Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations

Dug Hun Hong,Jae Duck Kim

doi:10.1155/2019/5080723

Abstract

The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.

Highlights

Introduction and PreliminariesA number of studies have examined the Sugeno integral since its introduction in 1974 [1]
This paper proposes Holder type inequalities and reverse Holder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions
Ralescu and Adams [2] generalized a range of fuzzy measures and provided several equivalent definitions of fuzzy integrals

Summary

Introduction and Preliminaries

A number of studies have examined the Sugeno integral since its introduction in 1974 [1]. Ouyang et al [14] generalized a Chebyshev type inequality for the fuzzy integral of monotone functions based on an arbitrary fuzzy measure. Hong [15] extended previous research by presenting a Hardy-type inequality for Sugeno integrals in [6]. Wu et al [21] considered Holder type inequalities for Sugeno integrals They did not examine their results under usual multiplication operations and did not make the essential assumption of 1 < p < ∞, 1/p + 1/q = 1 for the classical Holder inequality. We propose Holder type inequalities for Sugeno integrals and find optimal constants for which these inequalities hold for nonincreasing concave or convex functions under usual multiplication operations. We consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or Advances in Fuzzy Systems convex functions.

Hölder Type Inequalities

Reverse Hölder Type Inequalities