Abstract
We consider the modified Hermite–Hadamard inequality and related results on integral inequalities, in the context of fractional calculus using the Riemann–Liouville fractional integrals. Our results generalize and modify some existing results. Finally, some applications to special means of real numbers are given. Moreover, some error estimates for the midpoint formula are pointed out.
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1 Introduction
The generalization of certain integral inequalities to the fractional scope, in both continuous and discrete versions, have attracted many researchers in the recent few years and before [1, 19, 20]. In this article, our work is devoted to Hadamard–Hermite type for convex functions in the framework of Riemann–Liouville fractional type integrals.
A function \(g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}\) is said to be convex on the interval \(\mathcal{I}\), if the inequality
holds for all \(x,y\in \mathcal{I}\) and \(\ell \in [0,1]\). We say that g is concave if −g is convex.
For convex functions (1), many equalities and inequalities have been established by many authors; such as the Hardy type inequality [3], Ostrowski type inequality [7], Olsen type inequality [8], Gagliardo–Nirenberg type inequality [22], midpoint type inequality [10] and trapezoidal type inequality [14]. But the most important inequality is the Hermite–Hadamard type inequality [6], which is defined by
where \(g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}\) is assumed to be a convex function on \(\mathcal{I}\) where \(u,v\in \mathcal{I}\) with \(u< v\).
A number of mathematicians in the field of applied and pure mathematics have devoted their efforts to generalizing, refining, finding counterparts of, and extending the Hermite–Hadamard inequality (2) for different classes of convex functions and mappings. For more recent results obtained in view of inequality (2), we refer the reader to [2, 4, 6, 13, 16, 18].
In [21], Sarikaya et al. obtained the Hermite–Hadamard inequalities in fractional integral form:
where \(g:[u,v]\subseteq \mathbb{R}\to \mathbb{R}\) is assumed to be a positive convex function on \([u,v]\), \(g\in L_{1}[u,v]\) with \(u< v\), and \(\mathfrak{J}^{\vartheta }_{u^{+}}\) and \(\mathfrak{J}^{\vartheta } _{v^{-}}\) are the left-sided and right-sided Riemann–Liouville fractional integrals of order \(\vartheta >0\), which, respectively, are defined by [9]
It is clear that inequality (3) is a generalization of Hermite–Hadamard inequality (2). If we take \(\vartheta =1\) in (3) we obtain (2). Many inequalities have been established in view of inequality (3); for more details see [5, 10, 11, 14, 15, 21, 23].
Recently, in [12], Mehrez and Agarwal obtained a new modification of the Hermite–Hadamard inequality (2); this is given by
Furthermore, Mehrez and Agarwal obtained many inequalities in view of inequalities (4); for which we refer the reader to their interesting paper [12].
The aim of this paper is to establish new inequalities of Hermite–Hadamard type for convex functions via Riemann–Liouville fractional integrals.
2 Preliminary lemmas
In order to obtain our main results, we need some qualities which are stated in the following lemmas.
Lemma 1
([23])
Let\(g:[u,v]\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\((u,v)\)with\(u< v\). If\(g'\in L_{1}[u,v]\), then we have
where
Lemma 2
([5])
Let\(g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}\)be a twice differentiable function on\(\mathcal{I}^{o}\) (the interior of\(\mathcal{I}\)). Assume that\(u,v\in \mathcal{I}^{o}\)with\(u< v\). If\(g''\in L_{1}[u,v]\), then for\(\vartheta >0\)we have
Lemma 3
Let\(g:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable function on\(\mathcal{I}^{o}\)and\(g\in L_{1}[u,v]\). Ifgis a convex function on\([u,v]\). then for\(\vartheta >0\)we have
and
Proof
From the definition of Riemann–Liouville fractional integral, we have
By using the change of the variable \(x=\frac{3}{4}\ell +\frac{u+v}{4}\) for \(\ell \in [\frac{3u-v}{3},\frac{3v-u}{3} ]\), we obtain
Since g is convex on \([u,v]\), we have
It follows from this and (9) that
Again, by using the change of the variable \(z=\frac{3}{2}\ell \) for \(\ell \in [\frac{3u-v}{2},\frac{3v-u}{2} ]\), we obtain
Replace u by \(\frac{3u-v}{2}\) and v by \(\frac{3v-u}{2}\) in the right-hand side of inequality (3) and multiply both sides by \(\frac{1}{2}\), we get
From (10) and (11), we obtain the desired inequality (7) and from (7) we can easily obtain the inequality (8). These complete the proof of Lemma 3. □
Remark 1
If we use \(\vartheta =1\) in Lemma 3, then Lemma 3 reduces to Lemma 3 in [12]. In particular, inequalities (7) reduces to the inequalities (4).
