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

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Abstract

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.

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Correspondence to Sadegh Abbaszadeh.

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The authors declare that they have no conflict of interest.

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Communicated by A. Di Nola.

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Abbaszadeh, S., Eshaghi, M. A Hadamard-type inequality for fuzzy integrals based on r-convex functions. Soft Comput 20, 3117–3124 (2016). https://doi.org/10.1007/s00500-015-1934-8

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Keywords

  • Sugeno integral
  • The Hadamard inequality
  • r-convex function
  • Seminormed Sugeno integral