Abstract
In the paper, the authors establish some generalized fractional integral inequalities of the Hermite–Hadamard type for \((\alpha,m)\)-convex functions, show that one can find some Riemann–Liouville fractional integral inequalities and classical integral inequalities of the Hermite–Hadamard type, and generalize and extend some known results.
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1 Introduction
Let \(h: I\subseteq \mathbb{R}\to \mathbb{R}\) be a convex function with \(a< b\) and \(a, b\in I\). Then
Inequality (1.1) is well known in the literature as the Hermite–Hadamard inequality. A number of mathematicians have devoted their efforts to generalize, refine, counterpart, and extend the Hermite–Hadamard inequality (1.1) for different classes of convex functions and mappings. For several recent results concerning inequality (1.1), we may refer the interested reader to [1, 10, 14, 27, 33, 34, 39].
Let us recall some definitions and known results concerning convexity.
Definition 1.1
([33])
A function \(h: I\subseteq \mathbb{R}\to \mathbb{R}\) is said to be convex on an interval I if the inequality
holds for all \(x,y\in I\) and \(\lambda \in (0,1)\).
Definition 1.2
A function: \(h: [0,b]\to \mathbb{R}\) is said to be m-convex if
holds for all \(a,b\in [0,b]\) and \(\lambda \in [0,1]\) and for some \(m\in (0,1]\).
Definition 1.3
Let \((\alpha ,m)\in (0,1]^{2}\). A function: \(h: [0,b]\to \mathbb{R}\) is said to be \((\alpha ,m)\)-convex if
holds for all \(a,b\in [0,b]\) and \(\lambda \in [0,1]\) and for some \(m\in (0,1]\).
The Riemann–Liouville integrals \(J_{a+}^{\alpha }h(t)\) and \(J_{b-}^{\alpha }h(t)\) of order \(\alpha \ge 0\) are defined in [5] respectively by \(J_{a+}^{0} h(t)=J _{b-}^{0} h(t)=h(t)\),
and
for \(h \in L_{1}([a,b])\) and \(\alpha >0\), where Γ denotes the classical Euler gamma function which can be defined [17, 22] by
or by
Recently, the following integral identity and the Riemann–Liouville fractional integral inequalities of the Hermite–Hadamard type for \((\alpha ,m)\)-convex functions were obtained.
Lemma 1.1
([26, Lemma 2.1])
Let \(h:[a,b]\subseteq \mathbb{R}\to \mathbb{R}\) be differentiable on an interval \((a,b)\) with \(a< b\) such that \(h'\in L_{1}([a,b])\). Then
for \(\alpha >0\), where
Theorem 1.1
([26, Theorem 3.1])
Let \(h: [0,\infty )\to \mathbb{R}\) be differentiable on \([0,\infty )\) and \(h'\in L_{1}([a,b])\) for \(0\le a< b\) and \(\alpha >0\). If \(\vert h'\vert ^{q}\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\) and \(q\ge 1\), then
Theorem 1.2
([26, Theorem 3.2])
Let \(h: [0,\infty )\to \mathbb{R}\) be differentiable on \([0,\infty )\) and \(h'\in L_{1}([a,b])\) for \(0\le a< b\) and \(\alpha >0\). If \(\vert h'\vert ^{q}\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\) and for \(q>1\) and \(q\ge r\ge 0\), then
where \(B(s,t)\) denotes the classical beta function which can be defined [18, 19] by
For more information about the Hermite–Hadamard type inequalities for \((\alpha ,m)\)-convex functions, please refer to the papers [2, 3, 6, 15, 21, 26, 28,29,30, 32, 35, 38] and closely related references therein.
2 A review for generalized fractional integral operators
Now we recall some necessary definitions and mathematical preliminaries of the generalized fractional integrals which are defined by Sarikaya and Ertuğral in [24].
Let \(\varphi : [0,\infty )\to [0,\infty )\) satisfy the condition \(\int _{0}^{1} \frac{\varphi (t)}{t}\,\operatorname {d}t<\infty \).
