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The main purpose of this survey paper is to point out some very recent developments on Simpson’s inequality for strongly extended s-convex function. Firstly, the concept of strongly extended
*s*-convex function is introduced. Next a new identity is also established. Finally, by this identity and H
?lder’s inequality, some new Simpson type for the product of strongly extended
*s*-convex function are obtained.

Convex function is a kind of important function and has wide applications in pure and applied mathematics [

First, some definitions concerning various convex functions are listed.

Definition 1.1. A function

holds for all

The s-convex function was defined in [

Definition 1.2. A function

for some

If

In [

Definition 1.3. ( [

for some

In [

Definition 1.4. ( [

is valid for all

In [

Definition 1.5. A function

is valid all

The following inequalities of Hermite-Hadamard type were established for some of the above convex functions.

Theorem 1.1. ( [

(1) If

(2) If

Theorem 1.2. ( [

Theorem 1.3. ( [

In [

Theorem 1.4. Let

In this paper, the authors introduce the concept of strongly extended s-convex function and establish a new identity. By this identity and Hölder’s inequality, some new Simpson type for the product of strongly extended s-convex function and discussed and some results are obtained.

Now the concept of strongly extended s-convex function is introduced.

Definition 2.1. A function

is valid for all

For establishing new integral inequalities of Simpson type involving the strongly extended s-convex function, the following identity is needed:

Lemma 2.1. Let

Proof. By straightforward computation, the result is followed. The proof is completed.

Lemma 2.2. ( [

Theorem 3.1. Let

Proof. Using Lemma 2.1 and by Hölder’s inequality, the followings can be obtained:

where,

Again

Substituting the above (3.3)-(3.6) into the inequality (3.2) results in the inequality (3.1).

Theorem 3.1 is proved.

Corollary 3.2. Under conditions of Theorem 3.1, if

Theorem 3.3. Let

Proof. Since

Theorem 3.3 is proved.

Theorem 3.4. Let

Proof. By the Lemma 2.1 and using Hölder’s inequality, the followings can be obtained:

where,

Since

Substituting (3.10)-(3.13) into the inequality (3.9) yields (3.8). Theorem 3.4 is proved.

In this paper, the authors introduce the concept of strongly extended s-convex function and establish a new identity. Then by this identity and Hölder’s inequality, some new Simpson type for the product of strongly extended s-convex function are obtained.

This work was supported by the National Natural Science Foundation of China No. 11361038 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China.

Sun, Y.X. and Yin, H.P. (2016) Some Integral Inequalities of Simpson Type for Strongly Extended s-Con- vex Functions. Advances in Pure Mathematics, 6, 745-753. http://dx.doi.org/10.4236/apm.2016.611060