A Review on Some Fuzzy Integral Inequalities

In this paper, we introduce fuzzy measure and fuzzy integral concepts and express some of the fuzzy integral properties. The main purpose of this article is to reviewing of  some important mathematical inequalities that have many applications in modeling mathematical problems. Firstly, we prove  the related Gauss-Winkler type inequality for fuzzy integrals. Indeed, we prove fuzzy version provided by D. H. Hong. Another the famous mathematical inequality is Minkowski's inequality. It is an important inequality from both mathematical and application points of view. Here, we state  a Minkowski type inequality for fuzzy integrals. The established results are based on the classical Minkowski's inequality for integrals. In the continue, we showed that by an example the classical Pr'{e}kopa-Leindler type inequality is not valid for the Sugeno integral. We proved one version of the Pr'{e}kopa-Leindler type inequality by adding  concave fuzzy measure and quasi-concave fuzzy measure assumptions for the Sugeno integral  with different proofs. Also, we  obtained a derivation version of the Pr'{e}kopa-Leindler inequality and illustrated all of the main results by examples. Finally, we investigate the Thunsdorff's inequality for Sugeno integral. By an example, we show that the classical form of this inequality does not hold for the Sugeno integral. Then, by reviewing the initial conditions, we prove two main theorems for this inequality.By checking the special case of the aforementioned Thunsdorff's inequality, we prove Frank-Pick type inequality for the Sugeno integral and illustrate it by an example.