Abstract
In this paper, we give an upper bound for the Sugeno fuzzy integral of log-convex functions using the classical Hadamard integral inequality. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.
Highlights
1 Introduction Aggregation is a process of combining several numerical values into a single one which exists in many disciplines, such as image processing [ ], pattern recognition [ ] and decision making [, ]
We will establish an upper bound on the Sugeno fuzzy integral of logconvex functions
6 Conclusion In this paper, we have established an upper bound on the Sugeno fuzzy integral of logconvex functions which is a useful tool to estimate unsolvable integrals of this kind
Summary
Aggregation is a process of combining several numerical values into a single one which exists in many disciplines, such as image processing [ ], pattern recognition [ ] and decision making [ , ]. The main purpose is to estimate the upper bound of Sugeno fuzzy integral for log-convex functions using the classical Hadamard integral inequality. In Section , the upper bound of the Sugeno fuzzy integral for log-convex functions is investigated. The seminormed Sugeno fuzzy integral of a function f ∈ F+(X) over A ∈ with respect to T and the fuzzy measure μ is defined byf dμ =. The following Hadamard inequality provides an upper bound for the mean value of a log-convex function f : [a, b] −→ R (see [ ]): ˆb f (x) dx ≤ L f (a), f (b). Let f : [a, b] −→ ( , ∞) be a log-convex function with f (a) = f (b), Borel field and μ be the Lebesgue measure on X = R. Surely the exact solution is less than or equal to
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