nu_{2}\right)\right|^{q} -\left(\frac{3\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right], \end{multline}$

(2.10)

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

Proof. By taking modulus of identity (2.7) and using Hölder's inequality, we get

$\begin{multline*} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right)^{\vartheta }} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right)+}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ \left. \left. +\; J_{\left(\nu_{1}+\nu_{2}-x\right)-}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{4}\left( \int_{0}^{1}\eta^{\vartheta p}\right)^{\frac{1}{p}} \left\{\left(\int_{0}^{1} \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)\right|^{q}d\eta\right) ^{\frac{1}{q}}\right. \\ \left. +\left(\int_{0}^{1} \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\right|^{q}d\eta\right) ^{\frac{1}{q}}\right\}. \end{multline*}$

Then, by applying the Jensen-Mercer inequality with the convexity of $\left|\sigma'\right|^{q}$ , we can deduce

$\begin{align*} &\left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ &\left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ &\leq \frac{\left( y-x\right) }{4}\left( \int_{0}^{1}\eta^{\vartheta p}\right)^{\frac{1}{p}} \left\{ \left( \int_{0}^{1}\left| \sigma'\left( \nu_{1}\right)\right| ^{q}+\left| \sigma'\left( \nu_{2}\right) \right|^{q} -\left(\frac{1-\eta}{2}\left|\sigma'(x)\right|^{q}+\frac{1+\eta}{2}\left|\sigma'(y)\right|^{q}\right)\right)^{\frac{1}{q}}\right. \\ &\left. +\left( \int_{0}^{1}\left| \sigma'\left( \nu_{1}\right)\right| ^{q}+\left| \sigma'\left( \nu_{2}\right) \right|^{q} -\left(\frac{1+\eta}{2}\left| \sigma'\left( x\right)\right|^{q}+\frac{1-\eta}{2}\left| \sigma'\left( y\right) \right|^{q} \right) \right)^{\frac{1}{q}}\right\} \\ & = \frac{\left(y-x\right)}{4\sqrt[p]{\vartheta p+1}}\left[\left( \left| \sigma'\left( \nu_{1}\right) \right| ^{q} +\left|\sigma'\left( \nu_{2}\right) \right| ^{q}-\left( \frac{\left|\sigma'\left( x\right) \right|^{q} +3\left| \sigma'\left(y\right) \right|^{q} }{4}\right) \right) ^{\frac{1}{q}}\right. \\ &\left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left| \sigma'\left( \nu_{2}\right) \right| ^{q} -\left(\frac{3\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right], \end{align*}$

which completes the proof of Theorem 2.3.

Corollary 2.4. Theorem 2.3 with

● $\vartheta = 1$ becomes:

$\begin{multline*} \left|\frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma\left(w\right)dw -\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right| \\ \leq \frac{\left( y-x\right)}{4\sqrt[p]{p+1}}\left[\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q} +\left|\sigma'\left( \nu_{2}\right) \right| ^{q}-\left( \frac{\left|\sigma'\left( x\right) \right|^{q} +3\left|\sigma'\left(y\right) \right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left(\left|\sigma'\left( \nu_{1}\right)\right|^{q}+\left| \sigma'\left( \nu_{2}\right)\right|^{q} -\left(\frac{3\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q} }{4}\right)\right)^{\frac{1}{q}}\right]. \end{multline*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} &&\left\vert \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right\vert \leq \frac{\left( \nu_{2}-\nu_{1}\right) }{2^{\frac{2}{p}}}\left(\frac{1}{p+1}\right)^{\frac{1}{p}}\left[ \left\vert \sigma^{\prime }\left( \nu_{1}\right) \right\vert +\left\vert \sigma^{\prime }\left( \nu_{2}\right) \right\vert \right] . \end{align*}$

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} &\left\vert \frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(\nu_{2}-\nu_{1}\right)^{\vartheta}} \left[ J_{\nu_{1}+}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right. \right. \left. \left. +\; J_{\nu_{2}-}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right] -\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right\vert \\ &\leq \frac{2^{\vartheta-1-\frac{2}{q}}}{\nu_{2}-\nu_{1}}\left(\frac{1}{p+1}\right)^{\frac{1}{p}}\left[ \left\vert \sigma'\left( \nu_{1}\right) \right\vert +\left\vert \sigma'\left( \nu_{2}\right) \right\vert \right]. \end{align*}$

Theorem 2.4. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R}\to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $\left|\sigma'\right|^{q}, q\geq 1$ is convex on $[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right.\right. \\ \left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left(y-x\right)}{4\left(\vartheta+1\right)}\left[\left(\left|\sigma'(\nu_{1})\right|^{q} +\left|\sigma'\left(\nu_{2}\right)\right|^{q}-\left( \frac{\left| \sigma'\left(x\right)\right|^{q} +\left( 2\vartheta+3\right)\left|\sigma'(y)\right|^{q}}{2\left(\vartheta+2\right) }\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left( \left|\sigma'\left(\nu_{1}\right)\right|^{q}+\left| \sigma'\left( \nu_{2}\right)\right|^{q} -\left(\frac{\left(2\vartheta+3\right)\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{2\left( \vartheta+2\right) }\right) \right) ^{\frac{1}{q}}\right]. \end{multline}$

(2.11)

Proof. By taking modulus of identity (2.7) with the well-known power mean inequality, we can deduce

