Hermite – Jensen – Mercer Type Inequalities for Caputo Fractional Derivatives

Inequality (3) is known as the Jensen–Mercer inequality. Recently, inequality (3) has been generalized, see ([12–15]). For more recent and related results connected with Jensen–Mercer inequality, see ([11, 16–18]). *e previous era of fractional calculus is as old as the history of differential calculus. Several fractional operators are introduced that generalize ordinary integrals. However, the fractional derivatives have some basic properties than the corresponding classical ones. On the contrary, besides the smooth requirement, the Caputo derivative does not coincide with the classical derivative [19]. Caputo fractional derivatives are introduced by the Italian mathematician Caputo in 1967. Since then, a lot of research involves Caputo fractional derivatives [20–22]. *e Caputo fractional derivatives are defined as in [23–26].

e Jensen inequality [10] states that h is a convex function on the interval [u, v]; then, where ∀x r ∈ [u, v] and all μ r ∈ [0, 1], (r � 1, 2, . . . , n). e Hermite-Hadamard inequality asserts that if a mapping h: J⊆R ⟶ R is a convex function on J with u, v ∈ J, u < v, then e reverse direction in the above inequality holds when h is concave.
e Caputo fractional derivatives of order α are defined as follows: If α � n ∈ 1, 2, 3, . . . { } and usual derivatives of h of order n exist, then the Caputo fractional derivatives Specifically, we get where n � 1 and α � 0. In this article, by using the Jensen-Mercer inequality, we proved Hermite-Hadamard's inequalities for fractional integrals and established some Hermite-Hadamard type inequalities for differentiable mappings whose derivatives in absolute value are convex.

Hermite-Hadamard-Mercer Type Inequalities for Caputo Fractional Derivatives
By using Jensen-Mercer inequalities, Hermite-Hadamard type inequalities can be expressed in Caputo fractional derivatives as follows.

Theorem 2. Suppose that a positive function
is a convex function on [u, v], then the following inequalities for Caputo fractional derivatives hold: ∀x, y ∈ [u, v], α > 0, and Γ(·) is the gamma function.
Proof. To prove the first part of inequality, we use convexity of h (n) to get Multiplying both sides of (16) by τ n− α− 1 and then integrating with respect to τ over [0, 1], we get Further simplifying gives and so the first inequality of (14) is proved. Now, for the proof of the second inequality of (14), we first note that if h (n) is a convex function, then by employing Jensen-Mercer inequality (3) for τ ∈ [0, 1] gives By adding the inequalities of (19) and (20), we get Multiplying both sides by τ n− α− 1 and then integrating over τ ∈ [0, 1], we get After simplification, we get Now, concatenating (18) and (23), we get (14). Now, we introduce some new lemmas involving Caputo fractional derivatives. [u, v], then the following equality for Caputo fractional derivatives holds: where ∀x, y ∈ [u, v], α > 0, τ ∈ [0, 1], and Γ(·) is the gamma function.
Proof. It suffices to note that where and replacing the values of I 1 and I 2 in (25) Proof. It suffices to note that Journal of Function Spaces and combining (29) and (30) with (28), we get (27). □ Remark 2. If we take x � u and y � v in Lemma 2, then it reduces to Lemma 2 in [24].
Proof. By using Lemma 1 and the Jensen-Mercer inequality, we get where
Proof. By using Lemma 2 and applying the famous Hölder integral inequality, we get By applying Minkowski's inequality, we get Journal of Function Spaces 7 which completes the proof.

Conclusion
In this paper, we presented Hermite-Hadamard-Mercer inequalities for convex functions via Caputo fractional derivatives. We also developed some new bounds using Hölder-Iscan and improved power-mean integral inequalities. Our results will attract attentions of many researchers working in the field of inequalities and enable them to think further for other generalized convex functions. One may think to extend these results for higher order convex functions. e works above can also build up for a convex function of two variables. For further directions, we refer to [28][29][30][31][32][33].

Data Availability
All data are included within this paper.

Conflicts of Interest
e authors of this paper declare that they have no conflicts of interest.