On some new generalized fractional inequalities for twice differentiable functions

In this paper, we establish an identity involving Sarikaya fractional integrals for twice differentiable functions. We obtain some new generalized fractional inequalities for the functions whose second derivatives in absolute value are convex by utilizing obtained equality. Utilizing the new inequalities obtained, some new inequalities for Riemann–Liouville fractional integrals and k-Riemann–Liouville fractional integrals are obtained. In addition, some of these results generalize ones obtained in earlier works.

Many mathematicians have been working on twice differentiable functions recently. For instance, the inequalities for twice differentiable convex mappings associated with Hadamard's inequality in [3,4] were obtained. Moreover, Mohammed and Sarikaya established some new generalized fractional integral inequalities of midpoint and trapezoid type for twice differentiable convex functions in [19]. Sarıkaya and Aktan [22] established some new Simpson and the Hermite-Hadamard type inequalities for functions whose absolute values of derivatives are convex. In addition Hezenci et al. obtained several fractional Simpson's inequality for twice differentiable functions. In [8], some generalizations of integral inequalities of Bullen-type for twice differentiable functions involving Riemann-Liouville fractional integrals were obtained.
The goal of our research is, by using Sarikaya fractional integrals, to obtain new generalized inequalities for functions whose second derivatives in absolute values are convex functions. Some new results with some special choices will establish generalizations and connections for the classical midpoint inequalities and for the midpoint inequalities obtained for Riemann-Liouville, k-Riemann-Liouville, and different generalized fractional integrals.
Here, we give some definitions and notations which are used frequently in main section. The well-known gamma and beta are defined as follows: For 0 < x, y < ∞, and x, y ∈ R, The generalized fractional integrals were introduced by Sarikaya and Ertugral as follows: Definition 1.1 [24] Let us note that a function ϕ : [0, ∞) → [0, ∞) satisfy the following condition: We consider the following left-sided and right-sided Sarikaya fractional integral operators respectively.
The most significant feature of generalized fractional integrals is that they generalize some important types of fractional integrals such as Riemann-Liouville fractional integrals, k-Riemann-Liouville fractional integrals, Hadamard fractional integrals, Katugampola fractional integrals, conformable fractional integrals, etc.
These important special cases of the integral operators (1.1) and (1.2) are mentioned as follows: (1) Let us consider ϕ (t) = t. Then, the operators (1.1) and (1.2) reduce to the Riemann integral.

A new identity for twice differentiable functions
In this section, we obtain the equality with one real parameter for generalized fractional integrals and twice differentiable functions.
Proof By using the integration by parts, we obtain With help of the equality (2.1) and using the change of the variable u = t x + (1 − t) a for t ∈ [0, 1] it can be rewritten as follows Similarly, we get

From the Eqs. (2.1) and (2.2), we have
This ends the proof of Lemma 2.1.

Some generalized inequalities for Sarikaya fractional integrals
In this section, using Sarikaya fractional integrals, we will establish some generalized inequalities for functions whose absolute value of second derivatives are convex functions with various powers. We will also obtain some new results by special choices of main results.
Proof By taking modulus in Lemma 2.1, we have By using convexity of f , we obtain . This finishes the proof of Theorem 3.1.
Proof If we choose x = a+b 2 in Theorem 3.1, Remark 3.4 If we assign ϕ (t) = t in Corollary 3.3, then we have the following midpoint inequality which was given by Sarikaya et al. [22,23].
which was given by Mohammed and Sarikaya [19].
where A 1 , A 2 , B 1 and B 2 are defined by as in Lemma 2.1 and 1 p + 1 q = 1. Proof By using the Hölder inequality in inequality (3.2), we have With the help of the convexity of f q , we get This finishes the proof of Theorem 3.7. ∈ [a, b] in Theorem 3.7, then we have the following inequality

Corollary 3.9
If we assign x = a+b 2 in Theorem 3.7, then we have the following midpoint inequality where and defined by as in 3.3.
Proof If we assign x = a+b 2 in Theorem 3.7, then we have the following inequality This gives the fist inequality. For the proof of second inequality, let Using the facts that, Corollary 3.10 If we assign ϕ (t) = t in Corollary 3.9, then we have the following inequality

Corollary 3.11
If we take ϕ (t) = t α (α) , α > 0 in Corollary 3.9, then we obtain the following midpoint type inequality for Riemann-Liouville fractional integrals f (b) . b] in Corollary 3.9, then we have the following midpoint type inequality for k -Riemann-Liouville fractional integrals Then, we obtain the desired result of Theorem 3.13.
Corollary 3.14 If we choose ϕ(t) = t for all t ∈ [a, b] in Theorem 3.13, then we have the following inequality

Corollary 3.15
If we take x = a+b 2 in Theorem 3.13, then we have the following midpoint type inequality for generalized fractional integrals Remark 3. 16 If we choose ϕ(t) = t for all t ∈ [a, b] in Corollary 3.15, then we have the following midpoint inequality for Riemann integrals Corollary 3.18 By choosing ϕ(t) = t α k k k (α) , α, k > 0, for all t ∈ [a, b] in Corollary 3.15, then we have the following midpoint type inequality for k-Riemann-Liouville fractional integrals

Conclusion
In this research, some generalized inequalities for twice differentiable functions using Sarikaya fractional integrals are obtained. Moreover, we prove that our results generalize the inequalities obtained in some earlier works. What's more, we obtain new inequalities for Riemann-Liouville fractional integrals and k-Riemann-Liouville fractional integrals. In the future works, mathematicians can focus to generalize our results by utilizing some other kinds of convex function classes.