3 Hermite–Hadamard type inequalities
Our main results start from the following theorem.
Theorem 1
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'\in L_{1} [\frac{3u-v}{2},\frac{3v-u}{2} ]\)and\(g': [\frac{3u-v}{2},\frac{3v-u}{2} ]\to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2},\frac{3v-u}{2} ]\). If\(|g'|^{q}, q\geq 1\)is a convex function on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), then
Proof
First we prove the theorem for \(q=1\). By changing the variables \(u\to \frac{3u-v}{2}\) and \(v\to \frac{3v-u}{2}\) in Lemma 1, we get
Using the convexity of \(|g'|\) on \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), we obtain
Analogously, we obtain
Using (15) and (16) in (14), we get inequality (12) for \(q=1\).
For \(q>1\) we use the Hölder inequality and the convexity of \(|g'|^{q}\) on \([\frac{3u-v}{2},\frac{3v-u}{2} ]\) to obtain
Analogously,
Using (17) and (18) in (14), we get
where
Applying the formula
for (19) and then using the fact that \(\vert x_{1}^{r}+x _{2}^{r} \vert \leq \vert x_{1}+x_{2} \vert ^{r}\), \(x_{1}, x_{2}, r \in [0,1]\), we obtain the inequality (12). This completes the proof of Theorem 1. □
Corollary 1
With similar assumptions to Theorem1, if\(\vartheta =1\), then
which is obtained by Mehrez and Agarwal in [12, Theorem 1].
Remark 2
In [23], the following inequality has been established:
We show an analytical and numerical comparison between the left-hand side of inequalities (20) and (21).
- 1.
Let \(q=1\) and \(u,v\in \mathbb{R}\) with \(u< v\). Then:
- (a)
If the function \(|g'|\) is increasing on \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\). Since \(\frac{3u-v}{2}< u< v<\frac{3v-u}{2}\), we obtain
$$\begin{aligned} g' \biggl(\frac{3u-v}{2} \biggr)< g'(u)\quad \text{and}\quad g'(v)< g' \biggl( \frac{3v-u}{2} \biggr); \end{aligned}$$or if the function \(|g'|\) is decreasing on \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), we obtain
$$\begin{aligned} g'(u)< g' \biggl(\frac{3u-v}{2} \biggr)\quad \text{and}\quad g' \biggl(\frac{3v-u}{2} \biggr)< g'(v). \end{aligned}$$In those cases, comparison does not occur analytically between inequalities (20) and (21).
- (b)
If the function \(|g'|\) is increasing on \([\frac{3u-v}{2},u ]\), and decreasing on \([v, \frac{3v-u}{2} ]\), then we have
$$\begin{aligned} g' \biggl(\frac{3u-v}{2} \biggr)< g'(u)\quad \text{and}\quad g' \biggl(\frac{3v-u}{2} \biggr)< g'(v). \end{aligned}$$This tells us the right-hand side of inequality (20) is better than the right-hand side of inequality (21).
- (c)
If the function \(|g'|\) is decreasing on \([\frac{3u-v}{2},u ]\), and increasing on \([v, \frac{3v-u}{2} ]\), then we conclude that the right-hand side of inequality (21) is better than the right-hand side of inequality (20).
