Diaz-Metcalf type inequality for Sugeno and pseudo-integrals

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.