DOI: 10.1515/ms-2016-0219
Math. Slovaca 66 (2016), No. 5, 1235–1248
STOLARSKY’S INEQUALITY
FOR CHOQUET-LIKE EXPECTATION
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Hamzeh Agahi* — Radko Mesiar**
(Communicated by Anatolij Dvurečenskij )
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ABSTRACT. Expectation is the fundamental concept in statistics and probability. As two generalizations of expectation, Choquet and Choquet-like expectations are commonly used tools in generalized
probability theory. This paper considers the Stolarsky inequality for two classes of Choquet-like integrals. The first class generalizes the Choquet expectation and the second class is an extension of the
Sugeno integral. Moreover, a new Minkowski’s inequality without the comonotonicity condition for two
classes of Choquet-like integrals is introduced. Our results significantly generalize the previous results
in this field. Some examples are given to illustrate the results.
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2016
Mathematical Institute
Slovak Academy of Sciences
1. Introduction
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Choquet expectation [7] plays an important role in analyzing many problems in statistics and
probability, especially when dealing with non-additive data [26]. The classical integral inequalities
are used in numerous applications in information theory, engineering, economics and statistics.
Stolarsky’s inequality is one of the most fundamental inequalities in analysis and its applications.
The history and the development of this inequality are described in [12, 21]. Stolarsky’s inequality
is a useful tool in several theoretical and applied fields. For instance, it plays a major role for the
gamma function [21]. A weighted version of Stolarsky’s inequality was presented in [1, 12]. The
mixing of Choquet expectation and integral inequalities can be applied to find solutions of many
uncertain problems. Recently, Agahi et al. [2] proved the following Stolarsky type inequality for
Choquet expectation.
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1.1 ([2: Theorem 4.1]) Let B([0,1]) be the Borel σ-algebra over [0,1] and ([0,1], B([0,1]),µ)
be a monotone probability space such that µ is a lower-semicontinuous monotone probability absolutely continuous with respect to the Lebesgue measure λ, i.e., if λ(E) = 0 for some E ∈ B([0, 1])
then also µ(E) = 0. Then the Stolarsky inequality holds for the corresponding Choquet expectation,
i.e., for any nonincreasing function X : [0, 1] → [0, 1] it holds
1
1
1
EµC X(ω a+b ) ≥ EµC [X(ω a )]EµC [X(ω b )],
(1.1)
where a, b ∈ (0, ∞).
2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 28A12, 60A86.
K e y w o r d s: Choquet-like expectation, Stolarsky’s inequality, monotone probability, Minkowski’s inequality.
The second author was supported by grant VEGA 1/0171/12.
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HAMZEH AGAHI — RADKO MESIAR
Sugeno integral was firstly introduced by Sugeno [23] and then was exploited by many authors
[16, 17, 25]. The idea of mixing of Sugeno integral and integral inequalities was presented by
Román-Flores et al. [18] and then followed by the authors [3,9,14]. For example, in [9], Stolarsky’s
inequality for a Lebesgue measure-based Sugeno integral and a continuous and strictly monotone
function was obtained which has been extended by Agahi et al. [2].
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1.2 ([2: Theorem 3.5]) Let X : [0, 1] → [0, 1] be a nondecreasing function and
([0, 1], B[0, 1], µ) be a monotone probability space. Let β, γ be automorphisms on [0, 1] (i.e.,
β, γ : [0, 1] → [0, 1] are increasing bijections) and α = (β −1 γ −1 )−1 . If : [0, 1]2 → [0, 1] be
any continuous aggregation function which is jointly strictly increasing and bounded from above by
min, then
Suµ [X(α)] ≥ (Suµ [X(β)]) (Suµ [X (γ)]) .
In 1995, Mesiar [13] introduced two classes of Choquet-like integrals as generalizations of Choquet expectation and Sugeno integral. The first class is called “Choquet-like expectation” which
generalizes the Choquet expectation (see Definition 2.7) and the second class is an extension of
the Sugeno integral (see Definition 2.9). This motivates us to propose the following problem.
