Some Hermite–Jensen–Mercer type inequalities for k-Caputo-fractional derivatives and related results

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 ψ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\psi^{(n)}$\end{document} whose derivatives in the absolute values are convex.

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.
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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][10][11] and the references therein).
Definition 2 (See [12]) Diaz and Parigun have defined the k-Gamma function Γ 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 -t k k is the k-Gamma function given by 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.
is the k-Gamma function.

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.
Remark 1 If we take k = 1 in Theorem 2, then it reduces to Theorem 2 in [14].
along with the assumptions in A 1 , then the following inequalities for the Caputo k-fractional derivatives hold: Proof To prove the first part of the inequality, we use the convexity of ψ (n) , Multiplying both sides by λ n-α k -1 above and then integrating the resulting inequality with respect to λ over [0, 1], we have Now for the proof of second inequality of (5), we first note that if ψ (n) is a convex function, then for λ ∈ [0, 1], it yields and By adding the inequalities of (10) and (11), we have Multiplying both sides by λ n-α k -1 in above and then integrating the resulting inequality with respect to λ over [0, 1], we have This implies 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].
along with the assumptions in A 1 , then the following equality for Caputo k-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].
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].
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].
along with the assumptions in A 1 , then the following inequality for Caputo k-fractional derivatives holds: Proof By using Lemma 2 and the Jensen-Mercer inequality, we have by using calculus tools, we obtain This completes the proof.
Remark 9 If we take k = 1 in Theorem 5, then it reduces to Theorem 5 in [14].
Proof By using Lemma 2 and applying the Hölder integral inequality, we have By the convexity of |ψ (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].
Remark 11 If we take k = 1 in Theorem 7, then it reduces to Theorem 7 in [14].

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.