We now define the left-sided and right-sided generalized fractional integral operators \({}_{a^{+}}I_{\varphi }h(t)\) and \({}_{b^{-}}I_{ \varphi }h(t)\) by
and
The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as the Riemann–Liouville fractional integrals [25, 26, 31], the k-Riemann–Liouville fractional integrals [11, 36], the Katugampola fractional integrals [7, 8], conformable fractional integrals [23, 37], the Hadamard fractional integrals [16], and so on. These important special cases of the integral operators in (2.1) and (2.2) are mentioned below.
-
1.
If we take \(\varphi (u)=u\), the operators in (2.1) and (2.2) reduce to the Riemann integrals
$$ I_{a^{+}}h(t)= \int _{a}^{t} h(u)\,\operatorname {d}u, \quad t>a \quad \text{and} \quad I_{b^{-}}h(t)= \int _{t}^{b} h(u)\,\operatorname {d}u, \quad t< b. $$ -
2.
If we take \(\varphi (u)=\frac{u^{\alpha }}{\varGamma (\alpha )}\), the operators in (2.1) and (2.2) become the Riemann–Liouville fractional integrals
$$ I_{a^{+}}h(t)=\frac{1}{\varGamma (\alpha )} \int _{a}^{t} (t-u)^{\alpha -1}h(u)\,\operatorname {d}u, \quad t>a $$and
$$ I_{b^{-}}h(t)=\frac{1}{\varGamma (\alpha )} \int _{t}^{b} (u-t)^{\alpha -1}h(u)\,\operatorname {d}u, \quad t< b. $$ -
3.
If we take \(\varphi (u)=\frac{u^{\alpha /k}}{k\varGamma _{k}(\alpha )}\), the operators in (2.1) and (2.2) are the k-Riemann–Liouville fractional integrals
$$ I_{a^{+},k}h(t)=\frac{1}{k\varGamma _{k}(\alpha )} \int _{a}^{t} (t-u)^{ \alpha /k-1}h(u)\,\operatorname {d}u, \quad t>a $$and
$$ I_{b^{-},k}h(t)=\frac{1}{k\varGamma _{k}(\alpha )} \int _{t}^{b} (u-t)^{ \alpha /k-1}h(u)\,\operatorname {d}u, \quad t< b, $$where
$$ \varGamma _{k}(\alpha )= \int _{0}^{\infty }u^{\alpha -1}e^{-u^{k}/k}\,\operatorname {d}u, \quad \mathbb{R}(\alpha )>0 $$and
$$ \varGamma _{k}(\alpha )=k^{\alpha /k-1}\varGamma \biggl( \frac{\alpha }{k} \biggr),\quad \mathbb{R}(\alpha )>0, k>0 $$ -
4.
If we take \(\varphi (u)=\frac{u}{\alpha }\exp (-\frac{1-\alpha }{ \alpha }u )\), the operators in (2.1) and (2.2) reduce to the right-sided and left-sided fractional integral operators with exponential kernel for \(\alpha \in (0,1)\)
$$ \mathcal{I}_{a^{+}}^{\alpha }h(t)=\frac{1}{\alpha } \int _{a}^{t} \exp \biggl(-\frac{1-\alpha }{\alpha }(t-u) \biggr)h(u)\,\operatorname {d}u, \quad t>a $$and
$$ \mathcal{I}_{b^{-}}^{\alpha }h(t)=\frac{1}{\alpha } \int _{t}^{b} \exp \biggl(-\frac{1-\alpha }{\alpha }(u-t) \biggr)h(u)\,\operatorname {d}u, \quad t< b $$which are defined in [9].
Recently, Sarikaya and Ertuğral [24] established the following trapezoid inequalities for generalized fractional integrals.