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Under what conditions does the Stolarsky inequality hold for two classes of Choquetlike integrals?
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We will give the answer to the above problem in Section 3 which will imply a generalization of
[2, 9]. Our results expand the applicability of Stolarsky type inequality for Choquet expectation
by combining the properties of pseudo-analysis.
Recently, there were obtained some of the classical integral inequalities for integrals with respect to non-additive measures based on the concept of comonotonicity [4, 6, 15, 28]. For example,
Minkowski’s inequality for Sugeno integral has studied in [4]. In [4], the authors showed that
in general, the Minkowski inequality is not valid without the comonotonicity condition (See [4:
Example 3.6]). However, the major question that arises in mind is the following problem.
How can Minkowski’s inequality be explained without comonotonicity condition for
Sugeno integral?
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In this paper, we also give a new Minkowski’s inequality without the comonotonicity condition for two classes of Choquet-like integrals. In special cases, our results give a new version of
Minkowski’s inequality for Sugeno integral without the comonotonicity condition.
The paper is organized as follows: Section 2 recalls the concepts of Choquet-like integrals while
Sections 3, 4 present our main results. Finally, some concluding remarks are added.
2. Preliminaries
To prove our results, we shall first recall some basic definitions and previous results. For details,
we refer to [13] (see also [19]).
2.1 ([24]) An operation ⊕ : [0, ∞]2 → [0, ∞] is called a pseudo-addition if the following properties are satisfied:
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STOLARSKY’S INEQUALITY FOR CHOQUET-LIKE EXPECTATION
(P1) a ⊕ 0 = 0 ⊕ a = a (neutral element);
(P2) (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) (associativity);
(P3) a ≤ c and b ≤ d imply that a ⊕ b ≤ c ⊕ d (monotonicity);
(P4) an → a and bn → b imply that an ⊕ bn → a ⊕ b (continuity).
2.2 ([13, 24]) Let ⊕ be a given pseudo-addition on [0, ∞]. Another binary operation
⊗ on [0, ∞] is said to be a pseudo-multiplication corresponding to ⊕ if the following properties are
satisfied:
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(M1) a ⊗ (x ⊕ y) = (a ⊗ x) ⊕ (a ⊗ y) (left distributivity);
(M2) a ≤ b implies (a ⊗ x) ≤ (b ⊗ x) and (x ⊗ a) ≤ (x ⊗ b) (monotonicity);
(M3) a ⊗ x = 0 ⇔ a = 0 or x = 0 (absorbing element and no zero divizors);
(M4) exists e ∈ (0, ∞] (i.e., there exist the neutral element e) such that e ⊗ x = x ⊗ e = x for any
x ∈ [0, ∞] (neutral element);
(M5) an → a ∈ (0, ∞) and xn → x imply(an ⊗ xn ) → (a⊗ x) and ∞⊗ x = lim (a⊗ x) (continuity);
a→∞
(M7) (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) (associativity).
2.3
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(M6) a ⊗ x = x ⊗ a (commutativity);
([13]) Let ⊗ be a pseudo-multiplication corresponding to a given pseudo-addition
⊕ fulfilling axioms (M1)–(M7).
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(I) If its identity element e is not an idempotent of ⊕, then there is a unique continuous strictly
increasing function g : [0, ∞] → [0, ∞] with g(0) = 0 and g(∞) = ∞, such that g(e) = 1 and
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a ⊕ b = g −1 (g(a) + g(b))
a ⊗ b = g −1 (g(a) · g(b))
⊕ is called a g-addition,
⊗ is called a g-multiplication.
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(II) If the identity element e of the pseudo-multiplication is also an idempotent of ⊕ (i.e., e ⊕ e
= e), then ⊕ = ∨ (= sup, i.e., the logical addition).