$\begin{multline*} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ \left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{4}\left( \int_{0}^{1}\eta^{\vartheta }\right) ^{1-\frac{1}{q}} \left\{\left(\int_{0}^{1}\eta^{\vartheta} \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)\right|^{q}d\eta\right)^{\frac{1}{q}}\right. \\ \left. +\left( \int_{0}^{1}\eta^{\vartheta } \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\right|^{q}d\eta\right)^{\frac{1}{q}}\right\} . \end{multline*}$

By applying the Jensen-Mercer inequality with the convexity of $\left|\sigma'\right|^{q}$ , we can deduce

$\begin{align*} &\left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right)^{\vartheta}} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ &\left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ &\leq \frac{\left( y-x\right)}{4}\left(\int_{0}^{1}\eta^{\vartheta }\right)^{1-\frac{1}{q}} \left\{ \left( \int_{0}^{1}\eta^{\vartheta }\left[ \left|\sigma'\left( \nu_{1}\right) \right|^{q} +\left| \sigma'\left(\nu_{2}\right) \right| ^{q}-\left( \frac{1-\eta}{2}\left| \sigma'\left(x\right)\right|^{q} +\frac{1+\eta}{2}\left| \sigma'\left( y\right)\right| ^{q}\right) \right] \right) ^{\frac{1}{q}}\right. \\ &\left. +\left( \int_{0}^{1}\eta^{\vartheta }\left[ \left| \sigma'\left(\nu_{1}\right)\right|^{q} +\left| \sigma'\left( \nu_{2}\right) \right|^{q}-\left( \frac{1+\eta}{2}\left| \sigma'\left( x\right) \right|^{q} +\frac{1-\eta}{2}\left| \sigma'\left( y\right) \right|^{q}\right) \right] \right) ^{\frac{1}{q}}\right\} \\ & = \frac{\left( y-x\right) }{4\left( \vartheta +1\right) }\left[\left(\left|\sigma'\left(\nu_{1}\right)\right|^{q} +\left|\sigma'\left( \nu_{2}\right) \right| ^{q}-\left( \frac{\left| \sigma'\left(x\right) \right|^{q} +\left(2\vartheta +3\right)\left|\sigma'(y)\right|^{q}}{2\left(\vartheta+2\right)}\right)\right)^{\frac{1}{q}}\right. \\ &\left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left|\sigma'\left( \nu_{2}\right)\right|^{q} -\left( \frac{\left( 2\vartheta +3\right) \left| \sigma'\left( x\right) \right|^{q} +\left|\sigma'\left( y\right)\right|^{q}}{2\left(\vartheta+2\right)}\right)\right)^{\frac{1}{q}}\right], \end{align*}$

which completes the proof of Theorem 2.4.

Corollary 5. Theorem 2.4 with

● $q = 1$ becomes Theorem 2.2.

● $\vartheta = 1$ becomes:

$\begin{multline*} \left| \frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma\left(w\right) dw -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{8}\left[ \left( \left|\sigma'\left( \nu_{1}\right)\right|^{q} +\left| \sigma'\left( \nu_{2}\right)\right|^{q}-\left( \frac{\left| \sigma'\left( x\right)\right|^{q} +5\left|\sigma'\left(y\right)\right|^{q}}{6}\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left|\sigma'\left(\nu_{2}\right)\right|^{q} -\left(\frac{5\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{6}\right)\right)^{\frac{1}{q}}\right]. \end{multline*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \left| \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw -\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)\right| \\ \leq \frac{\left(y-x\right)}{8}\left[\left(\frac{5\left|\sigma'\left(\nu_{1}\right)\right|^{q} +\left| \sigma'\left( \nu_{2}\right)\right| ^{q}}{6}\right)^{\frac{1}{q}} +\left(\frac{\left|\sigma'(\nu_{1})\right|^{q}+5\left| \sigma'(\nu_{2})\right|^{q}}{6}\right)^{\frac{1}{q}}\right]. \end{multline*}$

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \left| \frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(\nu_{2}-\nu_{1}\right)^{\vartheta}} \left[J_{\nu_{1} +}^{\vartheta}\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)+\; J_{\nu_{2} -}^{\vartheta }\sigma\left( \frac{\nu_{1}+\nu_{2}}{2}\right)\right] -\sigma\left( \frac{\nu_{1}+\nu_{2}}{2}\right)\right| \\ \leq \frac{\left(\nu_{2}-\nu_{1}\right)}{4\left(\vartheta+1\right)} \left[\left(\frac{\left(2\vartheta+3\right)\left|\sigma'(\nu_{1})\right|^{q}+\left|\sigma'(\nu_{2})\right|^{q}}{2\left(\vartheta+2\right)}\right)^{\frac{1}{q}} +\left(\frac{\left|\sigma'(\nu_{1})\right|^{q}+\left(2\vartheta+3\right)\left|\sigma'(\nu_{2})\right|^{q}}{2\left(\vartheta+2\right)}\right) ^{\frac{1}{q}}\right]. \end{multline*}$

3. Applications to special means

Here, we consider the following special means:

● The arithmetic mean:

$\mathcal{A}(\nu_{1},\nu_{2}) = \frac{\nu_{1}+\nu_{2}}{2}, \quad \nu_{1}, \nu_{2}\geq 0.$

● The harmonic mean:

$\mathcal{H}(\nu_{1},\nu_{2}) = \frac{2\nu_{1}\nu_{2}}{\nu_{1}+\nu_{2}}, \quad \nu_{1}, \nu_{2} > 0.$

● The logarithmic mean:

