P\'olya-Knopp and Hardy-Knopp type inequalities for Sugeno integral

TL;DR

This paper extends classical inequalities like Pólya-Knopp and Hardy-Knopp to the context of Sugeno integrals, exploring cases with mixed Riemann and Sugeno integrals and providing illustrative examples.

Contribution

It introduces new Pólya-Knopp and Hardy-Knopp type inequalities specifically for Sugeno integrals, including cases with mixed integrals and comprehensive proofs.

Findings

Established Pólya-Knopp inequalities for Sugeno integrals

Proved Hardy-Knopp inequality for Sugeno integrals

Provided examples demonstrating the validity of the inequalities

Abstract

In this paper, we investigate the P\'olya-Knopp type inequality for Sugeno integrals in two cases. In the first case, we suppose that the inner integral is the standard Riemann integral and the remaining two integrals are of Sugeno type. In the second case, all involved integrals are Sugeno integral. We present several examples illustrating the validity of our theorems. Finally, we prove a Hardy-Knopp type inequality for Sugeno integral.