Integral Inequalities of Hermite – Hadamard Type for Extended s-Convex Functions and Applications

In the paper, the authors set up an identity for a function whose third derivative is integrable, establish by the Hölder inequality some new integral inequalities of the Hermite–Hadamard type for extended s-convex functions in the second sense, and apply these integral inequalities to construct inequalities for several special means.


Introduction
A function h : I ⊆ R → R is said to be convex if holds for all u, v ∈ I and µ ∈ [0, 1].For any convex function h(t) on a closed interval [a, b], the double inequality is classical and called as the Hermite-Hadamard integral inequality.
It is well known that the convex function is a basic concept in mathematics and mathematical sciences and there exists a theory on convex functions.For the theory of convex functions, please refer to two monographs [1,2], for example.Among the theory of convex functions, the Hermite-Hadamard type inequalities play important roles.For new development of the Hermite-Hadamard type inequalities, please refer to the monograph [3], for example.
We now introduce the s-convex function in the second sense and the extended s-convex function in the second sense.Definition 1 ([4]).Let s ∈ (0, 1].A function h : I ⊆ [0, ∞) → R is said to be s-convex in the second sense if holds for all u, v ∈ I and µ ∈ [0, 1].
Remark 1.We note that, the following five cases, and only the following five cases, are possible: 1. when s ∈ (0, 1], any s-convex function in the second sense is equivalent to an extended s-convex function in the second sense for the same s ∈ (0, 1]; 2. when s ∈ (0, 1] and h(t) is an s-convex function in the second sense, there exists a number s 0 ∈ [−1, 1] \ {s} such that h(t) is an extended s 0 -convex function in the second sense; 3.
when s ∈ (0, 1] and h(t) is an s-convex function in the second sense, there does not exist a number s 0 ∈ [−1, 1] \ {s} such that h(t) is an extended s 0 -convex function in the second sense; equivalently speaking, for any s ∈ (0, 1], there exists an s-convex function h(t) in the second sense, but h(x) is not an s 0 -convex function in the second sense for all s 0 ∈ [−1, 1] \ {s}; 4.
when s ∈ (0, 1] and h(t) is not an s-convex function in the second sense, there exists a number s 0 ∈ [−1, 1] \ {s} such that h(t) is an extended s 0 -convex function in the second sense; 5.
when s ∈ (0, 1] and h(t) is not an s-convex function in the second sense, there does not exist a number s 0 ∈ [−1, 1] \ {s} such that h(t) is an extended s 0 -convex function in the second sense; equivalently speaking, for any s ∈ (0, 1], there exists a function h(t) which is neither an s-convex function in the second sense nor an extended s 0 -convex function in the second sense for all s 0 ∈ [−1, 1] \ {s}.
We now construct an example as follows.Let h This means that h(t) is not an s-convex function in the second sense on [0, 1] for s ∈ (0, 1].On the other hand, it is easy to see that for every u, v ∈ [0, 1] and µ ∈ (0, 1).This means that h(t) is an extended −1-convex function in the second sense on [0, 1].
There have been some studies dedicated to generalizing the s-convex function in the second sense and the extended s-convex function in the second sense and to establishing their Hermite-Hadamard type inequalities.For more details, please refer to the papers [6][7][8][9][10][11][12][13][14][15] and closely related references therein.
The following are some of the Hermite-Hadamard type inequalities for s-convex functions in the second sense.
Theorem 3 ([18]).Let h : I ⊆ [0, ∞) → R be differentiable on I During the past two decades, starting from s-convex functions to generalizing different concepts of convex functions, there has been a continuous interest in the development of integral inequalities of the Hermite-Hadamard type.It seems that it is a rich machinery for obtaining results.For example, in the paper [19] and closely related references therein, a strong extension and refinement of the Hadamard inequality in the right hand side of (1) was generalized to nonlinear integrals.For more results on integral inequalities of the Hermite-Hadamard type for diverse convex functions, please refer to [20][21][22][23][24][25][26][27] and closely related references therein.
In [6], some integral inequalities of the Simpson type were established for a function h(t) whose third derivative h (t) belongs to L 1 ([a, b]) and |h (t)| q is an extended s-convex in the second sense for q ≥ 1.One of the key steps is to set up an identity In this paper, we will set up an identity different from (3), establish some integral inequalities for a function h(t) whose third derivative h (t) belongs to L 1 ([a, b]) and |h (t)| q is an extended s-convex in the second sense for q ≥ 1, and apply these integral inequalities to construct inequalities for several special means.

A Lemma
For attaining our main aim, we need a lemma below.Lemma 1.Let h : I ⊆ R → R be a three times differentiable function on I • and a, b Proof.When λ ∈ (0, 1), by integrating by parts, we have Substituting these two equalities into the right side of (4) and changing variables result in the required conclusion.The proof is complete.

Remark 2.
When taking λ = 0, 1 in (4) respectively, we derive The identity (3) mentioned above and the newly-established identity (4) can not be derived from each other.

Inequalities for Extended s-Convex Functions in the Second Sense
Now, we are in a position to establish some new integral inequalities of the Hermite-Hadmard type for extended s-convex functions in the second sense.when λ ∈ [0, 1] and −1 < s ≤ 1, we have (5)

2.
when λ ∈ (0, 1) and s = −1, we have Proof.When λ ∈ [0, 1] and −1 < s ≤ 1, since |h (t)| q is an extended s-convex function in the second sense on [a, b], by Lemma 1 and the Hölder inequality, we have When λ ∈ (0, 1) and s = −1, since |h (t)| q is an extended s-convex function in the second sense on [a, b], by Lemma 1 and the Hölder inequality, we have The proof of Theorem 4 is complete.
Corollary 2. If s = q = 1 in (5), then s ∈ (−1, 1], and q ≥ 1, then Proof.Since |h (t)| q is an extended s-convex function in the second sense on [a, b], then by Lemma 1 and the Hölder inequality, we have The proof of Theorem 5 is complete.

Corollary 4. Under assumptions of Theorem 5, if
Proof.Since |h (t)| q is an extended s-convex function in the second sense on [a, b], then by Lemma 1 and the Hölder inequality, we have The proof of Theorem 6 is complete.

Applications to Means
Making use of results for integral inequalities of the Hermite-Hadmard type for extended s-convex functions in the second sense in the above section, we construct some inequalities for means.
Let b > a > 0, the arithmetic mean A µ (a, b) and the generalized logarithmic mean L r (a, b) are defined [28,29] respectively as Consequently, applying (5) in Theorem 4 to |h (t)| q = t sq yields the following theorem.
Corollary 6.Under assumptions of Theorem 7, if λ = 1 2 and s = 1, then Applying Theorem 5 to the function |h (t)| q = t sq yields the following theorem.Applying Theorem 6 to |h (t)| q = t sq yields Theorem 9. Let b > a > 0, q > 1, −1 < sq ≤ 1, −1 < s ≤ 1, and λ ∈ [0, 1].Then Remark 3. Since the identity (3) in [6] and the newly-established identity (4) in Lemma 1 can not be derived from each other, we can be sure that all results in this paper and those in [6] can not be compared with each other.

Theorem 5 .
Let h : I ⊆ [0, ∞) → R be a three times differentiable function on I, a, b ∈ I with a < b, h ∈ L 1 ([a, b]), and λ ∈ [0, 1].If |h (t)| q is an extended s-convex function in the second sense on [a, b],
• , a, b ∈ I with a < b, and h ∈ L 1 ([a, b]).If |h (t)| is s-convex in the second sense on [a, b] for some fixed s ∈ (0, 1], then