(p)
For x ∈ [0, ∞] and p ∈ (0, ∞), we will introduce the pseudo-power x⊗ as follows: If p = n is a
(n)
natural number, then x⊗ = x ⊗ x ⊗ · · · ⊗ x. If p is not a natural number, then the corresponding
n−times
(p)
(m)
(n)
power is defined by x⊗ = sup y⊗ | y⊗ x, where m, n are natural numbers such that m
n p .
(p)
Evidently, if x ⊗ y = g −1 (g(x) · g(y)), then x⊗ = g −1 (g p (x)).
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2.4 ([11])
A monotone measure µ on a measurable space (Ω, F ) is a set function
µ : F → [0, ∞) satisfying
(i) µ (∅) = 0;
(ii) µ(Ω) > 0;
(iii) µ(A) ≤ µ(B) whenever A ⊆ B;
moreover, µ is called finite if µ = µ(Ω) < ∞. The triple (Ω, F , µ) is also called a monotone
measure space if µ is a monotone measure on F .
We call µ a monotone probability, if µ = 1. When µ is a monotone probability, the triple
(Ω, F , µ) is called a monotone probability space.
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HAMZEH AGAHI — RADKO MESIAR
)
2.5 Let (Ω, F ) be a measurable space. For each number a ∈ (0, ∞], M(Ω,F
denotes
a
the set of all monotone measures (in the sense of Definition 2.4) satisfying µ = a.
2.6 Let (Ω, F , µ) be a monotone measure space and X : Ω → [0, ∞) be an F -measur-
able function. The Choquet expectation of X over A ∈ F w.r.t. the monotone measure µ is defined
as
∞
µ
(2.1)
EC [XIA ] = µ(A ∩ {X ≥ t}) dt.
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0
where the integral on the right-hand side is the (improper) Riemann integral. In particular, if
A = Ω, then
∞
µ
EC [Xt] = µ({X ≥ t}) dt.
0
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Mesiar [13] has shown that there are two classes of Choquet-like integral: the Choquet-like
integral (denoted by EµCl,g ) based on a g-addition and a g-multiplication and the Choquet-like
integral based on ∨ and a corresponding pseudo-multiplication ⊗.
2.7 ([13]) Let (Ω, F ) be a measurable space and µ : F → [0, ∞] be a monotone
measure. Let ⊕ and ⊗ be generated by a generator g. The Choquet-like expectation of a nonnegative measurable function X over A ∈ F w.r.t. the monotone measure µ can be represented
as
∞
g(µ)
µ
−1
−1
gµ(A ∩ {g(X) ≥ t}) dt .
ECl,g [XIA ] = g (EC [g(X)IA ]) = g
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0
g(µ)
EµCl,g [X] = g −1 EC [g(X)] .
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In particular, if A = Ω, then
(2.2)
Remark 2.8 Notice that we sometimes call this kind of Choquet-like integral a g-Choquet integral (g-C-integral for short). It is plain that the g-C-integral is the original Choquet integral
(expectation) whenever g = i (the identity mapping).
2.9 ([13]) Let ⊗ be a pseudo-multiplication corresponding to ∨ and fulfilling
(M1)–(M7). Then the Choquet-like integral (so-called S⊗
µ integral) of a measurable function
X : Ω → [0, ∞) w.r.t. a finite monotone measure µ can be represented as
A
S⊗
µ [Xt] = sup (a ⊗ µ({X ≥ a}).
(2.3)
a∈[0,∞]
It is plain that the S⊗
µ integral is the Sugeno integral (denoted by Suµ [·]) whenever ⊗ = ∧ [25].
During this paper, we always consider the existence of all S⊗
µ [·].
Remark 2.10 Notice that when working on [0, 1], we mostly deal with e = 1, then ⊗ = is
a semicopula (t-seminorm), i.e., a binary operation : [0, 1]2 → [0, 1] which is non-decreasing in
both components and has 1 as neutral element. Then ⊗ = satisfies a ⊗ b ≤ min(a, b) for all
(a, b) ∈ [0, 1]2 , see [8].
