On Non-additive Probabilistic Inequalities of Hölder-type

Abstract

Non-additive measure is a generalization of additive probability measure. Integral inequalities play important roles in classical probability and measure theory. Some well-known inequalities such as the Minkowski inequality and the Hölder inequality play important roles not only in the theoretical area but also in application. Non-additive integrals are useful tools in several theoretical and applied statistics which have been built on non-additive measure. For instance, in decision theory and applied statistics, the use of the non-additive integrals can be envisaged from two points of view: decision under uncertainty and multi-criteria decision-making. In fact, the non-additive integrals provide useful tools in many problems in engineering and social choice where the aggregation of data is required. In this paper, Hölder and Minkowski type inequalities for semi(co)normed non-additive integrals are discussed. The main results of this paper generalize some previous results obtained by the authors. Mathematics Subject Classification (2000). Primary 99Z99; Secondary 00A00.

General form of Chebyshev type inequality for generalized Sugeno integral

“…for all α, β, where 1 , then all functions f | A and g| B are m-positively dependent with respect to △. For instance, all functions are m-positively dependent whenever m is a supermodular capacity 2 .…”

“…If (X, A) is a measurable space, then two A-measurable real functions f and g defined on X satisfy the inequality f g dP f dP g dP (1) for any probability measure P if and only if functions f, g are comonotone.…”

“…Nowadays, there is a huge number of papers dealing with inequalities for various non-additive integrals, see e.g. [1,2,13,15,20]. In this paper, we aim to provide a general form of the Chebyshev type inequality (necessary as well as sufficient conditions) for the generalized upper Sugeno integral for m-positively dependent functions, as well as for any comonotone functions and monotone measure, see Section 3.…”

“…It is well-known that the classical integral inequalities (including the Chebyshev one) need not hold in general when replacing in (1) the probability measure by a non-additive measure and the additive (Lebesgue) integral by a non-additive integral. Nowadays, there is a huge number of papers dealing with inequalities for various non-additive integrals, see e.g.…”

“…Assume that m ∈ M 1 (X,A) , △ : m(A) × m(A) → m(A), k ∈ (0, 1], S i ∈ S for i = 1, 2, 3 and ⋆, ♦ : [0, 1] 2 → [0, 1] are non-decreasing and ♦ is left-continuous. Let ϕ i : [0, 1] → [0,1] be functions such that ϕ 1 is non-decreasing and ϕ j are increasing and right-continuous, ψ i : [0, ϕ i (1)] → [0, 1] be non-decreasing and ψ j be left-continuous, where i = 1, 2, 3 and j = 2, 3.…”