Theorem 2.1
([24])
Let \(h: [a,b]\to \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(\vert h'\vert \) is convex on \([a,b]\), then
where
Theorem 2.2
([24])
Let \(h: [a,b]\to \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(\vert h'\vert ^{q}\) is convex on \([a,b]\) for \(p,q>1\) and \(\frac{1}{p}+ \frac{1}{q}=1\), then
In [4], Ertuğral and Sarikaya established the following trapezoid inequalities for generalized fractional integrals.
Theorem 2.3
([4])
Let \(h: [a,b]\to \mathbb{R}\) be absolutely continuous on \(I^{\circ }\) such that \(h'\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) with \(a< b\). If the mapping \(\vert h'\vert \) is convex on \([a,b]\), then
where
Theorem 2.4
([4])
Let \(h: [a,b]\to \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(\vert h'\vert ^{q}\) for \(q>1\) is convex on \([a,b]\), then
where \(\frac{1}{p}+\frac{1}{q}=1\).
Most recently, Mohammed and Sarikayain [12] established some generalized fractional integral inequalities of midpoint and trapezoid types for twice differential functions.
Theorem 2.5
([12])
Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be a twice differentiable function on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If the function \(\vert h''\vert \) is convex on \([a,b]\), then
where
Theorem 2.6
([12])
Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be twice differentiable on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If \(\vert h''\vert ^{q}\) for \(q>1\) is convex on \([a,b]\), then
where \(\frac{1}{p}+\frac{1}{q}=1\).
Theorem 2.7
([12])
Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be twice differentiable on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If \(\vert h''\vert \) is convex on \([a,b]\), then
Theorem 2.8
([12])
Let \(h:I\subseteq \mathbb{R}\to \mathbb{R}\) be twice differentiable on \(I^{\circ }\) such that \(h''\in L_{1}([a,b])\) with \(a,b\in I^{\circ }\) and \(a< b\). If \(\vert h''\vert ^{q}\) for \(q>1\) is convex on \([a,b]\), then
where \(\frac{1}{p}+\frac{1}{q}=1\).
3 A generalized fractional integral identity
Before stating and proving our main results, we formulate the following important fractional integral identity.
Lemma 3.1
Let \(f:[a,b]\to \mathbb{R}\) be a differentiable function on \((a,b)\) with \(a< b\) such that \(f\in L_{1}([a,b])\). Then
where
Proof
Integrating by parts gives
Changing the variable \(x=\frac{3a+b}{4}t+\frac{a+b}{2}(1-t)\) yields
Similarly, we obtain
and
where we used the fact that \(\Delta (1)=\nabla (0)\). Adding \(I_{1}\), \(I_{2}\), \(I_{3}\), and \(I_{4}\) results in identity (3.1). The proof is thus completed. □
Remark 3.1
Since \(\Delta (1)=\nabla (0)\), we can write identity (3.1) in Lemma 3.1 as
Remark 3.2
Under assumptions of Lemma 3.1, if \(\varphi (t)=t\), then identity (3.1) reduces to
which has been proved in [26].
Remark 3.3
Under assumptions of Lemma 3.1, if \(\varphi (t)=\frac{t^{ \alpha }}{\varGamma (\alpha )}\), then identity (3.1) reduces to identity (1.2).
Remark 3.4
Under assumptions of Lemma 3.1, if \(\varphi (t)=\frac{t^{ \alpha /k}}{{k\varGamma _{k}(\alpha )}}\), then
Remark 3.5
Under assumptions of Lemma 3.1, applying \(\varphi (t)=\frac{t}{ \alpha }\exp (-\frac{1-\alpha }{\alpha }t )\) gives
for \(A=\frac{1-\alpha }{\alpha }\frac{b-a}{2}\).
4 Generalized fractional integral inequalities of Hermite–Hadamard type
Now we are in a position to state and prove our main results.