$\begin{equation*} \mathcal{L}(\nu_{1},\nu_{2}) = \begin{cases} \frac{\nu_{2}-\nu_{1}}{\ln \nu_{2}-\ln \nu_{1}}, & \text{if}\; \nu_{1}\neq \nu_{2}, \\ \nu_{1}, & \text{if}\; \nu_{1} = \nu_{2}, \end{cases} \quad \nu_{1}, \nu_{2} > 0. \end{equation*}$

● The generalized logarithmic mean:

$\begin{equation*} \mathcal{L}_{n}(\nu_{1},\nu_{2}) = \begin{cases} \left[\frac{\nu_{2}^{n+1}-\nu_{1}^{n+1}}{\left(n+1\right)(\nu_{2}-\nu_{1})}\right]^{\frac{1}{n}}, & \text{if}\; \nu_{1}\neq \nu_{2} \\ \nu_{1}, & \text{if}\; \nu_{1} = \nu_{2}, \end{cases} \quad \nu_{1},\nu_{2} > 0;\; n\in\bf{Z}\setminus \{-1,0\}. \end{equation*}$

Proposition 3.1. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, for all $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

(3.1)

Proof. By applying Corollary 2.3 (first item) for the convex function $\sigma(x) = x^{n}, \, x > 0$ , one can obtain the result directly.

Proposition 3.2. Let $0 < \nu_{1} < \nu_{2}$ . Then, for all $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

(3.2)

Proof. By applying Corollary 2.3 (first item) for the convex function $\sigma(x) = \frac{1}{x}, \, x > 0$ , one can obtain the result directly.

Proposition 3.3. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, we have:

$\begin{equation} \left| \mathcal{L}_{n}^{n}(\nu_{1},\nu_{2}) -\mathcal{A}^{n}(\nu_{1},\nu_{2}) \right| \leq \frac{n(\nu_{2}-\nu_{1})}{4} \left[ \mathcal{A}\left(\nu_{1}^{n-1}, \nu_{2}^{n-1}\right) \right], \end{equation}$

(3.3)

and

$\begin{equation} \left| \mathcal{L}^{-1}(\nu_{1},\nu_{2}) -\mathcal{A}^{-1}(\nu_{1},\nu_{2}) \right| \leq \frac{(\nu_{2}-\nu_{1})}{4}\mathcal{H}^{-1}\left( \nu_{1}^{2},\nu_{2}^{2}\right) . \end{equation}$

(3.4)

Proof. By setting $x = \nu_{1}$ and $y = \nu_{2}$ in results of Proposition 3.1 and Proposition 3.2, one can obtain the Proposition 3.3.

Proposition 3.4. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, for $q > 1, \, \frac{1}{p}+\frac{1}{q} = 1$ and for all $x, y\in[\nu_{1}, \nu_{2}],$ we have:

$\begin{multline} \left| \mathcal{L}_{n}^{n}\left( \nu_{1}+\nu_{2}-y,\nu_{1}+\nu_{2}-x\right) -\left( 2\mathcal{A}(\nu_{1},\nu_{2}) -\mathcal{A}\left( x,y\right) \right) ^{n}\right| \\ \leq \frac{n(y-x)}{4\sqrt[p]{p+1}}\Bigg\{\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left(x^{q(n-1)}, 3y^{q(n-1)}\right) \right]^{\frac{1}{q}} \\ +\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left(3x^{q(n-1)}, y^{q(n-1)}\right) \right]^{\frac{1}{q}}\Bigg\}. \end{multline}$

(3.5)

Proof. By applying Corollary 2.4 (first item) for convex function $\sigma\left(x\right) = x^{n}, \, x > 0$ , one can obtain the result directly.

Proposition 3.5. Let $0 < \nu_{1} < \nu_{2}$ . Then, for $q > 1, \, \frac{1}{p}+\frac{1}{q} = 1$ and for all $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

$\begin{multline} \left| \mathcal{L}^{-1}\left( \nu_{1}+\nu_{2}-y,\nu_{1}+\nu_{2}-x\right) -\left( 2\mathcal{A}(\nu_{1},\nu_{2}) -\mathcal{A}\left(x,y\right)\right)^{-1}\right| \\ \leq \frac{\sqrt[q]{2}(y-x)}{4\sqrt[p]{p+1}}\Bigg\{\left[ \mathcal{H}^{-1}\left(\nu_{1}^{2q},\nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(x^{2q}, 3y^{2q}\right) \right]^{\frac{1}{q}} \\ +\left[\mathcal{H}^{-1}\left(\nu_{1}^{2q},\nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(3x^{2q}, y^{2q}\right) \right]^{\frac{1}{q}}\Bigg\}. \end{multline}$

(3.6)

Proof. By applying Corollary 2.4 (first item) for the convex function $\sigma\left(x\right) = \frac{1}{x}, \, x > 0$ , one can obtain the result directly.

Proposition 3.6. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, for $q > 1$ and $\frac{1}{p}+\frac{1}{q} = 1$ , we have:

$\begin{multline} \left| \mathcal{L}_{n}^{n}(\nu_{1},\nu_{2}) -\mathcal{A}^{n}(\nu_{1},\nu_{2}) \right| \leq \frac{n(\nu_{2}-\nu_{1})}{4\sqrt[p]{p+1}} \Bigg\{\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left( \nu_{1}^{q(n-1)}, 3\nu_{2}^{q(n-1)}\right) \right]^{\frac{1}{q}} \\ +\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left(3\nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) \right]^{\frac{1}{q}}\Bigg\}, \end{multline}$

