A Hadamard-type inequality for fuzzy integrals based on r-convex functions

In this paper, it is shown that the Hadamard integral inequality for r -convex functions is not satisfied in the fuzzy context. Using the classical Hadamard integral inequality, we give an upper bound for the Sugeno integral of r -convex functions. In addition, we generalize the results related to t...

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Bibliographic Details
Published inSoft computing (Berlin, Germany) Vol. 20; no. 8; pp. 3117 - 3124
Main Authors Abbaszadeh, Sadegh, Eshaghi, Madjid
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2016
Springer Nature B.V
Subjects
Artificial Intelligence
Computational Intelligence
Control
Convex analysis
Engineering
Foundations
Hypotheses
Inequality
Integrals
Mathematical Logic and Foundations
Mechatronics
Probability
Robotics
Upper bounds
Seminormed Sugeno integral
convex function
The Hadamard inequality
Sugeno integral
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Summary:In this paper, it is shown that the Hadamard integral inequality for r -convex functions is not satisfied in the fuzzy context. Using the classical Hadamard integral inequality, we give an upper bound for the Sugeno integral of r -convex functions. In addition, we generalize the results related to the Hadamard integral inequality for Sugeno integral from 1-convex functions (ordinary convex functions) to r -convex functions. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.
ISSN:1432-7643
1433-7479
DOI:10.1007/s00500-015-1934-8