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On some Hermite–Hadamard inequalities involving k-fractional operators
Journal of Inequalities and Applications volume 2021, Article number: 32 (2021)
Abstract
The main objective of this paper is to establish some new Hermite–Hadamard type inequalities involving k-Riemann–Liouville fractional integrals. Using the convexity of differentiable functions some related inequalities have been proved, which have deep connection with some known results. At the end, some applications of the obtained results in error estimations of quadrature formulas are also considered.
1 Introduction
In literature, inequalities are very important for convex functions especially the integral inequalities for convex functions originated form Hermite and Hadamard (see [11, p. 137]). The researchers have worked on Hermite–Hadamard type inequalities since 1893 [4]. The classical Hermite–Hadamard inequality reads as follows: if \(f :I\rightarrow \mathbb{R}\) is convex on the interval I of real numbers and \({a,b}\in I\) with \(a< b\), then
We note that the Hermite–Hadamard inequality may be assessed as a treatment of the conception of convexity. The Hermite–Hadamard inequality for convex functions has conferred revived awareness in the latest years and some unusual variations of essential and conclusion have been established (see, for example, [5, 6, 14, 17]). In the last few years, the theory of inequalities has progressed very fast. The evolution of the hypothesis associated with ancient inequalities has developed in a resumption of attentiveness in this field. In many classical inequalities, the Hermite–Hadamard inequality is one of the important inequality of analysis. Such an inequality has been applied for different types of problems of fractional calculus (see [1, 2, 8–10, 15, 16]). In this paper, as a continuation of the study of the Hermite–Hadamard inequality, we establish some results for k-Riemann–Liouville fractional integral by using the definition of convex functions via fractional calculus.
Below, let us recall first some basic concepts and some earlier results.
Definition 1
A function \(f :[a,b]\rightarrow \mathbb{R}\) is said to be convex on an interval \([a,b]\subseteq \mathbb{R}\), if
holds for \(\xi ,\eta \in [a,b]\) and \(\tau \in [0,1]\).
Definition 2
([13])
Let \(f \in L_{1}[a,b]\). The Riemann–Liouville integrals \(I_{a^{+}}^{\lambda }{f }\) and \(I_{b^{-}}^{\lambda }{f }\) of order \({\lambda }>0\) with \(a\geq 0\) are defined by
and
respectively, where Γ is the classical Gamma function and \(I_{a^{+}}^{0}{f }=I_{b^{-}}^{0}{f }=f (\xi )\).
Theorem 1
([14])
Consider \(f :[a,b]\rightarrow \mathbb{R}\) a positive mapping with \(0 \leq a< b\) and \(f \in L_{1}[a,b]\). If f is a convex function on \([a,b]\), then
Lemma 1.1
([14])
Suppose \(f :[a,b]\rightarrow \mathbb{R}\) is a differentiable function on \((a,b)\) with \(a< b\). If \(f ^{\prime }\in \L [a,b]\), then
Theorem 2
([14])
Assume that \(f :[a,b]\rightarrow \mathbb{R}\) is a differentiable function on \((a,b)\) with \(a< b\). If \(|f ^{\prime }|\) is convex on \([a,b]\), then
Theorem 3
([12])
Let \(f :[a,b]\rightarrow \mathbb{R}\) be a positive mapping with \(0 \leq a< b\) and \(f \in L_{1}[a,b]\). If f is a convex function on \([a,b]\), then
Lemma 1.2
([12])
Suppose \(f :[a,b]\rightarrow \mathbb{R}\) is a differentiable function on \((a,b)\) with \(a< b\). If \(f ^{\prime }\in \L [a,b]\), then
Theorem 4
([12])
Consider \(f :[a,b]\rightarrow \mathbb{R,}\) a differentiable function on \((a,b)\) with \(a< b\). If \(|f ^{\prime }|^{h}\) is convex on \([a,b]\) for \(h\geq {1}\), then
2 Hermite–Hadamard’s inequalities for k-fractional integrals
In [3], the k-gamma function was introduced by Diaz et al. as follows.
Definition 3
Let k and \(\mathbb{R}(v)\) be positive. Then the k-gamma function is defined by following integral:
Definition 4
([7])
If \(k>0\), Let \(f \in L_{1}(a,b)\), \(a\geq 0\), then k-Riemann–Liouville fractional integrals \(I_{a^{+},k}^{\lambda }{f }\) and \(I_{b^{-},k}^{\lambda }{f }\) of order \({\lambda }>0\) for a real-valued continuous function \(f(\mu )\) are defined by
and
respectively. Here \(\Gamma _{k}\) is the k-Gamma function.
