Some new inequalities of Hermite-Hadamard type for s-convex functions with applications

Abstract In this paper, we present several new and generalized Hermite-Hadamard type inequalities for s-convex as well as s-concave functions via classical and Riemann-Liouville fractional integrals. As applications, we provide new error estimations for the trapezoidal formula.


Introduction
Let I Â R be an interval. Then a real-valued function f W I ! R is said to be convex (concave) on I if the inequality f OE x C .1 /y Ä . / f .x/ C .1 /f .y/ holds for all x; y 2 I and 2 OE0; 1.
A large number of important properties and inequalities have been established for the class of convex (concave) functions since the convexity (concavity) was introduced more than a hundred years ago . But one of the most important inequalities for the convex (concave) function is the Hermite-Hadamard inequality [22], which can be stated as follows: Theorem 1.1. Let I Â R be an interval and f W I ! R be a convex function on I . Then the inequality holds for all a; b 2 I with a < b. Both inequalities given in (1) hold in the reversed direction if f is concave on the interval I .
Let s 2 .0; 1. Then the function f W OE0; 1/ ! R is said to be s-convex on the interval OE0; 1/ if the inequality f OE x C .1 /y Ä s f .x/ C .1 / s f .y/ (2) takes place for all x; y 2 OE0; 1/ and 2 OE0; 1. f is said to be s-concave if inequality (2) is reversed.
We clearly see that the s-convexity (concavity) defined on OE0; 1/ reduces to ordinary convexity (concavity) if s D 1.
In [43], the authors established the Hermite-Hadamard type inequality for the s-convex (concave) functions as follows.
Theorem 1.2 ([43]). Let s 2 .0; 1 and f W I Â OE0; 1/ ! R be an s-convex function on I . Then the double inequality holds for all a; b 2 I with a < b. Both inequalities given in (3) hold in the reversed direction if f is s-concave on the interval I .
Both of the upper and lower bounds given in (3) for the s-convex (concave) functions were improved by Jagers in [44]. Hussian et al. [45] provided the Hermite-Hadamard type inequalities for the twice differentiable functions by using the following Lemma 1.3.

Lemma 1.3 ([45]
). Let f W I ı Â R ! R be a differentiable mapping on I ı , and a; b 2 I ı with a < b. Then the identity is valid if f 00 2 LOEa; b, where and in what follows I ı denotes the interior of the interval I .
In [51], Chu et al. discovered a new identity for the twice differentiable function. Lemma 1.6 ([51]). Let f W I Â R ! R be a differentiable mapping on I ı , and a; b 2 I ı with a < b. Then the identity Next, we recall the definition of the fractional integrals [52]. Let 0 Ä a < b, Á > 0 and f 2 LOEa; b. Then the left-sided and right-sided Riemann-Liouville fractional integrals J Á a C f and J Á b f of order Á are defined by We clearly see that J 0 In particular, the fractional integral reduces to the classical integral if Á D 1.
holds if f is convex on OEa; b.
The following identity for the twice differentiable function, which was discovered by Chu et al. [51], will be used in the next section.
The main purpose of this paper is to establish several new Hermite-Hadamard type inequalities for s-convex (concave) functions via the classical and Riemann-Liouville fractional integrals, and provide the error estimatimations for the trapezoidal formula.
holds for all x 2 OEa; b if jf 00 j is s-convex on OEa; b and f 00 2 LOEa; b.
Proof. It follows from (4) and the triangular inequality together with the s-convexity of jf 00 j thať jf 00 .x/j :

Corollary 2.2. Under the assumptions of Theorem 2.1, one haš
.jf 00 .a/j C jf 00 .b/j/: , then the first inequality of (8) follows easily from (7). While the second inequality of (8) can be derived from the s-convexity of jf 00 j.
.jf 00 .a/j C jf 00 .b/j/: Proof. From (4) together with the triangular and Hölder inequalities we clearly see thať Making use of the s-convexity of jf 00 j q , we get Note that 0 Therefore, inequality (9) follows easily from the (10)-(13).