(3.7)

and

$\begin{multline} \left| \mathcal{L}^{-1}(\nu_{1},\nu_{2}) -\mathcal{A}^{-1}(\nu_{1},\nu_{2}) \right| \leq \frac{\sqrt[q]{2}(\nu_{2}-\nu_{1})}{4\sqrt[p]{p+1}} \Bigg\{\left[ \mathcal{H}^{-1}\left( \nu_{1}^{2q}, \nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(\nu_{1}^{2q}, 3\nu_{2}^{2q}\right) \right]^{\frac{1}{q}} \\ +\left[ \mathcal{H}^{-1}\left( \nu_{1}^{2q}, \nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(3\nu_{1}^{2q}, \nu_{2}^{2q}\right) \right]^{\frac{1}{q}}\Bigg\}. \end{multline}$

(3.8)

Proof. By setting $x = \nu_{1}$ and $y = \nu_{2}$ in results of Proposition 3.4 and Proposition 3.5, one can obtain the Proposition 3.6.

4. Conclusions

As we emphasized in the introduction, integral inequality is the most important field of mathematical analysis and fractional calculus. By using the well-known Jensen-Mercer and power mean inequalities, we have proved new inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional operators. In the last section, we have considered some propositions in the context of special functions; these confirm the efficiency of our results.

