Integral inequalities for s-convex functions via generalized conformable fractional integral operators

We introduce new operators, the so-called left and right generalized conformable fractional integral operators. By using these operators we establish new Hermite–Hadamard inequalities for s-convex functions and products of two s-convex functions in the second sense. Also, we obtain two interesting identities for a differentiable function involving a generalized conformable fractional integral operator. By applying these identities we give Hermite–Hadamard and midpoint-type integral inequalities for s-convex functions. Different special cases have been identified and some known results are recovered from our general results. These results may motivate further research in different areas of pure and applied sciences.


Introduction
The theory of inequalities is known to play an important role in almost all areas of pure and applied sciences. Richard Bellman stated succinctly, at the Second International Conference on Mathematical Inequalities, Oberwolfach, Germany, July 30- August 5, 1978, that "there are three reasons for the study of inequalities: practical, theoretical, and aesthetic". In the last few decades the theory of inequalities has attracted the attention of great number of researchers.
The interesting mean type inequality, known as the Hermite-Hadamard inequality for convex functions, is given by the following theorem. Theorem 1 Let h : I ⊂ R − → R be a convex function, and let λ 1 , λ 2 ∈ I with λ 1 < λ 2 . Then Inequality (2) is also acknowledged as the trapezium inequality.
The trapezium inequality has an extraordinary interest due to its wide applications in the field of mathematical analysis. Authors of recent decades have studied (2) in the premises of newly invented definitions due to the motivation of convex functions. The interested readers can see the references [2-7, 9, 12-15, 18, 19, 21, 22, 24-28, 30-32, 34-37].
The s-convex functions in the second sense are presented in [13]. Also, researchers started to study conformable fractional integrals; see [1,8,12,18,19,30,32]. Khalil et al. [17] defined the fractional integral only of order 0 < α ≤ 1, whereas Abdeljawad [1] generalized the definition of left and right conformable fractional integrals to any order α > 0. In 2017, Khan et al. implemented this definition by providing a class of Hermite-type inequalities.
Motivated by the literature cited, our paper is organized as follows: In Sect. 2, using the new operators, we establish the so-called left and right generalized conformable fractional integral operators, new Hermite-Hadamard inequalities for s-convex functions and products of two s-convex functions in the second sense. In Sect. 3, we obtain two interesting identities for differentiable function involving generalized conformable fractional integral operator. By applying these identities we give Hermite-Hadamard and midpoint-type integral inequalities for s-convex functions. Various particular cases will be identified, and some known results will be recaptured from our general results. In Sect. 4, we give a brief conclusion.
Multiplying both sides of inequality (18) (1θ ) ζ -1 and integrating the resulting inequality with respect to θ over [0, 1], we obtain which means that the left side of (16) is proved. To prove the right side of (16), since f is s-convex in the second sense on [λ 1 , λ 2 ], we have the inequalities and Adding (20) and (21), we get Multiplying both sides of inequality (22) by Φ(θ(λ 2 -λ 1 )) θ (1θ ) ζ -1 and integrating the resulting inequality with respect to t over [0, 1], we obtain which means that the right side of (16) is proved. The proof of Theorem 3 is completed.
Corollary 1 Taking s = 1 in Theorem 3, we get the following inequalities for convex functions via generalized conformable fractional integral operators: Remark 2 Taking ξ = s = 1 in Corollary 2, we get the well-known Hermite-Hadamard inequality (2).
Let us represent now Hermite-Hadamard inequalities for the product of two s-convex functions in the second sense via general conformable fractional integral operators.
Proof Let x, y ∈ [λ 1 , λ 2 ]. Since f and g are s-convex in the second sense on [λ 1 , λ 2 ], we have and 2 s g Multiplying both sides of inequalities (29) and (30), we obtain Multiplying both sides of inequality (31) by Φ(θ(λ 2 -λ 1 )) θ (1θ ) ζ -1 and integrating the resulting inequality with respect to θ over [0, 1], we have So, we get which means that the left side of (25) is proved. To prove the right side of (25), since f and g are s-convex in the second sense on [λ 1 , λ 2 ], we have the inequalities and Applying inequalities (33) to (36), we have Multiplying both sides of inequality (37) by Φ(θ(λ 2 -λ 1 )) θ (1θ ) ζ -1 and integrating the resulting inequality with respect to θ over [0, 1], we obtain So, we get which means that the right side of (25) is proved. The proof of Theorem 4 is completed.

Some other results
To establish the results of this section regarding general conformable fractional integral operators, we first prove the following two lemmas.
and |f | q is s-convex in the second sense with s ∈ (0, 1], then for q > 1 and 1 p + 1 q = 1, wehave the following inequality for generalized conformable fractional integrals: where Proof By Lemma 3.1, the s-convexity in the second sense of |f | q , the Hölder inequality, and properties of the modulus we have The proof of Theorem 5 is completed.
We point out some particular cases of Theorem 5.

Corollary 6
Taking s = 1 in Theorem 5, we have the following inequality for convex function via generalized conformable fractional integral operators: Corollary 7 Taking |f | ≤ K in Theorem 5, we obtain and |f | q is s-convex in the second sense with s ∈ (0, 1], then for q ≥ 1, we have the following inequality for generalized conformable fractional integrals: where and Ψ ζ Φ is defined by (15).
Proof By Lemma 3.1, the s-convexity in the second sense of |f | q , the well-known power mean inequality, and properties of the modulus we have The proof of Theorem 6 is completed.
We point out some particular cases of Theorem 6.

Corollary 9
Taking s = 1 in Theorem 6, we get the following inequality for convex function via generalized conformable fractional integral operators: where

Corollary 10
Taking |f | ≤ K in Theorem 6, we obtain Theorem 7 Let f : [λ 1 , λ 2 ] − → R be a differentiable function on (λ 1 , λ 2 ). If f ∈ L[λ 1 , λ 2 ] and |f | q is s-convex in the second sense with s ∈ (0, 1], then for q > 1 and 1 p + 1 q = 1, we have the following inequality for generalized conformable fractional integrals: where Proof By Lemma 3.2, the s-convexity in the second sense of |f | q , the Hölder inequality, and properties of the modulus we have The proof of Theorem 7 is completed.
We point out some particular cases of Theorem 7.