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Some Hermite–Jensen–Mercer type inequalities for k-Caputo-fractional derivatives and related results
Advances in Difference Equations volume 2020, Article number: 262 (2020)
Abstract
In this paper, certain Hermite–Hadamard–Mercer type inequalities are proved via k-Caputo fractional derivatives. We established some new k-Caputo fractional derivatives inequalities with Hermite–Hadamard–Mercer type inequalities for differentiable mapping \(\psi^{(n)}\) whose derivatives in the absolute values are convex.
1 Introduction
Let \(0< u_{1}\leq u_{2}\leq\cdots\leq u_{n}\) and let \(\mu= (\mu _{1},\mu_{2},\ldots,\mu_{n} )\) be non-negative weights such that \(\sum_{i=1}^{n} \mu_{i}=1\). The famous Jensen inequality [1] states that if ψ is a convex function on the interval \([\theta _{1},\theta_{2} ]\), then
for all \(u_{i}\in [ \theta_{1},\theta_{2} ]\) and \(\mu_{i}\in [ 0,1 ]\) (\(i=1,2,\ldots,n \)).
In 1883, the Hermite–Hadamard (H-H) inequality was considered the most useful inequality in mathematical analysis. It is also known as the classical H-H inequality.
The Hermite–Hadamard inequality asserts that if \(\psi:J\subseteq{R}\to {R}\) is a convex function defined on J and \(\theta_{1},\theta_{2}\in J\) such that \(\theta_{1}<\theta_{2}\), then
For recent results related with the Jensen–Mercer inequality, see [1–4].
Theorem 1
Ifψis a convex function on\([ \theta_{1},\theta_{2} ]\), then
\(\forall u_{i}\in [ \theta_{1},\theta_{2} ]\)and all\(\mu _{i}\in [ 0,1 ]\) (\(i=1,2,\ldots,n \)).
Inequality (2) is known as the Jensen–Mercer inequality. Recently, inequality (2) has been studied and generalized in [5–7].
Fractional calculus was generally a study kept for the best minds in mathematics. The early era of fractional calculus is as old as the history of differential calculus. One generalized the differential operators and ordinary integrals. However, the fractional derivatives have some more basic properties than the corresponding classical ones. On the other hand, besides the smooth requirement, the Caputo derivative does not coincide with the classical derivative [8]. It was introduced in 1967.
In the following, we give the definition of Caputo fractional derivatives (see [9–11] and the references therein).
Definition 1
Let \(\alpha>0\) and \(\alpha\notin \{ 1,2,3,\dots \}\), \(n= [ \alpha ] +1\), \(\psi\in C^{n} [ \theta_{1},\theta_{2} ] \). The Caputo fractional derivatives of order α are defined as follows:
and
If \(\alpha=n \in \{ 1,2,3,\ldots \}\) and the usual derivatives of ψ of order n exist, then the Caputo fractional derivatives \(( {}^{ c}D_{\theta_{1}^{+}}^{\alpha}\psi ) (u )\) coincide with \(\psi^{(n)} ( u ) \).
In particular, we have
where \(n=1\) and \(\alpha=0\).
Definition 2
(See [12])
Diaz and Parigun have defined the k-Gamma function \(\varGamma_{k}\), a generalization of the classical Gamma function, which is given by the following formula:
It is shown that the Mellin transform of the exponential function \(e^{-\frac{t^{k}}{k}} \) is the k-Gamma function given by
Obviously, \(\varGamma_{k}(x+k)=x \varGamma_{k}(x)\), \(\varGamma(x)=\lim_{k \rightarrow1}\varGamma_{k} (x)\) and \(\varGamma_{k}(x)=k^{\frac{x}{k}-1} \varGamma (\frac{x}{k})\).
Definition 3
([13])
Let \(\alpha>0\), \(k\geq1\) and \(\alpha\notin \{ 1,2,3,\dots \}\), \(n= [\alpha ] +1\), \(\psi\in C^{n} [\theta_{1},\theta_{2} ]\). The right-sided and left-sided Caputo k-fractional derivatives of order α are defined as follows:
and
For \(k=1\), Caputo k-fractional derivatives give the definition of Caputo fractional derivatives.
In this article, by using the Jensen–Mercer inequality, we prove Hermite–Hadamard inequalities for fractional integrals and we establish some new Caputo k-fractional derivatives connected with the left and right sides of Hermite–Hadamard type inequalities for differentiable mappings whose derivatives in absolute values are convex.
Throughout the paper, we need the following assumptions.
\(A_{1} = \forall u,v\in [ \theta_{1},\theta_{2} ]\), \(\alpha >0\), \(k\geq1 \) and \(\varGamma_{k} ( \cdot )\) is the k-Gamma function.
