Diaz-Metcalf type inequality for Sugeno and pseudo-integrals

Document Type : Research Paper


Department of Mathematics, University of Maragheh, Maragheh, Iran


In this paper, we have proved Diaz-Metcalf inequality for fuzzy integrals. More precisely:
If $f, g: [0, 1]\to\mathbb{R}$ are continuous and strictly increasing functions, then the fuzzy integral inequality
$$ - \hspace{-1em} \int_0^1 f^s d\mu\cdot  - \hspace{-1em} \int_0^1  g^sd\mu\le  - \hspace{-1em} \int_0^1\left(f\cdot g\right)^sd\mu,$$
holds, where $s>1$ and $\mu$ is  the Lebesgue measure on $\mathbb{R}$. In addition, we have shown this inequality for pseudo-integrals.


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