Acknowledgements

We would like to express our special thanks to the editor and referees. Also, the first author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	A. W. Bergen, N. Caporaso, Cigarette smoking, J. Natl. Cancer Inst., 91 (1999), 1365–1376. https://doi.org/10.1093/jnci/91.16.1365 doi: 10.1093/jnci/91.16.1365
[2]	N. J. Wald, A. K. Hackshaw, Cigarette smoking: an epidemiological overview, Brit. Med. Bull., 52 (1996), 3–11. https://doi.org/10.1093/oxfordjournals.bmb.a011530 doi: 10.1093/oxfordjournals.bmb.a011530
[3]	B. Lloyd, K. Lucas, Smoking in adolescence: images and identities, London: Routledge, 1998.
[4]	S. Cohen, E. Lichtenstein, Perceived stress, quitting smoking, and smoking relapse, Health Psychol., 9 (1990), 466–478. https://doi.org/10.1037//0278-6133.9.4.466 doi: 10.1037//0278-6133.9.4.466
[5]	A. H. Mokdad, J. S. Marks, D. F. Stroup, J. L. Gerberding, Actual causes of death in the United States, JAMA, 291 (2004), 1238–1245. https://doi.org/10.1001/jama.291.10.1238 doi: 10.1001/jama.291.10.1238
[6]	A. Zeb, G. Zaman, S. Momani, Square-root dynamics of a giving up smoking model, Appl. Math. Model., 37 (2013), 5326–5334. https://doi.org/10.1016/j.apm.2012.10.005 doi: 10.1016/j.apm.2012.10.005
[7]	O. Sharomi, A. B. Gumel, Curtailing smoking dynamics: A mathematical modeling approach, Appl. Math. Comput., 195 (2008), 475–499. https://doi.org/10.1016/j.amc.2007.05.012 doi: 10.1016/j.amc.2007.05.012
[8]	Z. Alkhudhari, S. Al-Sheikh, S. Al-Tuwairqi, Stability analysis of a giving up smoking model, Int. J. Appl. Math. Res., 3 (2014), 168–177. http://doi.org/10.14419/ijamr.v3i2.2239 doi: 10.14419/ijamr.v3i2.2239
[9]	N. H. Shah, F. A. Thakkar, B. M. Yeolekar, Stability analysis of tuberculosis due to smoking, Int. J. Innov. Sci. Res. Tech., 3 (2018), 230–237.
[10]	Q. Din, M. Ozair, T. Hussain, U. Saeed, Qualitative behavior of asmoking model, Adv. Differ. Equ., 2016 (2016), 96. https://doi.org/10.1186/s13662-016-0830-6 doi: 10.1186/s13662-016-0830-6
[11]	A. M. Pulecio-Montoya, L. E. Lopez-Montenegro, L. M. Benavides, Analysis of a mathematical model of smoking, Contemp. Eng. Sci., 12 (2019), 117–129. https://doi.org/10.12988/ces.2019.9517 doi: 10.12988/ces.2019.9517
[12]	Z. Zhang, R. Wei, W. Xia, Dynamical analysis of a giving up smoking model with time delay, Adv. Differ. Equ., 2019 (2019), 505. https://doi.org/10.1186/s13662-019-2450-4 doi: 10.1186/s13662-019-2450-4
[13]	S. A. Khan, K. Shah, G. Zaman, F. Jarad, Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative, Chaos, 29 (2019), 013128. https://doi.org/10.1063/1.5079644 doi: 10.1063/1.5079644
[14]	S. Ucar, E. Ucar, N. ozdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Soliton. Fract., 118 (2019), 300–306. https://doi.org/10.1016/j.chaos.2018.12.003 doi: 10.1016/j.chaos.2018.12.003
[15]	G. Rahman, R. P. Agarwal, Q. Din, Mathematical analysis of giving up smoking model via harmonic mean type incidence rate, Appl. Math. Comput., 354 (2019), 128–148. https://doi.org/10.1016/j.amc.2019.01.053 doi: 10.1016/j.amc.2019.01.053
[16]	C. Sun, J. Jia, Optimal control of a delayed smoking model with immigration, J. Biol. Dynam., 13 (2019), 447–460. https://doi.org/10.1080/17513758.2019.1629031 doi: 10.1080/17513758.2019.1629031
[17]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method, Math. Sci., 13 (2019), 115–128. https://doi.org/10.1007/s40096-019-0284-6 doi: 10.1007/s40096-019-0284-6
[18]	A. M. S. Mahdy, N. H. Sweilam, M. Higazy, Approximate solution for solving nonlinear fractional order smoking model, Alex. Eng. J., 59 (2020), 739–752. https://doi.org/10.1016/j.aej.2020.01.049 doi: 10.1016/j.aej.2020.01.049
[19]	A. A. Alshareef, H. A. Batarfi, Stability analysis of chain, mild and passive smoking model, Amer. J. Comput. Math., 10 (2020), 31–42. https://doi.org/10.4236/ajcm.2020.101003 doi: 10.4236/ajcm.2020.101003
[20]	Z. Zhang, J. Zou, R. K. Upadhyay, A. Pratap, Stability and Hopf bifurcation analysis of a delayed tobacco smoking model containing snuffing class, Adv. Differ. Equ., 2020 (2020), 349. https://doi.org/10.1186/s13662-020-02808-5 doi: 10.1186/s13662-020-02808-5
[21]	A. Bernoussi, Global stability analysis of an SEIR epidemic model with relapse and general incidence rates, Appl. Sci., 21 (2019), 54–68.
[22]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[23]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[24]	D. Baleanu, A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 444–462. https://doi.org/10.1016/j.cnsns.2017.12.003 doi: 10.1016/j.cnsns.2017.12.003
[25]	T. Abdeljawad, Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals, Chaos, 29 (2019), 023102. https://doi.org/10.1063/1.5085726 doi: 10.1063/1.5085726
[26]	T. Abdeljawad, D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Differ. Equ., 2018 (2018), 468. https://doi.org/10.1186/s13662-018-1914-2 doi: 10.1186/s13662-018-1914-2
[27]	H. Khan, F. Jarad, T. Abdeljawad, A. Khan, A singular ABC-fractional differential equation with p-Laplacian operator, Chaos Soliton. Fract., 129 (2019), 56–61. https://doi.org/10.1016/j.chaos.2019.08.017 doi: 10.1016/j.chaos.2019.08.017
[28]	S. Rezapour, S. Etemad, H. Mohammadi, A mathematical analysis of a system of Caputo-Fabrizio fractional differential equations for the anthrax disease model in animals, Adv. Differ. Equ., 2020 (2020), 481. https://doi.org/10.1186/s13662-020-02937-x doi: 10.1186/s13662-020-02937-x
[29]	H. M. Alshehri, A. Khan, A fractional order Hepatitis C mathematical model with Mittag-Leffler kernel, J. Funct. Space., 2021 (2021), 2524027. https://doi.org/10.1155/2021/2524027 doi: 10.1155/2021/2524027