- (a)
- 2.
Suppose that m and n represent the right-hand side of inequalities (20) and (21), respectively. Let \([u,v]= [-1,-\frac{1}{2} ]\) and \(g(x)=e^{x}\), then we obtain \(\mathbf{m}=0.199744\) and \(\mathbf{n}=0.167444\) when \(q=1\) and \(\mathbf{m}=0.155592\) when \(q=2\). Then we conclude that the right-hand side of inequality (20) is worse than the right-hand side of inequalities (21) when \(q=1\), but better when \(q=2\).
Theorem 2
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'\in L_{1} [\frac{3u-v}{2},\frac{3v-u}{2} ]\)and\(g': [\frac{3u-v}{2},\frac{3v-u}{2} ]\to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2},\frac{3v-u}{2} ]\). If\(|g'|^{q}, q>1\)is a convex function on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), then
where\(\frac{1}{p}+\frac{1}{q}=1\).
Proof
Applying Hölder’s inequality and the convexity of \(|g'|^{q}, q>1\) on \([\frac{3u-v}{2},\frac{3v-u}{2} ]\), we obtain
Since \((H_{1}-H_{2})^{q}\leq H_{1}^{q}-H_{2}^{q}\) for each \(H_{1}, H _{2}>0\) and \(q>1\), (23) becomes
In a similar manner, we get
Using (24) and (25) in (14), we get
which proves the first inequality of (22). Applying the fact
to the last inequality, we get the second inequality of (22). This completes the proof. □
Collecting both of Theorems 1 and 2 we obtain the following corollary.
Corollary 2
Let\(\frac{1}{p}+\frac{1}{q}=1\), then from Theorems1and2, we have
where\(\gamma _{1}= (\frac{2^{\vartheta }-1}{2^{\vartheta }( \vartheta +1)}+\frac{1}{2} )\)and\(\gamma _{2}= (\frac{2^{ \vartheta p}-1}{2^{\vartheta p+\frac{p}{q}}(\vartheta p+1)} +\frac{1}{2^{ \frac{p}{q}+1}} )^{\frac{1}{p}}\).
Corollary 3
With similar assumptions to Theorem2if\(\vartheta =1\), we have
which is obtained by Mehrez and Agarwal in [12, Theorem 2].
Theorem 3
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\). If\(|g''|^{q}, q\geq 1\)is a convex function on\([\frac{3u-v}{2},\frac{3v-u}{2} ]\), then
Proof
From Lemma 2 we have
Using the convexity of \(|g''|^{q}, q>1\) on \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\) and Hölder’s inequality, we have
Using (29) in (28) we get (26) for \(q>1\).
Now by using the convexity of \(|g''|\), we find
Substituting (30) into (28) we deduce that the inequality (26) holds true for \(q=1\). Hence the proof of Theorem 3 is completed. □
Corollary 4
With similar assumptions to Theorem3if\(\vartheta =1\), we have
Remark 3
In [12, Theorem 3], Mehrez and Agarwal obtained the following inequality:
The right-hand side of (31) confirms the modification of our work compared with (32).
Remark 4
If \(g''(x)\) is bounded on the interval \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), then Theorem 3 reduces to
for some \(M\in \mathbb{R}\).
Theorem 4
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\). If\(|g''|^{q}, q>1\)is a convex function on\([\frac{3u-v}{2},\frac{3v-u}{2} ]\), then
where\(\frac{1}{p}+\frac{1}{q}=1\).
Proof
From inequality (28) and the Hölder inequality, we have
Using the fact that \(|g''|^{q}, q>1\) is convex on \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), we have
Observe that \(\ell ^{p} (1-\ell ^{\vartheta } )^{p}\leq \ell ^{p} ([1-\ell ]^{\vartheta } )^{p} =\ell ^{p} (1- \ell )^{\vartheta p}\). So, (34) completes the proof of Theorem 4. □
Corollary 5
With similar assumptions to Theorem4if\(\vartheta =1\), we have
Proof
The proof of this corollary follows from the facts that
□
Remark 5
The right-hand side of inequality (35) confirms the modification of our work compared with the right-hand side of inequality (3.24) in [12, Theorem 4].
