# Some Integral Inequalities for h-Godunova-Levin Preinvexity

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 2. The h-Godunova–Levin Functions and Their Properties

**Definition**

**5.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 3. New Hermite–Hadamard Inequality for h-Godunova–Levin Convex Function

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

## 4. Hermite–Hadamard Inequalities for h-Godunova–Levin Preinvex Function

**Definition**

**6.**

**Lemma**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**3.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**4.**

## 5. Applications

#### 5.1. Applications to Numerical Integration

**Proposition**

**5.**

**Proof.**

#### 5.2. Applications to Special Means

- 1.
- The arithmetic mean:$A=A({u}_{1},{u}_{2})=\frac{{u}_{1}+{u}_{2}}{2}$; ${u}_{1},{u}_{2}\in \mathbb{R},$ with ${u}_{1},{u}_{2}>0.$
- 2.
- The generalized log-mean:$L}_{m}({u}_{1},{u}_{2})=[\frac{{u}_{2}^{m+1}-{u}_{1}^{m+1}}{(m+1)({u}_{2}-{u}_{1})}{]}^{\frac{1}{m}$, $m\ne -1,0$.

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Almutairi, O.; Kılıçman, A.
Some Integral Inequalities for *h*-Godunova-Levin Preinvexity. *Symmetry* **2019**, *11*, 1500.
https://doi.org/10.3390/sym11121500

**AMA Style**

Almutairi O, Kılıçman A.
Some Integral Inequalities for *h*-Godunova-Levin Preinvexity. *Symmetry*. 2019; 11(12):1500.
https://doi.org/10.3390/sym11121500

**Chicago/Turabian Style**

Almutairi, Ohud, and Adem Kılıçman.
2019. "Some Integral Inequalities for *h*-Godunova-Levin Preinvexity" *Symmetry* 11, no. 12: 1500.
https://doi.org/10.3390/sym11121500