2 Hermite–Hadamard–Mercer type inequalities for Caputo k-fractional derivatives
By using the Jensen–Mercer inequality, Hermite–Hadamard type inequalities can be expressed in Caputo k-fractional derivative form as follows.
Theorem 2
Suppose that if\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a positive function with\(0\leq\theta_{1}<\theta_{2}\)and\(\psi\in C^{n} [ \theta_{1},\theta_{2} ] \). If\(\psi^{(n)}\)is a convex function on\([\theta_{1},\theta_{2}]\)along with the assumptions in\(A_{1}\), then the following inequalities for Caputok-fractional derivatives hold:
Proof
Using the Jensen–Mercer inequality, we have
for all \(w,z \in [ \theta_{1},\theta_{2} ]\).
Now by change of variables \(w=\lambda u+(1-\lambda)v \) and \(z=(1-\lambda )u+\lambda v \), for all \(u,v\in [ \theta_{1},\theta_{2} ]\) and \(\lambda\in [ 0,1 ] \) in (6), we have
Multiplying both sides by \(\lambda^{n-\frac{\alpha}{k}-1}\) above and then integrating the resulting inequality with respect to λ over \([ 0,1 ]\), we have
hence
and so the first inequality of (5) is proved.
Now for the proof of second inequality of (5), we first note that if \(\psi^{(n)}\) is a convex function, then for \(\lambda\in [ 0,1 ]\), it gives
Multiplying both sides by \(\lambda^{n-\frac{\alpha}{k}-1}\) above and then integrating the resulting inequality with respect to λ over \([ 0,1 ]\), we have
hence
Multiplying by \((-1)\) on both sides, we have
Adding \(\psi^{(n)}(\theta_{1})+\psi^{(n)}(\theta_{2})\) in both sides in (7), we get the second inequality of (5). □
Remark 1
If we take \(k=1\) in Theorem 2, then it reduces to Theorem 2 in [14].
Theorem 3
Suppose that if\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a positive function with\(0\leq\theta_{1}<\theta_{2}\)and\(\psi\in C^{n} [ \theta_{1},\theta_{2} ] \). If\(\psi^{(n)}\)is a convex function on\([\theta_{1},\theta_{2}]\)along with the assumptions in\(A_{1}\), then the following inequalities for the Caputok-fractional derivatives hold:
Proof
To prove the first part of the inequality, we use the convexity of \(\psi^{(n)}\),
for all \(u_{1}, v_{1} \in[\theta_{1},\theta_{2}]\). Now by writing the variables \(u_{1}=\frac{\lambda}{2}u+\frac{2-\lambda}{2}v \) and \(v_{1}=\frac{2-\lambda}{2}u+\frac{\lambda}{2}v\), for \(u,v\in[\theta _{1},\theta_{2}]\) and \(\lambda\in[0,1]\), we get
Multiplying both sides by \(\lambda^{n-\frac{\alpha}{k}-1}\) above and then integrating the resulting inequality with respect to λ over \([ 0,1 ]\), we have
hence
and so the first inequality of (8) is proved.
Now for the proof of second inequality of (5), we first note that if \(\psi^{(n)}\) is a convex function, then for \(\lambda\in [ 0,1 ]\), it yields
and
By adding the inequalities of (10) and (11), we have
Multiplying both sides by \(\lambda^{n-\frac{\alpha}{k}-1}\) in above and then integrating the resulting inequality with respect to λ over \([ 0,1 ]\), we have
This implies
Multiplying (12) by \(\frac{ (n-\frac{\alpha}{k} )}{2}\),
Combining (9) and (13), we get (8). □
Remark 2
If we take \(k=1\) in Theorem 3, then it reduces to Theorem 3 in [14].
Lemma 1
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(0\leq \theta_{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \)along with the assumptions in\(A_{1}\), then the following equality for Caputok-fractional derivatives holds:
Proof
It suffices to note that
where
and
Combining (16) and (17) with (15) and get (14). □
Remark 3
If we take \(k=1\) in Lemma 1, then it reduces to Lemma 1 in [14].
Remark 4
If we take \(u=a\) and \(v=b\) in Lemma 1, then it reduces to Remark 2.5 in [11].
Lemma 2
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(0\leq \theta_{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \)along with the assumptions in\(A_{1}\), then the following equality for Caputok-fractional derivatives holds:
Proof
It suffices to note that
where
and
Combining (20) and (21) with (19), we get (18). □
Remark 5
If we take \(k=1\) in Lemma 2, then it reduces to Lemma 2 in [14].
Remark 6
If we take \(u=a\) and \(v=b\) in Lemma 2, then it reduces to Lemma 2 in [10].