[30]	C. T. Deressa, S. Etemad, S. Rezapour, On a new four-dimensional model of memristor-based chaotic circuit in the context of nonsingular Atangana-Baleanu-Caputo operators, Adv. Differ. Equ., 2021 (2021), 444. https://doi.org/10.1186/s13662-021-03600-9 doi: 10.1186/s13662-021-03600-9
[31]	C. T. Deressa, S. Etemad, M. K. A. Kaabar, S. Rezapour, Qualitative analysis of a hyperchaotic Lorenz-Stenflo mathematical model via the Caputo fractional operator, J. Funct. Space., 2022 (2022), 4975104. https://doi.org/10.1155/2022/4975104 doi: 10.1155/2022/4975104
[32]	P. Kumar, V. S. Erturk, Environmental persistence influences infection dynamics for a butterfly pathogen via new generalised Caputo type fractional derivative, Chaos Soliton. Fract., 144 (2021), 110672. https://doi.org/10.1016/j.chaos.2021.110672 doi: 10.1016/j.chaos.2021.110672
[33]	A. Devi, A. Kumar, T. Abdeljawad, A. Khan, Stability analysis of solutions and existence theory of fractional Lagevin equation, Alex. Eng. J., 60 (2021), 3641–3647. https://doi.org/10.1016/j.aej.2021.02.011 doi: 10.1016/j.aej.2021.02.011
[34]	A. Pratap, R. Raja, R. P. Agarwal, J. Alzabut, M. Niezabitowski, E. Hincal, Further results on asymptotic and finite-time stability analysis of fractional-order time-delayed genetic regulatory networks, Neurocomputing, 475 (2022), 26–37. https://doi.org/10.1016/j.neucom.2021.11.088 doi: 10.1016/j.neucom.2021.11.088
[35]	H. Mohammadi, S. Kumar, S. Rezapour, S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Soliton. Fract., 144 (2021), 110668. https://doi.org/10.1016/j.chaos.2021.110668 doi: 10.1016/j.chaos.2021.110668
[36]	R. Begum, O. Tunc, H. Khan, H. Gulzar, A. Khan, A fractional order Zika virus model with Mittag-Leffler kernel, Chaos Soliton. Fract., 146 (2021), 110898. https://doi.org/10.1016/j.chaos.2021.110898 doi: 10.1016/j.chaos.2021.110898
[37]	A. Ali, Q. Iqbal, J. K. K. Asamoah, S. Islam, Mathematical modeling for the transmission potential of Zika virus with optimal control strategies, Eur. Phys. J. Plus, 137 (2022), 146. https://doi.org/10.1140/epjp/s13360-022-02368-5 doi: 10.1140/epjp/s13360-022-02368-5
[38]	P. Kumar, V. S. Erturk, H. Almusawa, Mathematical structure of mosaic disease using microbial biostimulants via Caputo and Atangana-Baleanu derivatives, Results Phys., 24 (2021), 104186. https://doi.org/10.1016/j.rinp.2021.104186 doi: 10.1016/j.rinp.2021.104186
[39]	R. Zarin, H. Khaliq, A. Khan, D. Khan, A. Akgul, U. W. Humphries, Deterministic and fractional modeling of a computer virus propagation, Results Phys., 33 (2022), 105130. https://doi.org/10.1016/j.rinp.2021.105130 doi: 10.1016/j.rinp.2021.105130
[40]	D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Bound. Value Probl., 2020 (2020), 64. https://doi.org/10.1186/s13661-020-01361-0 doi: 10.1186/s13661-020-01361-0
[41]	C. Thaiprayoon, W. Sudsutad, J. Alzabut, S. Etemad, S. Rezapour, On the qualitative analysis of the fractional boundary value problem describing thermostat control model via $\psi$ -Hilfer fractional operator, Adv. Differ. Equ., 2021 (2021), 201. https://doi.org/10.1186/s13662-021-03359-z doi: 10.1186/s13662-021-03359-z
[42]	J. Alzabut, G. M. Selvam, R. A. El-Nabulsi, D. Vignesh, M. E. Samei, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions, Symmetry, 13 (2021), 473. https://doi.org/10.3390/sym13030473 doi: 10.3390/sym13030473
[43]	S. T. M. Thabet, S. Etemad, S. Rezapour, On a new structure of the pantograph inclusion problem in the Caputo conformable setting, Bound. Value Probl., 2020 (2020), 171. https://doi.org/10.1186/s13661-020-01468-4 doi: 10.1186/s13661-020-01468-4
[44]	P. Kumar, V. S. Erturk, A. Yusuf, K. S. Nisar, S. F. Abdelwahab, A study on canine distemper virus (CDV) and rabies epidemics in the red fox population via fractional derivatives, Results Phys., 25 (2021), 104281. https://doi.org/10.1016/j.rinp.2021.104281 doi: 10.1016/j.rinp.2021.104281
[45]	J. K. K. Asamoah, E. Okyere, E. Yankson, A. A. Opoku, A. Adom-Konadu, E. Acheampong, et al., Non-fractional and fractional mathematical analysis and simulations for Q fever, Chaos Soliton. Fract., 156 (2022), 111821. https://doi.org/10.1016/j.chaos.2022.111821 doi: 10.1016/j.chaos.2022.111821
[46]	H. Khan, C. Tunc, W. Chen, A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with p-Laplacial operator, J. Appl. Anal. Comput., 8 (2018), 1211–1226. https://doi.org/10.11948/2018.1211 doi: 10.11948/2018.1211
[47]	A. Omame, U. K. Nwajeri, M. Abbas, C. P. Onyenegecha, A fractional order control model for Diabetes and COVID-19 co-dynamics with Mittag-Leffler function, Alex. Eng. J., 61 (2022), 7619–7635. https://doi.org/10.1016/j.aej.2022.01.012 doi: 10.1016/j.aej.2022.01.012
[48]	D. Baleanu, S. Etemad, H. Mohammadi, S. Rezapour, A novel modeling of boundary value problems on the glucose graph, Commun. Nonlinear Sci. Numer. Simul., 100 (2021), 105844. https://doi.org/10.1016/j.cnsns.2021.105844 doi: 10.1016/j.cnsns.2021.105844
[49]	S. Rezapour, B. Tellab, C. T. Deressa, S. Etemad, K. Nonlaopon, H-U-type stability and numerical solutions for a nonlinear model of the coupled systems of Navier BVPs via the generalized differential transform method, Fractal Fract., 5 (2021), 166. https://doi.org/10.3390/fractalfract5040166 doi: 10.3390/fractalfract5040166
[50]	E. Ucar, N. Özdemir, E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), 308. https://doi.org/10.1051/mmnp/2019002 doi: 10.1051/mmnp/2019002
[51]	E. Ucar, S. Ucar, F. Evirgen, N. Özdemir, A fractional SAIDR model in the frame of Atangana-Baleanu derivative, Fractal Fract., 50 (2021), 32. https://doi.org/10.3390/fractalfract5020032 doi: 10.3390/fractalfract5020032
[52]	S. Ucar, Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives, Discrete Cont. Dyn. Syst. S, 14 (2021), 2571–2589. https://doi.org/10.3934/dcdss.2020178 doi: 10.3934/dcdss.2020178