Remark 6
If \(g''(x)\) is bounded on the interval \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), then Theorem 4 reduces to
for \(\frac{1}{p}=1-\frac{1}{q}\) and for some \(M\in \mathbb{R}\).
Theorem 5
With similar assumptions to Theorem4, we have
Proof
By using the Hölder inequality and the convexity of \(|g''|^{q}, q>1\) on \([\frac{3u-v}{2},\frac{3v-u}{2} ]\), we have
Using (37) in (28), we obtain the required inequality (36). □
Corollary 6
With similar assumptions to Theorem5if\(\vartheta =1\), we have
Remark 7
The right-hand side of inequality (38) confirms the modification of our work compared with the right-hand side of inequality (3.27) in [12, Theorem 5].
Theorem 6
With similar assumptions to Theorem3, we have
Proof
Let \(q>1\), then, by using the Hölder inequality and the convexity of \(|g''|^{q}\) on \([\frac{3u-v}{2},\frac{3v-u}{2} ]\), we have
Using (40) in (28) we obtain the inequality (39) for \(q>1\).
Now, using the convexity of \(|g''|\) and the properties of the modulus, we find
Substituting (41) into (28) we deduce that the inequality (39) holds true for \(q=1\). Thus (40), (41) and (28) complete the proof of Theorem 6. □
Corollary 7
With similar assumptions to Theorem5if\(\vartheta =1\), we have
Remark 8
The right-hand side of inequality (42) confirms the modification of our work compared with the right-hand side of inequality (3.29) in [12, Theorem 6].
Collecting Theorems 3–6 we obtain the following corollary.
Corollary 8
From Theorems3–6we deduce that
where
for\(q>1\).
A few results for concave functions will be extended in the following theorems.
Theorem 7
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\). If\(|g''|^{q}\)is concave on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\), then
where\(q=\frac{p}{p-1}\)such that\(p\in \mathbb{R}\), \(p>1\).
Proof
Applying Hölder’s inequality to (28), we get
By using the concavity of \(|g''|^{q}\) on \([\frac{3u-v}{2}, \frac{3v-u}{2} ]\) and the integral Jensen’s inequality, we get
and analogously
Thus substituting the obtained results of (45) and (46) in (44), we get (43) as desired. □
Theorem 8
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'': [\frac{3u-v}{2},\frac{3v-u}{2} ] \to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\). Assume that\(\frac{1}{p}+\frac{1}{q}=1\)with\(p\geq 1\)such that\(|g''|^{q}\)is concave on\([\frac{3u-v}{2}, \frac{3v-u}{2} ]\). Then
Proof
From the concavity of \(|g''|^{q}\) and the power-mean inequality, we have
for all \(x,y\in [\frac{3u-v}{2},\frac{3v-u}{2} ]\) and \(\ell \in [0,1]\). This also gives
this means that \(|g''|\) is also concave. Again by the Jensen integral inequality, we obtain
and analogously
Thus substituting the obtained results of (48) and (49) in (44), we get (47) as desired. □
4 Applications
In this section some applications are presented to demonstrate the usefulness of our obtained results in the previous sections.
4.1 Applications to special means
Let u and v are two arbitrary positive real numbers such that \(u\neq v\), we consider the following special means [17].
- (i)
The arithmetic mean:
$$A=A(u,v)=\frac{u+v}{2}. $$ - (ii)
The inverse arithmetic mean:
$$H=H(u,v)=\frac{2}{\frac{1}{u}+\frac{1}{v}}, \quad u,v\neq 0. $$ - (iii)
The geometric mean:
$$G=G(u,v)=\sqrt{u v}. $$ - (iv)
The logarithmic mean:
$$L(u,v)=\frac{v-u}{\log (v)-\log (u)}, \quad u\neq v. $$ - (v)
The generalized logarithmic mean:
$$L_{n}(u,v)= \biggl[\frac{v^{n+1}-u^{n+1}}{(v-u)(n+1)} \biggr]^{ \frac{1}{n}}, \quad n\in \mathbb{Z}\setminus \{-1,0\}. $$
Proposition 1
Let\(u,v\in \mathbb{R}\)with\(0< u< v\)and\(n\in \mathbb{Z}/\{0,-1\}\), then we have
where\(p=\frac{q}{q-1}\).