Theorem 5
Let \(k>0\), \(I^{\lambda }_{a^{+},k}f \) and \(I^{\lambda }_{b^{-},k}f \) be the left and right sided k-Riemann–Liouville fractional integral of order \({\lambda }>0\). Let \(f :[a,b]\rightarrow \mathbb{R}\) be positive mapping with \(0\leq a< b\), \(f \in L_{1}[a,b]\). If f is convex on \([a,b]\), then
Proof
As f is convex mapping on \([a,b]\), we have, for \(\xi ,\eta \in [a,b]\) with \(\tau =\frac{1}{2}\) in (1),
Now let \(\xi =\mu a+(1-\mu )b\) and \(\eta =\mu b+(1-\mu )a\), then (9) becomes
Multiplying both sides of (10) by \(\mu ^{\frac{\lambda }{k}-1}\), then integrating with respect to μ over \({[0,1]}\), we get
where
and
By taking \(\mu a+(1-\mu )b=\phi \) in \(I_{1}\) and \(\mu b+(1-\mu )a=\omega \) in \(I_{2}\), we get
and
Substituting the values of \(I_{1}\) and \(I_{2}\) from (12) and (13) in (11), we get
which implies that
This completes the first inequality in (8). To complete the second inequality, we note that if f is convex, then, for \({\tau }\in [0,1]\), it yields that
and
By adding the above two inequalities, we get
Multiplying by \(\mu ^{\frac{\lambda }{k}-1}\) on both sides of (15), then integrating with regard to μ over \([0,1]\), we get
We denote
and
Putting \(\phi =\mu a+(1-\mu )b\) in \(K_{1}\), and \(\omega =\mu b+(1-\mu )a\) in \(K_{2}\), we obtain
and
Substituting the values of \(K_{1}\) and \(K_{2}\) from (17) and (18) in (16), we get
which implies that
By combining (17), and (19), we get (8). □
Lemma 2.1
Let \(k>0\), \(I^{\lambda }_{a^{+},k}f \) and \(I^{\lambda }_{b^{-},k}f \) be defined as Definition 4. Let \(f :[a,b]\rightarrow \mathbb{R}\) be a differentiable mapping on \((a,b)\) with \(a< b\). If \(f ^{\prime }\in \L [a,b]\), then
Proof
Let us consider
which, we can write as
Integrating \(I_{1}\) by parts, we get
Setting \(\xi =\mu a+(1-\mu )b\), then after some calculation, we get
Now integrating \(I_{2}\) by parts to get
Setting \(\xi =(\mu a+(1-\mu )b)\), after some calculation, we get
Applying (22) and (23) in (21), it follows that
or
Multiplying both sides of (24) by \(\frac{b-a}{2}\) to get the required result. □
Theorem 6
Let \(k>0\), \(I^{\lambda }_{a^{+},k}f \) and \(I^{\lambda }_{b^{-},k}f \) be defined as Definition 4. Let \(f :[a,b]\rightarrow \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(|f ^{\prime }|\) is convex on \([a,b]\), then
Proof
By using Lemma 2.1 and the definition of a convex function of \(|f ^{\prime }|\), we have
where
and
We calculate \(K_{1}\) to get
Similarly we can calculate \(K_{2}\) and get
Substituting the values of \(K_{1}\) and \(K_{2}\) in (26) and after some calculations, we get (25). □
3 Some more fractional inequalities for convex functions
Definition 5
Let \(f \in L_{1}[a,b]\). The k-Riemann–Liouville integrals \(I^{\lambda }_{(\frac{a+b}{2})^{+}, k}f \) and \(I^{\lambda }_{(\frac{a+b}{2})^{-}, k}f \) of order \({\lambda }>0\) and \(k>0\) with \(a\geq 0\) are defined by
and
respectively. Here \({\Gamma _{k}{(\lambda )}}\) is the k-Gamma function.
