Research On Theory Of Several Kinds Of Fuzzy Integral Inequality

With the development of fuzzy measure theory, the fuzzy integral inequalities are extensively investigated. Integral inequalities are useful tools in several theoretical and applied fields. Recently, the authors have been showed that several integral inequalities hold not only in the classical context but for the fuzzy context. On the basis of previous authors, some fuzzy integral inequalities are studied in this paper.This paper mainly investigates four classes of fuzzy integral inequalities, namely includes Sandor’s type Inequality for Sugeno Integral based on (α,m)-Convex Func-tion, Some General inequalities for Choquet Integral, Stolarsky type inequality for U-niversal integral and Barnes-Godunova-Levin type Inequality for extremal universal Integral based on (α, m)-concave Function. This paper is divided into six chapters. Main results of each chapter are described as follow:In Chapter 2, Sandor’s type inequality for Sugeno integral based on (α, m)-convex function is studied. Firstly, Sandor’s type inequality for Sugeno integral based on (α,m)-convex function was discussed, when/is a (α,m)-convex function and for x∈[a,b], satisfies f(a)≤f(b). Then, considering Sandor’s type inequality for Sugeno integral based on (α,m)-Convex Function, when f is a (α,m)-convex func-tion and for x∈[a,b], satisfies f(a)> f (b). Some examples are given to illustrate the validity of these results. Finally, Sugeno integral based on some special (α, m)-concave functions are discussed and some important results are obtained.In Chapter 3, Holder type inequality, Minkowski type inequality and Lyapunov type inequality based on Choquet integral are studied.In Chapter 4, a Stolarsky type inequality for Universal Integral was studied. First-ly, the form of a Stolarsky type inequality for Universal Integral was discussed when the integrand function f is a strictly increasing in the interval. Then, considering the form of a Stolarsky type inequality for Universal Integral when the integrand function f is non-decreasing in the interval. Some theorems are given as special cases of Universal Integral.In Chapter 5, Barnes-Godunova-Levin type inequality for extremal universal in-tegral based on (α,m)-concave function is studied. Firstly, the form of Barnes-Godunova-Levin type Inequality for extremal universal integral based on (α,m)-concave function was discussed, when (α,m) ∈(0,1)2. Then, some theorems are given as special cases of (α, m)-concave functions. Finally, some examples are given to illustrate the validity of these results.In chapter 6, the conclusions and prospects of this paper are summarized.