Theorem 4.1
Let \(f:[a,b]\to \mathbb{R}\) be a differentiable function on \((a,b)\) and \(f'\in L_{1}([a,b])\) for \(0\le a< b\) and \(\alpha >0\). If the mapping \(\vert f'\vert ^{q}\) for \(q\ge 1\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\), then
where the constants \(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\) are defined by
Proof
Using Lemma 3.1, the well-known power mean inequality, and the \((\alpha _{1},m)\)-convexity of \(\vert f'\vert ^{q}\) on \([0,\frac{b}{m} ]\) gives
This completes the proof. □
Remark 4.1
Under assumptions of Theorem 4.1, if \(\varphi (t)=t\) and \(m=\alpha _{1}=1\), then
which was proved in [26].
Remark 4.2
Under assumptions of Theorem 4.1, if \(\varphi (t)=\frac{t^{ \alpha }}{\varGamma (\alpha )}\), then the inequality in Theorem 4.1 reduces to inequality (1.3).
Corollary 4.1
Under assumptions of Theorem 4.1, if \(\varphi (t)=\frac{t^{ \alpha /k}}{{k\varGamma _{k}(\alpha )}}\), then
Corollary 4.2
Under assumptions of Theorem 4.1, if \(\alpha _{1}=1\) and \(\varphi (t)=\frac{t}{\alpha }\exp (-\frac{1-\alpha }{\alpha }t )\), then
where
Theorem 4.2
Let \(f:[a,b]\to \mathbb{R}\) be a differentiable function on \((a,b)\) and \(f'\in L_{1}([a,b])\) for \(0\le a< b\). If the mapping \(\vert f'\vert ^{q}\) is \((\alpha _{1},m)\)-convex on \([0,\frac{b}{m} ]\) for some \((\alpha _{1},m)\in (0,1]^{2}\), \(q\ge 1\), and \(q\ge r\ge 0\), then
where the constants \(C_{1}\), \(C_{2}\), \(D_{1}\), and \(D_{2}\) are defined by
Proof
By Lemma 3.1, the well-known Hölder inequality, and the \((\alpha _{1},m)\)-convexity of \(\vert f'\vert ^{q}\) on \([0,\frac{b}{m} ]\), we have
The required proof is complete. □
Remark 4.3
Under assumptions of Theorem 4.2, if \(\varphi (t)=t\), then inequality (4.1) reduces to
which was proved in [26].
Remark 4.4
Under assumptions of Theorem 4.2, if \(\varphi (t)=\frac{t^{ \alpha }}{\varGamma (\alpha )}\), then inequality (4.1) reduces to inequality (1.4).
Corollary 4.3
Under assumptions of Theorem 4.2, if \(\varphi (t)=\frac{t^{ \alpha /k}}{{k\varGamma _{k}(\alpha )}}\), then
Corollary 4.4
Under assumptions of Theorem 4.2, if \(r=0\) and \(\varphi (t)=\frac{t}{ \alpha }\exp (-\frac{1-\alpha }{\alpha }t )\), then
where \(\frac{1}{p}+\frac{1}{q}=1\) and \(A=\frac{1-\alpha }{\alpha } \frac{b-a}{2}\).
Remark 4.5
Under assumptions of Theorem 4.2, if \(A=\frac{1-\alpha }{ \alpha }\frac{b-a}{2}\), \(\alpha _{1}=r=1\), and \(\varphi (t)=\frac{t}{ \alpha }\exp (-\frac{1-\alpha }{\alpha }t )\), then Theorem 4.2 reduces to Corollary 4.2.
5 Conclusions
In this work, we establish generalized fractional integral inequalities, the Riemann–Liouville fractional integral inequalities, and some classical integral inequalities of the Hermite–Hadamard type for \((\alpha ,m)\)-convex functions. The results presented in this paper would provide generalizations and extensions of those given in earlier works.
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Qi, F., Mohammed, P.O., Yao, JC. et al. Generalized fractional integral inequalities of Hermite–Hadamard type for \({(\alpha,m)}\)-convex functions. J Inequal Appl 2019, 135 (2019). https://doi.org/10.1186/s13660-019-2079-6
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DOI: https://doi.org/10.1186/s13660-019-2079-6