Theorem 7
Let \(k>0\), \(I^{\lambda }_{(\frac{a+b}{2})^{+}, k}f \) and \(I^{\lambda }_{(\frac{a+b}{2})^{-}, k}f \) be defined in Definition 5. Let \(f :[a,b]\rightarrow \mathbb{R}\) be positive mapping with \(0\leq a< b\) and \(f \in L_{1}[a,b]\). If f is a convex function on \([a,b]\), then
Proof
As f is convex function on \([a,b]\), we have, for \(\xi ,\eta \in [a,b]\) with \(\tau =\frac{1}{2}\),
Putting \(\xi =\frac{\mu a}{2}+\frac{(2-\mu )b}{2}\) and \(\eta =\frac{ \mu b}{2}+\frac{(2-\mu )a}{2}\), then (30) becomes
Multiplying by \(\mu ^{\frac{\lambda }{k}-1}\) on both sides of (31), then integrating with respect to μ over \({[0,1]}\), we get
We set
Taking \(\phi =\frac{\mu a}{2}+\frac{(2-\mu )b}{2}\), after some calculations we get
and we set
Putting \(\omega =\frac{\mu b}{2}+\frac{(2-\mu )a}{2}\) to get
Substituting the values of \(I_{1}\) and \(I_{2}\) from (33) and (35) in (32), we get
The first part of the inequality is proved. To complete the second inequality, we note that if f is convex function, then, for \({\tau }\in [0,1]\), showing
and
By adding the above two inequalities, we get
Multiplying by \(\mu ^{\frac{\lambda }{k}-1}\) on both sides of (37) and integrating inequalities with respect to μ over \([0,1]\), we get
We take
and choose \(\phi =\frac{\mu a}{2}+\frac{(2-\mu )b}{2}\), we get after some simple calculations
Likewise we take
and choose \(\omega =\frac{\mu b}{2}+\frac{(2-\mu )a}{2}\), we get
Substituting the values of \(L_{1}\) and \(L_{2}\) from (39) and (40) in (38), we get
This implies that
From (36) and (41), we get the required result. □
Lemma 3.1
Let \(k>0\), \(I^{\lambda }_{(\frac{a+b}{2})^{+}, k}f \) and \(I^{\lambda }_{(\frac{a+b}{2})^{-}, k}f \) be defined as Definition 5. Let \(f :[a,b]\rightarrow \mathbb{R}\) be differentiable on \((a,b)\) with \(a< b\). If \(f ^{\prime }\in \L [a,b]\), then
Proof
Let
Note that
Substituting \(\xi =\frac{\mu a}{2}+\frac{(2-\mu )b}{2}\), we get after some computations
Similarly we can write for \(I_{2}\)
Taking \(\xi =\frac{\mu b}{2}+\frac{(2-\mu )a}{2}\), we get
By using (43) and (44), it follows that
Thus, multiplying \(\frac{b-a}{4}\) on both sides of the above, we get (42). □
Theorem 8
Let \(k>0\), \(I^{\lambda }_{(\frac{a+b}{2})^{+}, k}f \) and \(I^{\lambda }_{(\frac{a+b}{2})^{-}, k}f \) be defined as Definition 5. Let \(f :[a,b]\rightarrow \mathbb{R}\) be a differentiable function on \((a,b)\) with \(a< b\). If \(|f ^{\prime }|^{h}\) is convex on \([a,b]\) for \(h\geq {1}\), then
Proof
First, we consider the case of \(h=1\). By using Lemma 3.1, and the definition of convex function of \(|f ^{\prime }|\), we obtain
Now we consider the case of \(h>1\). By using Lemma 3.1, the Holder inequality and the definition of convex function of \(|f ^{\prime }|^{h}\), we get
This completes the proof. □
Remark 1
Our results described in the above theorems coincide with the results of [14] and [12] by replacing \(I_{a^{+},k}^{\lambda }{f (\xi )}\) by \(I_{a^{+}}^{\lambda }{f (\xi )}\).
4 Applications to quadrature formulas
In this section we apply the obtained results in to the error estimations of quadrature formulas. It is shown that our main results contain as special cases results such as mid-point inequality and trapezoid inequality. Also, the Hermite–Hadamard inequality can be deduced directly from our main results.
Proposition 1
(Hermite–Hadamard inequality)
By using the assumptions of Theorem 5with \(\lambda =1\) and \(k=1\), we get the following
Proposition 2
(Mid-point inequality)
By using the assumptions of Theorem 8with \(\lambda =1\), \(h=1\) and \(k=1\), we get the following mid-point type inequality:
Proposition 3
(Trapezoid inequality)
By using the assumptions of Theorem 6with \(\lambda =1\) and \(k=1\), we get the following trapezoid inequality:
5 Conclusion
We have discussed some Hermite–Hadamard type inequalities for k-Riemann–Liouville fractional integral using the convexity of differentiable functions. We stated our main results by Theorems 5, 6, 7 and 8, and showed that our results contain some existing results as special cases. As applications we have established two inequalities involving the error estimates of quadrature formulas.
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All authors are thankful to the careful referee and editor for their suggestions which help us to improve the final version of paper.
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This work was supported by the Natural Science Foundation of Fujian Province of China (No. 2016J01023).
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Wu, S., Iqbal, S., Aamir, M. et al. On some Hermite–Hadamard inequalities involving k-fractional operators. J Inequal Appl 2021, 32 (2021). https://doi.org/10.1186/s13660-020-02527-1
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DOI: https://doi.org/10.1186/s13660-020-02527-1