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General Fritz Carlson’s type inequality for Sugeno integrals. (English) Zbl 1219.26024

Fritz Carlson’s inequality for a real function \(f\) claims that
\[ \int_0^{\infty} f(x)\,dx \leq \sqrt{\pi} \left ( \int_0^{\infty} f^2(x)\,dx \right ) ^{\frac 14} \left ( \int_0^{\infty} x^2 f^2(x)\,dx \right ) ^{\frac 14}. \]
Its generalization for the Sugeno integral and \(\mu\) a Lebesgue measure by Caballero and Sadarangani reads
\[ \int_0^1 f(x)\,d\mu \leq \sqrt{2} \left ( \int_0^1 f^2(x)\,d\mu \right ) ^{\frac 14} \left ( \int_0^1 x^2 f^2(x)\,d\mu \right ) ^{\frac 14}. \]
The authors show a version of this inequality on a fuzzy measure space in the following form for comonotone functions \(f\) and \(h\), with Sugeno integrals of \(f, g, h\) not exceeding one:
\[ \int_A f(x) \,d\mu \leq \frac 1K \left ( \int_A f^p(x)g^p(x) \,d\mu \right ) ^{\frac{1}{p+q}} \left ( \int_A f^q(x) h^q(x) \,d\mu \right ) ^{\frac {1}{p+q}}. \]
Corollaries are derived for the cases of monotonicity and convexity of the considered functions.

MSC:

26E50 Fuzzy real analysis
26D15 Inequalities for sums, series and integrals
28E10 Fuzzy measure theory
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References:

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