New Integral Inequalities via the Katugampola Fractional Integrals for Functions Whose Second Derivatives Are Strongly η-Convex

In this paper, we introduced some new integral inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute values at certain powers are strongly η-convex functions via the Katugampola fractional integrals.


Introduction
Let I be an interval in R. A function f : I → R is said to be convex on I if for all x, y ∈ I and t ∈ [0, 1].The following inequalities which hold for convex functions is known in the literature as the Hermite-Hadamard type inequality.
Remark 1.If we take η(x, y) = x − y in Definition 1, then we recover the classical definition of convex functions.
Remark 2. If η(x, y) = x − y in Definition 2, then we have the class of strongly convex functions.
For some recent results related to the class of η-convex functions, we refer the interested reader to the papers [8,[12][13][14][15][16]. Definition 3 ([17]).The left-and right-sided Riemann-Liouville fractional integrals of order α > 0 of f are defined by Definition 4 ([18]).The left-and right-sided Hadamard fractional integrals of order α > 0 of f are defined by In 2011, Katugampola [19] introduced a new fractional integral operator which generalizes the Riemann-Liouville and Hadamard fractional integrals as follows: Definition 6.Let [a, b] ⊂ R be a finite interval.Then, the left-and right-sided Katugampola fractional integrals of order α > 0 of f ∈ X p c (a, b) are defined by Remark 3. It is shown in [19] that the Katugampola fractional integral operators are well-defined on X p c (a, b).
For more information about the Katugampola fractional integrals and related results, we refer the interested reader to the papers [19][20][21].Recently, Chen and Katugampola [20] introduced several integral inequalities of Hermite-Hadamard type for functions whose first derivatives in absolute value are convex functions via the Katugampola fractional integrals.We present two of their results here for the purpose of our discussion.The first result of importance to us employs the following lemma.Lemma 1 ([20]).Let α > 0, ρ > 0 and f : [a ρ , b ρ ] → R be a differentiable mapping on (a ρ , b ρ ) with 0 ≤ a < b.Then the following equality holds if the fractional integrals exist: By using Lemma 1, the authors proved the following result.
then the following inequality holds: The second result of importance to us also uses the following lemma.
then the following inequality holds: It is important to note that Lemmas 1 and 2 are corrected versions of [20] (Lemma 2.4 and Equation ( 14)).
Our purpose in this paper is to provide some new estimates for the right hand side of the inequalities in Theorems 3 and 4 for functions whose second derivatives in absolute value at some powers are strongly η-convex.

Main Results
To prove the main results of this paper, we need the following lemmas which are extensions of Lemmas 1 and 2 for the second derivative case of the function f .
Then the following equality holds if the fractional integrals exist:

Proof.
Let By using integration by parts we have that By a similar argument, one gets: Using ( 4) and ( 5), we have The desired identity in (3) follows from (6) by using (1) and rearranging the terms.
Then the following equality holds if the fractional integrals exist: Proof.We start by considering the following computation which is a direct application of integration by parts.
The intended identity in (7) follows from (8) by using (2) and rearranging the terms.
We are now in a position to prove our main results.
Proof.Using Lemma 3, the well-known power mean inequality and the strong η-convexity of | f | q , we obtain .
The desired inequality follows from the above estimation and observing that: This completes the proof of Theorem 5.
for q ≥ 1, then the following inequality holds: .
Proof.The result follows directly from Theorem 5 if we take η(x, y) = x − y and µ = 0. Theorem 6.Let α > 0, ρ > 0 and f : [a ρ , b ρ ] → R be a twice differentiable mapping on (a ρ , b ρ ) with 0 ≤ a < b.If | f | q is strongly η-convex with modulus µ ≥ 0 for q > 1, then the following inequalities hold: Proof.Using Lemma 3, the Hölder's inequality and the strong η-convexity of | f | q , we obtain . This proves the first inequality.To prove the second inequality, we observe that for any A > B ≥ 0 and s ≥ 1, we have (A − B) s ≤ A s − B s .Thus, it follows that 1 − u α+1 s ≤ 1 − u s(α+1) for all u ∈ [0, 1].Hence, we have that This completes the proof.
then the following inequalities hold: Proof.The result follows directly from Theorem 6 if we take η(x, y) = x − y and µ = 0.
with modulus µ ≥ 0 for q ≥ 1, then the following inequality holds: , where B(•, •) denotes the beta function defined by B(x, y) = Proof.Using Lemma 4, the power mean inequality and the strong η-convexity of | f | q , we obtain .
The desired result follows from the above inequality and using the following computations: This completes the proof of the theorem. .
Proof.The result follows directly from Theorem 7 if we take η(x, y) = x − y and µ = 0.
with modulus µ ≥ 0 for q > 1, then the following inequalities hold: Proof.Using Lemma 4, the Hölder's inequality and the strong η-convexity of | f | q , we obtain This proves the first inequality.Using a similar argument as in the proof of Theorem 6, we obtain This completes the proof of the theorem.
Proof.The result follows directly from Theorem 8 if we take η(x, y) = x − y and µ = 0.