[53]	S. Ucar, E. Ucar, N. Özdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Soliton. Fract., 118 (2019), 300–306. https://doi.org/10.1016/j.chaos.2018.12.003 doi: 10.1016/j.chaos.2018.12.003
[54]	A. Khan, H. M. Alshehri, J. F. Gómez-Aguilar, Z. A. Khan, G. Fernández-Anaya, A predator-prey model involving variable-order fractional differential equations with Mittag-Leffler kernel, Adv. Differ. Equ., 2021 (2021), 183. https://doi.org/10.1186/s13662-021-03340-w doi: 10.1186/s13662-021-03340-w
[55]	H. M. Alshehri, A. Khan, A fractional order Hepatitis C mathematical model with Mittag-Leffler kernel, J. Funct. Space., 2021 (2021), 2524027. https://doi.org/10.1155/2021/2524027 doi: 10.1155/2021/2524027
[56]	P. Bedi, A. Kumar, A. Khan, Controllability of neutral impulsive fractional differential equations with Atangana-Baleanu-Caputo derivatives, Chaos Soliton. Fract., 150 (2021), 111153. https://doi.org/10.1016/j.chaos.2021.111153 doi: 10.1016/j.chaos.2021.111153
[57]	W. Chen, Time-space fabric underlying anomalous diffusion, Chaos Soliton. Fract., 28 (2006), 923–929. https://doi.org/10.1016/j.chaos.2005.08.199 doi: 10.1016/j.chaos.2005.08.199
[58]	R. Kanno, Representation of random walk in fractal space-time, Physica A, 248 (1998), 165–175. https://doi.org/10.1016/S0378-4371(97)00422-6 doi: 10.1016/S0378-4371(97)00422-6
[59]	W. Chen, H. G. Sun, X. Zhang, D. Korosak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754–1758. https://doi.org/10.1016/j.camwa.2009.08.020 doi: 10.1016/j.camwa.2009.08.020
[60]	K. M. Owolabi, A. Atangana, A. Akgul, Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model, Alex. Eng. J., 59 (2020), 2477–2490. https://doi.org/10.1016/j.aej.2020.03.022 doi: 10.1016/j.aej.2020.03.022
[61]	Z. Ali, F. Rabiei, K. Shah, T. Khodadadi, Modeling and analysis of novel COVID-19 under fractal-fractional derivative with case study of malaysia, Fractals, 29 (2021), 2150020. https://doi.org/10.1142/S0218348X21500201 doi: 10.1142/S0218348X21500201
[62]	E. Bonyah, M. Yavuz, D. Baleanu, S. Kumar, A robust study on the listeriosis disease by adopting fractal-fractional operators, Alex. Eng. J., 61 (2022), 2016–2028. https://doi.org/10.1016/j.aej.2021.07.010 doi: 10.1016/j.aej.2021.07.010
[63]	M. Alqhtani, K. M. Saad, Numerical solutions of space-fractional diffusion equations via the exponential decay kernel, AIMS Mathematics, 7 (2022), 6535–6549. https://doi.org/10.3934/math.2022364 doi: 10.3934/math.2022364
[64]	M. Alqhtani, K. M. Saad, Fractal-fractional Michaelis–Menten enzymatic reaction model via different kernels, Fractal Fract., 6 (2021), 13. https://doi.org/10.3390/fractalfract6010013 doi: 10.3390/fractalfract6010013
[65]	K. M. Saad, J. F. Gomez-Aguilar, A. A. Almadiy, A fractional numerical study on a chronic hepatitis C virus infection model with immune response, Chaos Soliton. Fract., 139 (2020), 110062. https://doi.org/10.1016/j.chaos.2020.110062 doi: 10.1016/j.chaos.2020.110062
[66]	K. M. Saad, M. Alqhtani, Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear, AIMS Mathematics, 6 (2021), 3788–3804. https://doi.org/10.3934/math.2021225 doi: 10.3934/math.2021225
[67]	A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
[68]	M. Arfan, K. Shah, A. Ullah, Fractal-fractional mathematical model of four species comprising of prey-predation, Phys. Scr., 96 (2021), 124053. https://doi.org/10.1088/1402-4896/ac2f37 doi: 10.1088/1402-4896/ac2f37
[69]	M. Abdulwasaa, M. S. Abdo, K. Shah, T. A. Nofal, S. K. Panchal, S. V. Kawale, et al., Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India, Results Phys., 20 (2021), 103702. https://doi.org/10.1016/j.rinp.2020.103702 doi: 10.1016/j.rinp.2020.103702
[70]	K. Shah, M. Arfan, I. Mahariq, A. Ahmadian, S. Salahshour, M. Ferrara, Fractal-fractional mathematical model addressing the situation of Corona virus in Pakistan, Results Phys., 19 (2020), 103560. https://doi.org/10.1016/j.rinp.2020.103560 doi: 10.1016/j.rinp.2020.103560
[71]	Z. A. Khan, M. ur Rahman, K. Shah, Study of a fractal-fractional smoking models with relapse and harmonic mean type incidence rate, J. Funct. Space., 2021 (2021) 6344079. https://doi.org/10.1155/2021/6344079 doi: 10.1155/2021/6344079
[72]	M. Arif, P. Kumam, W. Kumam, A. Akgul, T. Sutthibutpong, Analysis of newly developed fractal-fractional derivative with power law kernel for MHD couple stress fluid in channel embedded in a porous medium, Sci. Rep., 11 (2021), 20858. https://doi.org/10.1038/s41598-021-00163-3 doi: 10.1038/s41598-021-00163-3
[73]	H. Najafi, S. Etemad, N. Patanarapeelert, J. K. K. Asamoah, S. Rezapour, T. Sitthiwirattham, A study on dynamics of CD4 $^+$ T-cells under the effect of HIV-1 infection based on a mathematical fractal-fractional model via the Adams-Bashforth scheme and Newton polynomials. Mathematics, 10 (2022), 1366. https://doi.org/10.3390/math10091366 doi: 10.3390/math10091366
[74]	H. Khan, K. Alam, H. Gulzar, S. Etemad, S. Rezapour, A case study of fractal-fractional tuberculosis model in China: Existence and stability theories along with numerical simulations, Math. Comput. Simul., 198 (2022), 455–473. https://doi.org/10.1016/j.matcom.2022.03.009 doi: 10.1016/j.matcom.2022.03.009
[75]	A. Atangana, A. Akgul, K. M. Owolabi, Analysis of fractal fractional differential equations, Alex. Eng. J., 59 (2020), 1117–1134. https://doi.org/10.1016/j.aej.2020.01.005 doi: 10.1016/j.aej.2020.01.005
[76]	A. U. Awan, A. Sharif, K. A. Abro, M. Ozair, T. Hussain, Dynamical aspects of smoking model with cravings to smoke, Nonlinear Eng., 10 (2021), 91–108. http://doi.org/10.1515/nleng-2021-0008 doi: 10.1515/nleng-2021-0008