Theorem 4
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(0\leq \theta_{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \). If\(|\psi^{(n+1)}|\)is a convex function on\([\theta_{1},\theta_{2}] \)along with the assumptions in\(A_{1}\), then the following inequality for Caputok-fractional derivatives holds:
Proof
By using Lemma 1 and the Jensen–Mercer inequality, we have
where
and
Combining (24) and (25) with (23) and we get (22). This completes the proof. □
Remark 7
If we take \(k=1\) in Theorem 4, then it reduces to Theorem 4 in [14].
Remark 8
If we take \(u=a\) and \(v=b\) in Theorem 4, then it reduces to Corollary 2.7 in [11].
Theorem 5
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(0\leq \theta_{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \). If\(|\psi^{(n+1)}|\)is a convex function on\([\theta_{1},\theta_{2}] \)along with the assumptions in\(A_{1}\), then the following inequality for Caputok-fractional derivatives holds:
Proof
By using Lemma 2 and the Jensen–Mercer inequality, we have
This completes the proof. □
Remark 9
If we take \(k=1\) in Theorem 5, then it reduces to Theorem 5 in [14].
Theorem 6
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(0\leq \theta_{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \). If\(|\psi^{(n+1)}|^{q}\)is a convex function on\([\theta_{1},\theta _{2}]\), \(q>1\)and along with the assumptions in\(A_{1}\), then the following inequality for Caputok-fractional derivatives holds:
Proof
By using Lemma 2 and applying the Hölder integral inequality, we have
By the convexity of \(|\psi^{(n+1)}|^{q}\), we have
This completes the proof. □
Remark 10
If we take \(k=1\) in Theorem 6, then it reduces to Theorem 6 in [14].
3 New Hölder and improved İşcan inequalities
Theorem 7
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(0\leq \theta_{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \). If\(|\psi^{(n+1)}|^{q}\)is a convex function on\([\theta_{1},\theta _{2}]\), \(q>1\)and along with the assumptions in\(A_{1}\), then the following inequality for Caputok-fractional derivatives holds:
Proof
By using Lemma 2 with Jensen–Mercer inequality and applying the Hölder–İşcan integral inequality [Theorem 1.4, [15]], we have
By the convexity of \(|\psi^{(n+1)}|^{q}\)
It is easy to see that
and
and
and
and
and
By combining (31), (32), (33), (34), (35), (36), with (29) we get (28).
This completes the proof. □
Remark 11
If we take \(k=1\) in Theorem 7, then it reduces to Theorem 7 in [14].
Theorem 8
Suppose that\(\psi: [ \theta_{1},\theta_{2} ] \rightarrow R\)is a differentiable mapping on\((\theta_{1},\theta_{2} )\)with\(\theta _{1}<\theta_{2}\)and\(\psi\in C^{n+1} [ \theta_{1},\theta_{2} ] \). If\(|\psi^{(n+1)}|^{q}\)is a convex function on\([\theta_{1},\theta_{2}]\), \(q\geq1\)and along with the assumptions in\(A_{1}\), then the following inequality for Caputok-fractional derivatives holds:
Proof
By using Lemma 2 with the Jensen–Mercer inequality and applying the improved power-mean integral inequality [Theorem1.5, [15]], we have
It is easy to see that
and
and
and
and
and
By combining (39), (40), (41), /(42), (43), (44) with (38) we get (37), which completes the proof. □
4 Conclusion
In this article, we show Hermite–Hadamard type inequalities can be expressed in Caputo k-fractional derivative form by employing the Jensen–Mercer inequality. New Hermite–Jensen–Mercer type inequalities using Caputo k-fractional derivatives are established for differentiable mappings whose derivatives in absolute values are convex. Some known results are recaptured as special cases of our results. We hope that our new idea and technique may inspire many researcher in this fascinating field.
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Acknowledgements
The authors are thankful to the reviewers for their valuable comments.
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This work was sponsored in part by The Key Research Base Project of Sichuan Provincial Education Department(KJJR2019-001);the Sichuan Provincial Philosophy and Social Science Planning Project of China(SC19B110);the Sichuan Provincial Quality Engineering Project(2019JWC024). The research of Saad Ihsan Butt has been fully supported by H.E.C. of Pakistan under NPRU project 7906.
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Zhao, S., Butt, S.I., Nazeer, W. et al. Some Hermite–Jensen–Mercer type inequalities for k-Caputo-fractional derivatives and related results. Adv Differ Equ 2020, 262 (2020). https://doi.org/10.1186/s13662-020-02693-y
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DOI: https://doi.org/10.1186/s13662-020-02693-y
Keywords
- Convex functions
- Hermite–Hadamard inequalities
- Jensen inequality
- Jensen–Mercer inequality
- k-Caputo fractional derivatives