Proof
The proof of this proposition follows from Corollary 2 with \(\vartheta =1\) and \(g(x)=x^{n}\). □
Proposition 2
Let\(u,v\in \mathbb{R}\)with\(0< u< v\)and\(n\in \mathbb{Z}/\{0,-1\}\), then we have
for\(q\geq 1\).
Proof
The assertion follows from Corollary 2 with \(\vartheta =1\) and \(g(x)=\frac{1}{x}\). □
Proposition 3
Let\(|n|\geq 3\)and\(u, v\in \mathbb{R}\)with\(0< u< v\), then
and
for\(q\geq 1\).
Proof
Observe that \(A^{-1} (u^{-1},v^{-1} )=H(u,v)=\frac{2}{ \frac{1}{u}+\frac{1}{v}}\). So, making the change of variables \(u\to v^{-1}\) and \(v\to u^{-1}\) in the inequalities (50) and (51), we can deduce the desired inequalities (52) and (53), respectively. □
Proposition 4
Let\(u,v\in \mathbb{R}\)with\(0< u< v\)and\(n\in \mathbb{Z}/\{0,-1\}\), then we have
where\(p=\frac{q}{q-1}\).
Proof
The proof of this proposition follows from Corollary 2 with \(\vartheta =1\) and \(g(x)=\frac{1}{x^{2}}\). □
4.2 The midpoint formula
Let d be a partition of the interval \([u,v]\) such that \(u=x_{0}< x _{1}< x_{2}<\cdots <x_{m-1}<x_{m}=v\). Consider the quadrature formula [17]
where \(\mathcal{E}(g,d)\) represents the associated approximation error and
is the midpoint version.
Proposition 5
Let\(g:\mathcal{I}^{o}\subseteq \mathbb{R}\to \mathbb{R}\)be a differentiable mapping on\(\mathcal{I}^{o}\), \(u,v\in \mathcal{I}^{o}\)with\(u< v\). Let\(g'\in L_{1} [\frac{3u-v}{2},\frac{3v-u}{2} ]\)and\(g': [\frac{3u-v}{2},\frac{3v-u}{2} ]\to \mathbb{R}\)be a continuous function on\([\frac{3u-v}{2},\frac{3v-u}{2} ]\). If\(|g'|^{q}, q\geq 1\)is a convex function, then, for every partitiondof\([\frac{3u-v}{2},\frac{3v-u}{2} ]\)in (54), we have
Proof
Applying Corollary 2 with \(\vartheta =1\) on the subinterval \([\frac{3x_{k}-x_{k+1}}{2},\frac{3x_{k+1}-x_{k}}{2} ]\) (\(k=0,1,\ldots, m-1\)) of the partition d, we obtain
Summing over k from 0 to \(m-1\) and taking into account that \(|g'|\) is convex, we obtain, by the triangle inequality,
This completes the proof of (55). □
5 Conclusion
In this paper, we generalized the modified Hermite–Hadamard inequality obtained by Mehrez and Agarwal in [12], it can be found in Lemma 3 and Theorems 1–6. Corollaries 4–7 confirm that our results modified the existing results of [12]. Furthermore, Theorems 7–8 modified the existing Theorems 5–6 of [5].