Corollary 2.5. Under the assumptions of Theorem 2.4, we havě
, then inequality (9) leads to the first inequality of (14) immediately. While the second inequality of (14) can be derived easily from the s-convexity of jf 00 j q and the elementary inequality ;˛n;ˇn 0 and 0 Ä Ä 1.
Remark 2.6. If s D 1, then the second inequality of (14) becomeš Theorem 2.7. Let s 2 .0; 1, p; q > 1 with 1=p C 1=q D 1, f W I Â OE0; 1/ ! R be a twice differentiable mapping on I ı , and a; b 2 I ı with a < b. Then the inequality Proof. It follows from the s-concavity of jf 00 j q and (3) that Therefore, inequality (15) follows from (4), (13), (16) and (17) together with the Hölder inequalities Proof. Let x D .a C b/=2, then inequality (15) leads to the first inequality of (18) immediately. While the second inequality of (18) can be obtained by the s-concavity of jf 00 j due to the fact that jf 00 j q is s-concave, indeed, the s-concavity of jf 00 j q leads to the conclusion that Remark 2.9. Let s D 1, then from the second inequality of (18), we geť  Proof. It follows from (4) and the power-mean inequality thať From the s-convexity of jf 00 j q on OEa; b we get Therefore, inequality (19) follows from (20)-(23). Proof. Let x D .a C b/=2, then the first inequality of (24) can be obtained from inequality (19) immediately. While the second inequality of (24) follows from the s-convexity of jf 00 j q and the inequality n X kD1 .˛k Cˇk/ Ä n X kD1˛ k C n X kD1ˇ k for˛1;ˇ1;˛2;ˇ2; ;˛n;ˇn 0 and 0 Ä Ä 1.

Corollary 3.3. Under the assumptions of Theorem 3.1, we havě
, then inequality (25) leads to the first inequality of (26). While the second inequality of (26) can be derived from the s-convexity of jf 00 j.

Corollary 3.7.
Under the assumptions of Theorem 3.5, we have the inequality as follows: Remark 3.8. Let s D 1, then the second inequality of (30) leads tǒ Proof. Theorem 3.9 follows easily from (16), (17), (28) and (29). Letting x D .a C b/=2 and making use of the s-concavity of jf 00 j, then inequality (31) leads to Corollary 3.11 immediately.
Corollary 3.11. Under the assumptions of Theorem 3.9, one haš Remark 3.12. Let s D 1 in the second inequality of (32), then we geť Theorem 3.13. Let Á > 0, s 2 .0; 1, q > 1, f W I Â OE0; 1/ ! R be a twice differentiable mapping on I ı , and a; b 2 I ı with a < b. Then the inequality holds for all x 2 OEa; b if f 00 2 LOEa; b and jf 00 j q is s-convex on OEa; b.
Proof. By use of (6) and the power-mean inequality, we havě It follows from the s-convexity of jf 00 j q on OEa; b that Note that Therefore, Theorem 3.13 follows from (34)- (37).

Applications to trapezoidal formula
Let d be a division a D x 0 < x 1 < x 2 < < x n 1 < x n D b of the interval OEa; b and consider the quadrature formula where is the trapezoidal version and E.f; d / denotes the associated approximation error.
holds if f 00 2 LOEa; b and jf 00 j is s-convex on OEa; b.
x iC1 x i / 3 OEjf 00 .x i /j C jf 00 .x iC1 /j 8 : Making use of the similar arguments as in Theorem 4.1, we can get Theorems 4.2-4.4 from the Corollaries 2.5, 2.8 and 2.11 immediately.
Theorem 4.2. Let s 2 .0; 1, p; q > 1 with 1=p C 1=q D 1, f W I Â OE0; 1/ ! R be a twice differentiable mapping on I ı , a; b 2 I ı with a < b and d be a division a D x 0 < x 1 < x 2 < < x n 1 < x n D b of the interval OEa; b. Then the inequality .x i C1 x i / 3 OEjf 00 .x i /j C jf 00 .x i C1 /j 16.s C 1/ 1=q holds if f 00 2 LOEa; b and jf 00 j q is s-convex on OEa; b. Theorem 4.3. Let s 2 .0; 1, p; q > 1 with 1=p C 1=q D 1, f W I Â OE0; 1/ ! R be a twice differentiable mapping on I ı , a; b 2 I ı with a < b and d be a division a D x 0 < x 1 < x 2 < < x n 1 < x n D b of the interval OEa; b. Then the inequality Ãȟ olds if f 00 2 LOEa; b and jf 00 j q is s-concave on OEa; b.

Conclusion
In the article, we present several new Hermite-Hadamard type inequalities and error estimatimations for the trapezoidal formula involving the s-convex and s-concave functions for the classical and Riemann-Liouville fractional integrals.