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STOLARSKY’S INEQUALITY FOR CHOQUET-LIKE EXPECTATION
2.11
The S⊗
µ integral on the [0, 1] scale related to the semicopula is given by
S
µ [X] = sup (a µ({X ≥ a}).
a∈[0,1]
This type of integral was called seminormed integral in [22].
[5: Theorem 2.13] helps us to reach the main results.
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Remark 2.12 For a fixed strict t-norm T , the corresponding STµ integral is the so-called SugenoWeber integral [27]. If is the standard product, then the Shilkret integral [20] (denoted by
Shµ [·]) can be recognized. Notice that the original Sugeno integral (denoted by Suµ [·]) which was
introduced by Sugeno [23] in 1974 is a special seminormed integral when = min.
2.13 Let X, Y : Ω → [0, ∞) be two nondecreasing functions and (Ω, F , µ) a monotone
measure space and : [0, ∞)2 → [0, ∞) be continuous and nondecreasing in both arguments and
ϕi : [0, ∞) → [0, ∞), i = 0, 1, 2 be continuous and strictly increasing functions. If ⊗ is a pseudomultiplication (with neutral element e = µ) corresponding to ∨ satisfying
(Ω,F )
then for any monotone measure µ ∈ Me
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−1
−1
ϕ−1
0 (ϕ0 (p1 p2 ) ⊗ c) ≥ [ϕ1 [(ϕ1 (p1 )) ⊗ c] p2 ] ∨ [p1 ϕ2 [ϕ2 (p2 ) ⊗ c]]
such that S⊗
µ [·] < ∞, the inequality
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−1 ⊗
−1 ⊗
⊗
ϕ−1
0 (Sµ [ϕ0 (X Y )]) ≥ ϕ1 (Sµ [ϕ1 (X)]) ϕ2 (Sµ [ϕ2 (Y )])
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holds.
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3. Stolarsky’s inequality
The purpose of this section is to prove the Stolarsky inequality for two classes of Choquet-like
integrals. Theorem 3.1 gives us the Stolarsky’s inequality for the first class Choquet-like integrals,
i.e., for Choquet-like expectation. Afterwards, we will obtain the Stolarsky’s inequality for the
second class in Theorem 3.3.
3.1 Let ([0, 1], B([0, 1]), µ) be a monotone probability space and let the pseudo-operations
A
be generated by a generator g : [0, ∞] → [0, ∞], g(1) = 1. Let a, b > 0. Then the Stolarsky inequality
holds for the corresponding Choquet-like expectation, i.e., for any nonincreasing real-valued function
X : [0, 1] → [0, 1] it holds
1
1 1 EµCl,g X(ω a+b ) ≥ EµCl,g X(ω a ) ⊗ EµCl,g X(ω b ) ,
(3.1)
where g(µ) is a lower-semicontinuous monotone probability absolutely continuous with respect to
the Lebesgue measure λ, i.e., if λ(E) = 0 for some E ∈ B([0, 1]) then also g(µ(E)) = 0.
P r o o f. Observe that
g(µ) 1
1
EµCl,g [X(ω a+b )] = g −1 EC g X(ω a+b ) .
(3.2)
From (3.2) and using the Stolarsky inequality for Choquet expectation (1.1), we have
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HAMZEH AGAHI — RADKO MESIAR
g(µ) 1
1 1 g(µ) EµCl,g [X(ω a+b )] ≥ g −1 EC g X(ω a ) · EC g X(ω b )
−1 g µ −1 g µ 1 1 −1
g g
EC
g X(ω a )
EC
g X(ω b )
·g g
=g
1 1 = g −1 g EµCl,g X(ω a ) · g EµCl,g X(ω b )
1 1 = EµCl,g X(ω a ) ⊗ EµCl,g X(ω b ) .
This completes the proof.
