# Sandor Type Fuzzy Inequality Based on the (s,m)-Convex Function in the Second Sense

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- $\mu (\mathrm{\Phi})=0$.
- (2)
- $A,B\in \Sigma $ and $A\subset B$ imply $\mu (A)\le \mu (B)$.
- (3)
- For any $n\ge 1$, ${A}_{n}\in \Sigma $ and ${A}_{1}\subset {A}_{2}\subset \cdots $, imply $\mu ({\displaystyle \underset{n=1}{\overset{\infty}{\cup}}{A}_{n}})=\underset{n\to \infty}{\mathrm{lim}}\mu ({A}_{n})$.
- (4)
- For any $n\ge 1$, ${A}_{n}\in \Sigma $, ${A}_{1}\supset {A}_{2}\supset \cdots $ and $\mu ({A}_{1})<+\infty $, imply $\mu ({\displaystyle \underset{n=1}{\overset{\infty}{\cap}}{A}_{n}})=\underset{n\to \infty}{\mathrm{lim}}\mu ({A}_{n})$.

**Remark**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

- (1)
- $(S){\displaystyle {\int}_{A}fd\mu}\le \mu (A)$;
- (2)
- For any constant $k\in {R}_{+}$, then $(S){\displaystyle {\int}_{A}kd\mu}\le k\wedge \mu (A)$;
- (3)
- If $f\le g$ on the set $A$, then $(S){\displaystyle {\int}_{A}fd\mu}\le (S){\displaystyle {\int}_{A}gd\mu}$;
- (4)
- If $A,B\in \Sigma $ and $A\subset B$, then $(S){\displaystyle {\int}_{A}fd\mu}\le (S){\displaystyle {\int}_{B}fd\mu}$;
- (5)
- If $\mu (A\cap \{f\ge \alpha \})\ge \alpha $, then $(S){\displaystyle {\int}_{A}fd\mu}\ge \alpha $;
- (6)
- If $\mu (A\cap \{f\ge \alpha \})\le \alpha $, then $(S){\displaystyle {\int}_{A}fd\mu}\le \alpha $;
- (7)
- $(S){\displaystyle {\int}_{A}fd\mu}<\alpha $ $\iff $ There exists $\gamma <\alpha ,s.t.\mu (A\cap \{f\ge \gamma \})<\alpha $;
- (8)
- $(S){\displaystyle {\int}_{A}fd\mu}>\alpha $ $\iff $ There exists $\gamma >\alpha ,s.t.\mu (A\cap \{f\ge \gamma \})>\alpha $;
- (9)
- $\mu (A)<+\infty $, then $(S){\displaystyle {\int}_{A}fd\mu}\ge \alpha $ $\iff $ $\mu (A\cap \{f\ge \alpha \})\ge \alpha $.

**Remark**

**2.**

**Definition**

**3.**

**Remark**

**3.**

**Lemma**

**1.**

**Proof.**

**Remark**

**4.**

**Lemma**

**2.**

**Proof.**

## 3. Sandor Type Inequalities for the Sugeno Integral Based on the(s,m)-Convex in the Second Sense

**Example**

**1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**5.**

**Theorem**

**2.**

**Proof.**

**Remark**

**6.**

**Remark**

**7.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Ren, H.; Wang, G.; Luo, L.
Sandor Type Fuzzy Inequality Based on the (s,m)-Convex Function in the Second Sense. *Symmetry* **2017**, *9*, 181.
https://doi.org/10.3390/sym9090181

**AMA Style**

Ren H, Wang G, Luo L.
Sandor Type Fuzzy Inequality Based on the (s,m)-Convex Function in the Second Sense. *Symmetry*. 2017; 9(9):181.
https://doi.org/10.3390/sym9090181

**Chicago/Turabian Style**

Ren, Haiping, Guofu Wang, and Laijun Luo.
2017. "Sandor Type Fuzzy Inequality Based on the (s,m)-Convex Function in the Second Sense" *Symmetry* 9, no. 9: 181.
https://doi.org/10.3390/sym9090181