#### Abstract

The author establish several Hermite-Hadamard and Simpson-like type inequalities for mappings whose first derivative in absolute value aroused to the th () power are ()-convex. Some applications to special means of positive real numbers are also given.

#### 1. Introduction

Recall that, for some fixed and , a mapping is said to be -convex on an interval if the inequality holds for , and .

Denote by the set of all -convex mappings on . For recent results and generalizations concerning -convex and -convex mappings, see [1β4].

For the simplicities of notations, for , let us denote

In [1, 3], KlariΔiΔ Bakula and Γzdemir et al., proved the following Hadamardβs inequalities for mappings whose second derivative in absolute value aroused to the -th power are -convex.

Theorem 1.1. Let be a twice differentiable mapping on the interior of an interval such that , where with and . If is -convex on for and with , then the following inequality holds: where where where

Theorem 1.2. Under the same notations in Theorem 2.2, if is -convex on for and with , then the following inequality holds:

Note that for one obtains the following classes of functions: increasing, -starshaped, starshaped, -convex, convex, and -convex. For the definitions and elementary properties of these classes, see [4β8].

For recent years, many authors present some new results about Simpsonβs inequality for -convex mappings and have established error estimations for the Simpsonβs inequality: for refinements, counterparts, generalizations, and new Simpsonβs type inequalities, see [1β3, 6].

In [9], Dragomir et al. proved the following theorem.

Theorem 1.3. Let be an absolutely continuous mapping on such that , where with . Then the following inequality holds:

The readers can estimate the in the generalized Simpsonβs formula without going through its higher derivatives which may not exist, not be bounded, or may be hard to find.

In this paper, the author establishes some generalizations of Hermite-Hadamard and Simpson-like type inequalities based on differentiable -convex mappings by using the following new identity in Lemma 2.1 and by using these results, obtain some applications to special means of positive real numbers.

#### 2. Generalizations of Simpson-Like Type Inequalities on πΎπΌπ(π)

In order to generalize the classical Simpson-like type inequalities and prove them, we need the following lemma [6].

Lemma 2.1. Let be a differentiable mapping on the interior of an interval , where with and . If , then, for and with , the following equality holds: for and each , where

By the similar way as Theorems 1.1β1.3, we obtain the following theorems.

Theorem 2.2. Let be a differentiable mapping on such that , where with and . If , for some and , then, for any , the following inequality holds: where

Proof. From Lemma 2.1 and using the properties of the modulus, we have the following:
Since is -convex on , we know that for any
By (2.5) and (2.6), we get the following: which completes the proof.

Corollary 2.3. In Theorem 2.2, (i) if we choose and , then we have the following: and (ii) if we choose and , then we have the following

Theorem 2.4. Under the same notations in Theorem 2.2, if , for some , and with , then, for any , the following inequality holds:

Proof. From Lemma 2.1 and using the properties of modulus, we have the following:
Using the power-mean integral inequality and -convexity of for any , we have the following(a)(b)(c)(d)
By the similar way as the above inequalities (a)β(d), we have the following:(aβ²)(bβ²)(cβ²)(dβ²)
By (2.11) and (2.16)β(2.19) the assertion (2.10) holds.

Corollary 2.5. In Theorem 2.4, (i) if we choose and , then we have that and (ii) if we choose and , then we have that

Theorem 2.6. Under the same notations in Theorem 2.2, if , for some , and with , then, for any , the following inequality holds: where

Proof. Suppose that . From Lemma 2.1, using the HΓΆlder integral inequality, we get the following: where we have used the fact that .
Since for some fixed and , we have the followings:
Hence, if we combine the inequalities in (2.24)-(2.25), we get the desired result.

Corollary 2.7. In Theorem 2.6, (i) if we choose and , then we have that and (ii) if we choose and , then we have where we have used the fact that .

#### 3. Applications to Special Means

Now using the results of Section 2, we give some applications to the following special means of positive real numbers with .(1) The arithmetic mean: .(2)The geometric mean: .(3)The logarithmic mean: ) for .(4)The harmonic mean: .(5)The power mean: , .(6)The generalized logarithmic mean: (7)The identric mean:

Proposition 3.1. For and with , we have the following inequalities:

Proof. The assertions follow from Corollary 2.3 for .

Proposition 3.2. For , we have the following inequalities:

Proof. The assertions follow from Corollary 2.3 for .

Proposition 3.3. For and , we have the following inequalities:

Proof. The assertions follow from Corollary 2.5 for .

Proposition 3.4. For and , we have the following inequalities:

Proof. The assertions follow from Corollary 2.7 for .

Proposition 3.5. For , we have the following inequalities:

Proof. The assertions follow from Corollary 2.7 for and .

#### Acknowledgment

The author is so indebted to the referee who read carefully through the paper very well and mentioned many scientifically and expressional mistakes.