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Example 3.2
√
(i) Let g(x) = xα , α > 0. The corresponding pseudo-operations are x ⊕ y = α xα + y α and
x ⊗ y = xy. Then (3.1) reduces on the following inequality
α 1
α
α
1
1
α
EµC (X(ω a+b ))α α EµC (X(ω a ))α · α EµC (X(ω b ))α .
µ
E2C
−1
2X(ω
1
a+b
)
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(ii) Let g(x) = 2x − 1. The corresponding pseudo-operations are x ⊕ y = ln12 ln (2x + 2y − 1)
and x ⊗ y = ln12 ln ((2x − 1) (2y − 1) + 1) . Then (3.1) reduces on the following inequality
1
1
µ
µ
− 1 E2C −1 2X(ω a ) − 1 · E2C −1 2X(ω b ) − 1 .
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Now we consider the second class of the Choquet-like integral where is based on ∨ and a
corresponding pseudo-multiplication ⊗ with neutral element e = µ.
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3.3 Let X : [0, ∞) → [0, ∞) be a nondecreasing function, ([0, ∞), B[0, ∞), µ) be a
monotone measure space, : [0, ∞)2 → [0, ∞) be continuous and nondecreasing in both arguments
and bounded from above by min and ϕi : [0, ∞) → [0, ∞), i = 0, 1, 2 be continuous and strictly
increasing functions. Let α, β, γ : [0, ∞) → [0, ∞) be increasing bijections such that α max {β, γ}.
If ⊗ is a pseudo-multiplication (with neutral element e = µ) corresponding to ∨ satisfying
−1
−1
ϕ−1
0 (ϕ0 (p1 p2 ) ⊗ c) ≥ ϕ1 [(ϕ1 (p1 )) ⊗ c] p2 ∨ p1 ϕ2 [ϕ2 (p2 ) ⊗ c] ,
([0,∞),B[0,∞))
such that S⊗
then for any monotone measure µ ∈ Me
µ [·] < ∞, the inequality
⊗
−1
⊗
S⊗
ϕ−1
Sµ ϕ2 (X(γ))
ϕ−1
µ ϕ1 X(β)
0 (Sµ ϕ0 (X(α)) ) ≥ ϕ1
2
holds.
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P r o o f. Since α ≥ β, α ≥ γ and X is nondecreasing, we have X(α) ≥ X(β) ∧ X(γ) ≥ X(β) X(γ). Note that all the three functions X(α), X(β) and X(γ) are nondecreasing. Now applying
Theorem 2.13, by the monotonicity of S⊗
µ integral there holds
−1 ⊗
⊗
ϕ−1
0 (Sµ [ϕ0 (X (α))]) ≥ ϕ0 (Sµ [ϕ0 (X(β) X(γ))])
⊗
⊗
≥ ϕ−1
Sµ [ϕ1 (X (β))] ϕ−1
Sµ [ϕ2 (X (γ))] .
1
2
Since we work on the [0, 1] scale in Theorem 3.3 then ⊗ = is semicopula (t-seminorm) and
the following result holds.
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STOLARSKY’S INEQUALITY FOR CHOQUET-LIKE EXPECTATION
3.4
Let X : [0, 1] → [0, 1] be a nondecreasing function, ([0, 1], B[0, 1], µ) be a monotone probability space, ∗ : [0, 1]2 → [0, 1] be continuous and nondecreasing in both arguments and
bounded from above by min and ϕi : [0, 1] → [0, 1], i = 0, 1, 2 be continuous and strictly increasing functions. Let α, β, γ : [0, 1] → [0, 1] be increasing bijections such that α max {β, γ}. If
semicopula satisfying
−1
−1
ϕ−1
0 (ϕ0 (p1 p2 ) c) ≥ ϕ1 [(ϕ1 (p1 )) c] p2 ∨ p1 ϕ2 [ϕ2 (p2 ) c] ,
then the inequality
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−1
ϕ−1
Sµ [ϕ1 (X (β))] ϕ−1
Sµ [ϕ2 (X (γ))]
0 (Sµ [ϕ0 (X(α))]) ≥ ϕ1
2
holds.
Let ϕ0 (x) = xr , 0 < r < ∞, and ϕ1 (x) = ϕ2 (x) = xs , 0 < s < ∞, then for all 0 < s < ∞, then
we get the following result.
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3.5 Let X : [0, 1] → [0, 1] be a nondecreasing function, ([0, 1], B[0, 1], µ) be a monotone probability space and : [0, 1]2 → [0, 1] be continuous and nondecreasing in both arguments
and bounded from above by min. Let α, β, γ : [0, 1] → [0, 1] be increasing bijections such that
α max {β, γ} . If semicopula satisfying
1
1
1
((a b)r c)) r ≥ (as c) s b ∨ a (bs c) s ,
then the inequality
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1
1
r 1r
s s
s s
(S
S
µ [(X(α)) ]) ≥ Sµ [(X (β)) ]
µ [(X (γ)) ]
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holds for all 0 < s < ∞ and 0 < r < ∞.
Specially, when s = 1, we have the following result.
3.6
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Let X : [0, 1] → [0, 1] be a nondecreasing function, ([0, 1], B[0, 1], µ) be a monotone probability space and : [0, 1]2 → [0, 1] be continuous and nondecreasing in both arguments
and bounded from above by min. Let α, β, γ : [0, 1] → [0, 1] be increasing bijections such that
α max {β, γ}. If semicopula satisfying
then the inequality
1
((a b)r c) r ≥ [(a c) b] ∨ [a (b c)] ,
(3.3)
r 1r
(S
µ [(X(α)) ]) ≥ Sµ [X(β)] Sµ [X (γ)]
(3.4)
A
holds for all 0 < r < ∞.
√
Example 3.7 Let nondecreasing function X be defined as X(ω) = ω. Let and be the
standard product. Then for r = 1, (3.3) holds readily. If µ is the Lebesgue measure, then
1
4
2√
and
Shµ X ω 4 =
.
Shµ [X (ω)] =
3,
Shµ X ω 2 =
9
4
27
Therefore,
2√
S
3
µ [X(ω)] = Shµ [X(ω)] =
9
2 4 1
= Shµ X ω 2 · Shµ X ω 4 = S
Sµ X ω
.
µ X ω
27
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HAMZEH AGAHI — RADKO MESIAR
Example 3.8 Let (x, y) = (x + y − 1) ∨ 0 in Example 3.7. Then a straightforward calculus
shows that
1
4
2√
+
− 1 ∨ 0 = Shµ X ω 2 + Shµ X ω 4 − 1 ∨ 0
S
3
µ [X(ω)] =
9
4 27
4 2
.
Sµ X ω
= S
µ X(ω )
Let = min in Corollary 3.6. Then we get the following result.
3.9 ([2: Theorem 3.5]) Let X : [0, 1] → [0, 1] be a nondecreasing function and
([0, 1], B[0, 1], µ) a monotone probability space. Let β, γ be automorphisms on [0, 1] (i.e.,
β, γ : [0, 1] → [0, 1] are increasing bijections) and α = (β −1 γ −1 )−1 . If : [0, 1]2 → [0, 1] be
any continuous aggregation function which is jointly strictly increasing and bounded from above by
min, then
(3.5)
Suµ [X(α)] ≥ (Suµ [X(β)]) (Suµ [X (γ)]) .
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P r o o f. As is bounded from above by min, α−1 = β −1 γ −1 ≤ β −1 ∧ γ −1 and thus α ≥ β,
α ≥ γ. Since X is nondecreasing, we have X(α) ≥ X(β) ∧ X(γ) ≥ X(β) X(γ). Notice that if is bounded from above by min, then for r = 1, (3.3) holds readily. Applying the Corollary 3.6, the
proof is completed.
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4. A new inequality
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This section intends to present a new Minkowski’s inequality without the comonotonicity condition for two classes of Choquet-like integrals. Theorems 4.1 and 4.3 are the main results of this
section. During this section, we always consider the existence of all Choquet expectation.
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4.1 Let X, Y be two non-negative measurable functions and let the pseudo-operations
be generated by a generator g. If
mg (Y (ω)) g(X(ω)) M g (Y (ω))
for all
ω ∈ Ω,
(4.1)
where 0 < m M , then for the Choquet-like expectation given by (2.2), the following inequalities
(s) ( 1s )
( 1s ) ( 1s ) (s)
(s)
EµCl,g X ⊕ Y
K1 ⊗ EµCl,g X⊗
⊕ EµCl,g Y⊗
,
(4.2)
A
⊗
⊗
( )
(s)
EµCl,g (X ⊕ Y )
K2 ⊗
1
s
⊗
hold where s 1 and the constants K1 = g −1
⊗
( ) ( 1s ) (s)
(s)
EµCl,g X⊗
⊕ EµCl,g Y⊗
,
1
s
⊗
M(m+1)+(M+1)
(M+1)(m+1)
−1 , K2 = g −1
⊗
m(M+1)+(m+1)
(m+1)(M+1)
are independent of µ.
P r o o f. We will prove inequality (4.2) and the other case is similar. By (4.1), we have
g (X(ω)) M (g (X(ω)) + g (Y (ω))) − M g (X(ω)) .
So
s
s
s
(M + 1) (g(X(ω))) M s (g(X(ω)) + g(Y (ω))) ,
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−1 STOLARSKY’S INEQUALITY FOR CHOQUET-LIKE EXPECTATION
and then
1s
g(µ)
EC [g (X s )]
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1s
M g(µ)
EC [(g(X) + g(Y ))s ] .
M +1
1
1
(g(X(ω)) + g (Y (ω))) − m
g (Y
Also, since mg (Y (ω)) g (X(ω)), we have g (Y (ω)) m
So
s
1 s
1
s
s
+ 1 (g (Y (ω))) (g (X(ω)) + g (Y (ω))) ,
m
m
and then,
1s
1
1 g(µ)
g(µ)
s s
EC [(g(X) + g(Y )) ] .
EC [g(Y s )]
m+1
Therefore, (4.4) and (4.5) give us the following result:
1s 1s 1
g(µ)
g(µ)
g(µ)
s s
s
s
g(K1 ) EC [g(X )] + EC [g(Y )]
EC [(g(X) + g(Y )) ] .
(4.4)
(ω)).
(4.5)
(4.6)
Now, observe that
( 1s )
1s g(µ)
EµCl,g (X ⊕ Y )(s)
g (X ⊕ Y )(s)
= g −1 EC
⊗
⊗
g(µ)
1 s s
g g −1 ((g(X ⊕ Y )) )
g(µ)
1s .
[((g ◦ X) + (g ◦ Y ))s ]
EC
EC
R
= g −1
C
= g −1
O
By using (4.6), we have
1s g(µ)
s
g −1 EC [((g ◦ X) + (g ◦ Y )) ]
1 g(µ)
1 g(µ)
g −1 g(K1 ) EC [(g ◦ X)s ] s + EC [(g ◦ Y )s ] s
U
TH
g(µ)
g(µ)
1 1 = g −1 g(K1 ) g g −1 EC [(g ◦ X)s ] s + g g −1 EC [(g ◦ Y )s ] s
1 1 g(µ)
g(µ)
+ g g −1 EC [(g ◦ Y )s ] s
= g −1 g(K1 ) g g −1 EC [(g ◦ X)s ] s
= K1 ⊗ g −1
g(µ)
g(µ)
1 1 g g −1 EC [(g ◦ X)s ] s + g g −1 EC [(g ◦ Y )s ] s
A
1s 1s g(µ) −1
g(µ) −1
⊕ g −1 EC
g g ((g ◦ X)s )
g g ((g ◦ Y )s )
= K1 ⊗ g −1 EC
g(µ) (s) 1s
g(µ) (s) 1s
⊕ g −1 EC g Y⊗
)
= K1 ⊗ g −1 EC g X⊗
g(µ) (s) 1s g(µ) (s) 1s = K1 ⊗ g −1 g g −1 EC g X⊗ ]
⊕ g −1 g g −1 EC g Y⊗
(s) 1s (s) 1s ⊕ g −1 g EµCl,g Y⊗
= K1 ⊗ g −1 g EµCl,g X⊗
= K1 ⊗
(s) ( 1 ) (s) ( 1s ) .