This article has been cited by:

1.	Tariq A. Aljaaidi, Deepak B. Pachpatte, Ram N. Mohapatra, The Hermite–Hadamard–Mercer Type Inequalities via Generalized Proportional Fractional Integral Concerning Another Function, 2022, 2022, 1687-0425, 1, 10.1155/2022/6716830
2.	Saad Ihsan Butt, Ahmet Ocak Akdemir, Muhammad Nadeem, Nabil Mlaiki, İşcan İmdat, Thabet Abdeljawad, $(m, n)$ -Harmonically polynomial convex functions and some Hadamard type inequalities on the co-ordinates, 2021, 6, 2473-6988, 4677, 10.3934/math.2021275
3.	Ifra Bashir Sial, Nichaphat Patanarapeelert, Muhammad Aamir Ali, Hüseyin Budak, Thanin Sitthiwirattham, On Some New Ostrowski–Mercer-Type Inequalities for Differentiable Functions, 2022, 11, 2075-1680, 132, 10.3390/axioms11030132
4.	Deniz Uçar, Inequalities for different type of functions via Caputo fractional derivative, 2022, 7, 2473-6988, 12815, 10.3934/math.2022709
5.	Soubhagya Kumar Sahoo, Y.S. Hamed, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Kamsing Nonlaopon, New midpoint type Hermite-Hadamard-Mercer inequalities pertaining to Caputo-Fabrizio fractional operators, 2023, 65, 11100168, 689, 10.1016/j.aej.2022.10.019
6.	Muhammad Imran Asjad, Waqas Ali Faridi, Mohammed M. Al-Shomrani, Abdullahi Yusuf, The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator, 2022, 7, 2473-6988, 7040, 10.3934/math.2022392
7.	Churong Chen, Discrete Caputo Delta Fractional Economic Cobweb Models, 2023, 22, 1575-5460, 10.1007/s12346-022-00708-5
8.	Soubhagya Kumar Sahoo, Ravi P. Agarwal, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Kamsing Nonlaopon, Khadijah M. Abualnaja, Hadamard–Mercer, Dragomir–Agarwal–Mercer, and Pachpatte–Mercer Type Fractional Inclusions for Convex Functions with an Exponential Kernel and Their Applications, 2022, 14, 2073-8994, 836, 10.3390/sym14040836
9.	Muhammad Tariq, Sotiris K. Ntouyas, Asif Ali Shaikh, A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators, 2023, 11, 2227-7390, 1953, 10.3390/math11081953
10.	Loredana Ciurdariu, Eugenia Grecu, Hermite–Hadamard–Mercer-Type Inequalities for Three-Times Differentiable Functions, 2024, 13, 2075-1680, 413, 10.3390/axioms13060413
11.	Muhammad Aamir Ali, Thanin Sitthiwirattham, Elisabeth Köbis, Asma Hanif, Hermite–Hadamard–Mercer Inequalities Associated with Twice-Differentiable Functions with Applications, 2024, 13, 2075-1680, 114, 10.3390/axioms13020114
12.	Muhammad Aamir Ali, Christopher S. Goodrich, On some new inequalities of Hermite–Hadamard–Mercer midpoint and trapezoidal type in q-calculus, 2024, 44, 0174-4747, 35, 10.1515/anly-2023-0019
13.	Thanin Sitthiwirattham, Ifra Sial, Muhammad Ali, Hüseyin Budak, Jiraporn Reunsumrit, A new variant of Jensen inclusion and Hermite-Hadamard type inclusions for interval-valued functions, 2023, 37, 0354-5180, 5553, 10.2298/FIL2317553S
14.	Muhammad Aamir Ali, Zhiyue Zhang, Michal Fečkan, GENERALIZATION OF HERMITE–HADAMARD–MERCER AND TRAPEZOID FORMULA TYPE INEQUALITIES INVOLVING THE BETA FUNCTION, 2024, 54, 0035-7596, 10.1216/rmj.2024.54.331
15.	Bahtiyar Bayraktar, Péter Kórus, Juan Eduardo Nápoles Valdés, Some New Jensen–Mercer Type Integral Inequalities via Fractional Operators, 2023, 12, 2075-1680, 517, 10.3390/axioms12060517
16.	THANIN SITTHIWIRATTHAM, MIGUEL VIVAS-CORTEZ, MUHAMMAD AAMIR ALI, HÜSEYIN BUDAK, İBRAHIM AVCI, A STUDY OF FRACTIONAL HERMITE–HADAMARD–MERCER INEQUALITIES FOR DIFFERENTIABLE FUNCTIONS, 2024, 32, 0218-348X, 10.1142/S0218348X24400164
17.	Muhammad Ali, Hüseyin Budak, Elisabeth Köbis, Some new and general versions of q-Hermite-Hadamard-Mercer inequalities, 2023, 37, 0354-5180, 4531, 10.2298/FIL2314531A

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2280) PDF downloads(152) Cited by(33)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(3)

AIMS Mathematics

A study on the fractal-fractional tobacco smoking model

Related Papers:

Abstract

1. Introduction

2. Main results

3. Applications to special means

4. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

A study on the fractal-fractional tobacco smoking model

Related Papers:

Abstract

1. Introduction

2. Main results

3. Applications to special means

4. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog