Hermite-Hadamard inequality involving Caputo-Fabrizio fractional integrals and related inequalities via $s$-convex functions in the second sense

1.   Introduction

Fractional calculus has been a fascinating field in the last decades. Some researchers introduced various fractional derivatives, integrals, and their properties with singular or non-singular kernels to exploit its applications in real world problems ^{[3,4,5,9,10,11,12,13,14,15]}. These fractional derivatives and fractional integrals have been used in inequalities as well. In inequalities, Hermite-Hadamard inequality is the renowned and is given in the following theorem.

Theorem 1.1. ^[1] Let $\xi:I \to \mathbb{R}$ is a convex function on the interval $I\subseteq \mathbb{R}$ and $l_{1}, l_{2} \in I$ with $l_{1} < l_{2}$, then

In ^[7,8] this inequality was generalized for s-convex functions. In the field of fractional calculus, the main focus of a researcher is to give a definition of new operators and to develop their applications and solution to problems by involving these operators. Singularity and locality are the properties in that the operators are distinct from each other. Whereas, the kernel of operators is based on the type of functions including power law, the exponential function, and the Mittag-Lefler function. The distinct property of the Caputo-Fabrizio operator is that its kernel is non-singular. In this paper, we develop the Hermite-Hadamard inequality for $s$-convex functions via Caputo-Fabrizio fractional integral. First we define Caputo-Fabrizio fracional derivative and integral ^[3].

Definition 1.1. Let $\xi\in H^{1}(l_{1}, l_{2})$(class of first order differentiable functions), $l_{1} < l_{2}$, $0\le\beta\le1,$ then the left fractional derivative in the Caputo-Fabrizio sense is

and the fractional integral corresponding to this operator is

Right fractional derivative is defined as

and the fractional integral corresponding to this operator is

The Caputo-Fabrizio derivative is obtained by changind the singular kernel $(t-w)^{-\beta}$ of Caputo fractional derivative by non-singular kernel $e^{\frac{-\beta(t-w)^{\beta}}{1-\beta}}$ and $\frac{1}{\Gamma(1-\beta)}$ by $\frac{B(\beta)}{1-\beta}$, where $B(\beta) > 0$ is a normalization function satisfying $B(0) = B(1) = 1$ given in ^[3].

Definition 1.2. ^[6] A function $\xi:I\subseteq\mathbb{R^{+}}\to \mathbb{R^{+}}$ is said to be $s$-convex function in second sense on $I$ if

holds for all $l_{1}, l_{2}\in I$, and $t\in [0, 1]$, for some fix $s \in (0, 1]$.

Lemma 1.1. [2,Lemma 1] Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$, the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$. Let $\xi:I^{0}\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{0}, l_{1}, l_{2}\in I$ with $l_{1} < l_{2}$. If $\xi{'}\in L([l_{1}, l_{2}])$, then the following equality holds:

Lemma 1.2. [2,Lemma 2] Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$, the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$. Let $\xi:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable function on $I^{0}$, $l_{1}, l_{2}\in I$ with $l_{1} < l_{2}$. If $\xi{'}\in L([l_{1}, l_{2}])$ and $0\le\beta\le1$, then following inequality holds:

where $k\in[l_{1}, l_{2}]$ and $B(\beta) > 0$ is normalization function.

2.   Generalization of Hermite-Hadamard inequality for s-convex function via Caputo-Fabrizio fractional operator

Here we present Hermite-Hadamard inequality for s-convex functions via Caputo-Fabrizio fractional operator which is stated in the following theorem.

Theorem 2.1. Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$; the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$. Let a function $\xi:[l_{1}, l_{2}]\subseteq \mathbb{R}\to \mathbb{R}$ be s-convex on $[l_{1}, l_{2}]$ for $s\in (0, 1)$ and $\xi\in L([l_{1}, l_{2}])$. If $0\le\beta \le1$, then we have the following double inequality:

where $k\in[l_{1}, l_{2}]$ and $B(\beta) > 0$ is normalization function.

Proof. Since $\xi$ is s-convex, taking $\lambda = \frac{1}{2}$ we get

On making change of variables and integrating over $[0, 1]$, we get

Multiplying both sides by $\frac{\beta(l_{2}-l_{1})}{B(\beta)}$ we get

and

Adding $\frac{2(1-\beta)}{B(\beta)}\xi(k)$ on both sides we have

which means that

i.e.,

which proves first inequality. Now, we prove second inequality, since

As $\xi$ is s-convex in the second sense, we get

Multiplying both sides by $\frac{\beta(l_{2}-l_{1})}{B(\beta)}$ we get

Adding $\frac{2(1-\beta)}{B(\beta)}\xi(k)$ on both sides

and this gives that

From (9) and (10) we get (8), which is required.

Remark 2.1. For $s = 1$ we get Theorem 2 of ^[2].

Theorem 2.2. Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$; the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$.

Let $\xi_{1}:[l_{1}, l{2}]\subseteq \mathbb{R}\to \mathbb{R}$ be a convex non-negative and $\xi_{2}:[l_{1}, l_{2}]\subseteq \mathbb{R}\to \mathbb{R}$ be an s-convex functions on $I$. If $\xi_{1}\xi_{2}\in L([l_{1}, l_{2}])$, then we have the following inequality:

where $M(l_{1}, l_{2}) = \xi_{1}(l_{1})\xi_{2}(l_{1})+\xi_{1}(l_{2})\xi_{2}(l_{2})$, $N(l_{1}, l_{2}) = \xi_{1}(l_{1})\xi_{2}(l_{2})+\xi_{1}(l_{2})\xi_{2}(l_{1})$.

Proof. Since $\xi_{1}$ is convex and $\xi_{2}$ is s-convex then we have

Multiplying these inequalities we get

Integrating over $t\in[0, 1]$, then making change of variables we get

Multiplying both sides by $\frac{\beta (l_{2}-l_{1})}{B(\beta)}$ and adding $\frac{2(1-\beta)}{B(\beta)}\xi_{1}(k)\xi_{2}(k)$, we get

On simplifying the last inequality, we get

Remark 2.2. (i) For $\beta = 1, B(\beta) = B(1) = 1$, we get Theorem 5 of ^[6].

(ii) For $s = 1$ we get Theorem 3 of ^[2].

Theorem 2.3. Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$; the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$.

Let $\xi_{1}, \xi_{2} :[l_{1}, l_{2}]\subseteq \mathbb{R}\to \mathbb{R}$, be non-negative functions such that $\xi_{1}, \xi_{2}$ and $\xi_{1}\xi_{2}$ are in $L([l_{1}, l_{2}])$. If $\xi_{1}$ is $s_{1}$-convex and $\xi_{2}$ is $s_{2}$-convex on $[l_{1}, l_{2}]$ for some fixed $s_{1}, s_{2}\in (0, 1]$, then

where $M(l_{1}, l_{2}) = \xi_{1}(l_{1}) \xi_{2}(l_{1})+\xi_{1}(l_{2}) \xi_{2}(l_{2})$, and $N(l_{1}, l_{2}) = \xi_{1}(l_{1}) \xi_{2}(l_{2})+\xi_{1}(l_{2}) \xi_{2}(l_{1})$, and $\Gamma.$ is the Gamma function.

Proof. As $\xi_{1}$ is $s_{1}$-convex and $\xi_{2}$ is $s_{2}$-convex we have

Multiplying these inequalities we obtain

Integrating with respect to $t$ over $[0, 1]$ and making change of variables, we get

Multiplying both sides by $\frac{\beta (l_{2}-l_{1})}{B(\beta)}$ and adding $\frac{2(1-\beta)}{B(\beta)}\xi_{1}(k)\xi_{2}(k)$ we get

Remark 2.3. (i) For $\beta = 1, B(\beta) = B(1) = 1$, we get [5,Theorem 6].

(ii) For $s_{1} = s_{2} = 1$ we get [2,Theorem 3].

Theorem 2.4. Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$; the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$.

Let $\xi_{1}, \xi_{2}:[l_{1}, l_{2}]\subseteq \mathbb{R}\to \mathbb{R}$, be the function such that $\xi_{1}, \xi_{2}, \xi_{1}\xi_{2}\in L([l_{1}, l_{2}])$. If $\xi_{1}$ is convex and non-negative on $[l_{1}, l_{2}]$ and $\xi_{2}$ is s-convex on $[l_{1}, l_{2}]$ for some fixed $s\in (0, 1]$, then

Where $M(l_{1}, l_{2}) = \xi_{1}(l_{1})\xi_{2}(l_{1})+\xi_{1}(l_{2})\xi_{2}(l_{2})$, and $N(l_{1}, l_{2}) = \xi_{1}(l_{1})\xi_{2}(l_{2})+\xi_{1}(l_{2})\xi_{2}(l_{1})$.

Proof. As

and

Using convexity of $\xi_{1}$ and s-convexity of $\xi_{2}$, we get

Integrating with respect to $t$ over $[0, 1]$ and making change of variable we get

Multiplying both sides by $\frac{\beta(l_{2}-l_{1})}{B(\beta)}$ and subtracting $\frac{2(1-\beta)}{B(\beta)}\xi_{1}(k)\xi_{2}(k)$, we obtain

After simplification we get

Corollary 2.1. For $\beta = 1$ we get Theorem 7 of ^[6].

3.   Some inequalities of Hermite-Hadamard type via Caputo-Fabrizio fractional operator

In this section, we present Hermite-Hadamard type inequalities for s-convex functions via Caputo-Fabrizio fractional operator.

Theorem 3.1. Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$; the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$. Let $\xi:I\subseteq \mathbb{R}\to \mathbb{R}$ be a positive differentiable mapping on $I^{0}$ and $|\xi'|$ is s-convex on $[l_{1}, l_{2}]$. If $\xi'\in L([l_{1}, l_{2}])$ and $\beta \in[0, 1]$, then we have the following inequality:

Proof. By Lemma 1.3 and the property of modulus we obtain,

Using s-convexity of $|\xi'|$ we get

Remark 3.1. Setting $s = 1$ in Theorem 3.1, we recapture Theorem 5 in ^[2].

Next theorem gives new upper bound of the left-Hadamard inequality for s-convex function via Caputto-Fabrizio fractional integral.

Theorem 3.2. Let $I$ be a real interval such that ${\rm{ł}}_1, {\rm{ł}}_2 \in I^{\circ}$; the interior of $I$ with ${\rm{ł}}_1 < {\rm{ł}}_2$. Let $\xi:I\subseteq \mathbb{R}\to \mathbb{R}$ be a positive differentiable mapping on $I^{0}$ and $|\xi{'}|^{q}$ is s-convex on $[l_{1}, l_{2}]$, where $p > 1, p^{-1}+q^{-1} = 1$. If $\xi{'}\in L([l_{1}, l_{2}])$ and $\beta\in [0, 1]$, then we have the following inequality:

Proof. Using similar arguments as the proof of Theorem 3.1, but this time we use Hölder inequality and s-convexity of $|\xi'|^{q}$ we have

Remark 3.2. For s = 1 we get Theorem 6 of ^[2].

4.   Application to special means

In this section, first we recall some mathematical means of two real numbers, and then present application of previous results on these means of real numbers.

(ⅰ) The arithmetic mean

(ⅱ) The generalized logarithmic mean

Proposition 1. Let $w_{1}, w_{2}\in \mathbb{R}^{+}$, $w_{1} < w_{2}$, $0 < s < 1$, then

Proof. By setting $\xi(z) = z^{s}$ with $\beta = 1$ and $B(\beta) = 1$ in Theorem 3.1 we get required result.

Proposition 2. Let $w_{1}, w_{2}\in \mathbb{R}^{+}$ with $w_{1}<w_{2}$, and $0<s<1$ and for all $q>1$, we have

Proof. By setting $\xi(z) = z^{s}$ with $\beta = 1$ and $B(\beta) = 1$ in Theorem 3.2 we get required result.

5.   Conclusions

The theory of convex functions is considered an important tool in the theory of optimization. Fractional calculus has a wide range of applications in Engineering and Applied Sciences. Caputo-Fabrizio fractional integral, with a non-singular kernel, is one of the major operators which is used to develop different mathematical models. In this paper, Hermite-Hadamard type inequalities for s-convex functions in the second sense via Caputo-Fabrizio integral operator are established. The obtained results are generalizations. In literature, the existing results become particular cases for these results which are given in remarks. The results of this paper may stimulate further research for the researchers working in this field.

Acknowledgments

This research article is supported by the National University of Sciences and Technology (NUST), Islamabad, Pakistan.

Conflict of interest

The authors declare that they have no competing interests.