References
Abdeljawad, T., Al-Mdallal, Q.M.: Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality. J. Comput. Appl. Math. 339, 218–230 (2018). https://doi.org/10.1016/j.cam.2017.10.021
Avci, M., Kavurmaci, H., Odemir, M.E.: New inequalities of Hermite–Hadamard type via s-convex functions in the second sense with applications. Appl. Math. Comput. 217, 5171–5176 (2011)
Ciatti, P., Cowling, M.G., Ricci, F.: Hardy and uncertainty inequalities on stratified Lie groups. Adv. Math. 277, 365–387 (2015)
Dragomir, S.S.: Hermite–Hadamard’s type inequalities for operator convex functions. Appl. Math. Comput. 218, 766–772 (2011)
Dragomir, S.S., Bhatti, M.I., Iqbal, M., Muddassar, M.: Some new Hermite–Hadamard’s type fractional integral inequalities. J. Comput. Anal. Appl. 18(4), 655–661 (2015)
Dragomir, S.S., Pearce, C.E.M.: Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, Victoria University (2000)
Gavrea, B., Gavrea, I.: On some Ostrowski type inequalities. Gen. Math. 18(1), 33–44 (2010)
Gunawan, H., Eridani: Fractional integrals and generalized Olsen inequalities. Kyungpook Math. J. 49, 31–39 (2009)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kırmacı, U.S.: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147, 137–146 (2004)
Lian, T., Tang, W., Zhou, R.: Fractional Hermite–Hadamard inequalities for \((s,m)\)-convex or s-concave functions. J. Inequal. Appl. 2018, 240 (2018)
Mehrez, K., Agarwal, P.: New Hermite–Hadamard type integral inequalities for convex functions and their applications. J. Comput. Appl. Math. 350, 274–285 (2019)
Mohammed, P.O.: Inequalities of type Hermite–Hadamard for fractional integrals via differentiable convex functions. Turk. J. Anal. Number Theory 4(5), 135–139 (2016)
Mohammed, P.O.: On new trapezoid type inequalities for h-convex functions via generalized fractional integral. Turk. J. Anal. Number Theory 6(4), 125–128 (2018)
Mohammed, P.O., Sarikaya, M.Z.: Hermite–Hadamard type inequalities for F-convex function involving fractional integrals. J. Inequal. Appl. 2018, 359 (2018)
Odemir, M.E., Avci, M., Set, E.: On some inequalities of Hermite–Hadamard type via m-convexity. Appl. Math. Lett. 23, 1065–1070 (2010)
Pearce, C.E.M., Pečarić, J.E.: Inequalities for differentiable mappings with application to special means and quadrature formula. Appl. Math. Lett. 13, 51–55 (2000)
Qi, F., Mohammed, P.O., Yao, J.-C., Yao, Y.-H.: Generalized fractional integral inequalities of Hermite–Hadamard type for \((\alpha ,m)\)-convex functions. J. Inequal. Appl. 2019, 135 (2019)
Rahman, G., Abdeljawad, T., Jarad, F., Khan, A., Nisar, K.S.: Certain inequalities via generalized proportional Hadamard fractional integral operators. Adv. Differ. Equ. 2019, 454 (2019)
Rashid, S., Abdeljawad, T., Jarad, F., Noor, M.A.: Some estimates for generalized Riemann–Liouville fractional integrals of exponentially convex functions and their applications. Mathematics 7, 807 (2019). https://doi.org/10.3390/math7090807
Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)
Sawano, Y., Wadade, H.: On the Gagliardo–Nirenberg type inequality in the critical Sobolev–Morrey space. J. Fourier Anal. Appl. 19(1), 20–47 (2013)
Zhu, C., Feckan, M., Wang, J.: Factional integral inequalities for differential convex mappings and applications to special means and a midpoint formula. J. Appl. Math. Stat. Inform. 8(2), 21–28 (2012)
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The authors would like to express their special thanks to the editor and the referees of this manuscript. The second 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).
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Mohammed, P.O., Abdeljawad, T. Modification of certain fractional integral inequalities for convex functions. Adv Differ Equ 2020, 69 (2020). https://doi.org/10.1186/s13662-020-2541-2
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DOI: https://doi.org/10.1186/s13662-020-2541-2