EµCl,g X⊗ ⊗s ⊕ EµCl,g Y⊗
⊗
This completes the proof.
1243
HAMZEH AGAHI — RADKO MESIAR
Example 4.2
(i) Let g(x) = ex . The corresponding pseudo-operations are x⊕y = ln(ex +ey ) and x⊗y = x+y.
Then (4.2) and (4.3) reduce on the following inequalities
µ µ s 1s
1 µ 1 K1 + ln EeC esX s + EeC esY s ,
ln EeC eX + eY
√
α
xα + y α and
O
PY
µ µ s 1s
1 µ 1 ln EeC eX + eY
K2 + ln EeC esX s + EeC esY s ,
where K1 = − ln M(m+1)+(M+1)
, K2 = − ln m(M+1)+(m+1)
, s 1.
(M+1)(m+1)
(m+1)(M+1)
(ii) Let g(x) = xα , α > 0. The corresponding pseudo-operations are x ⊕ y =
x ⊗ y = xy. Then (4.2) and (4.3) reduce on the following inequalities
α
1 α
1 s
αs
µα
α
α
EC [(X + Y ) ] K1 α EµC [X αs ] s + EµC [Y αs ] s ,
α
1 α
1 α
s
EµC [(X α + Y α ) ] K2 α EµC [X αs ] s + EµC [Y αs ] s ,
αs
M(m+1)+(M+1)
(M+1)(m+1)
− α1
, K2 =
m(M+1)+(m+1)
(m+1)(M+1)
− α1
, s 1.
C
where K1 =
Now, we focus on the second class of Choquet-like integrals.
Let X, Y be two non-negative measurable functions such that
R
4.3
0<m
X(ω)
M
Y (ω)
for all
ω ∈ Ω.
U
TH
O
Let s > 0. If ⊗ is a pseudo-multiplication satisfying
(m + 1)s ms a
a
s
(a ⊗ b) min
⊗ b , (M + 1)
s ⊗b
ms
(m + 1)s
(M + 1)
(4.7)
then, for any monotone measure µ, the S⊗
µ integral (2.3) satisfies the inequality
1
⊗ s 1s ⊗ s 1s
s s
Sµ [X ] + Sµ [Y ] K S⊗
µ [(X + Y ) ]
where K =
m(M+1)+(m+1)
(m+1)(M+1) .
P r o o f. Since X(ω) m (X(ω) + Y (ω)) − mX (ω), we have
s
s
(m + 1) (X(ω))s ms (X(ω) + Y (ω)) .
A
The monotonicity of S⊗
µ integral and (4.7) imply that
1s
m s
1
⊗ s 1s
m ⊗
s
s Sµ [(X + Y ) ] s .
(X + Y )
Sµ [X ] S⊗
µ
m+1
m+1
Also, since M X(ω)
Y (ω)
(4.8)
1
1
we have Y (ω) M
(X(ω) + Y (ω)) − M
Y (ω). So
1
s
1 s
+ 1 (Y (ω))s (X(ω) + Y (ω))s ,
M
M
and then,
⊗ s 1s
Sµ [Y ] S⊗
µ
1244
1s
1
1
1 ⊗
s
s Sµ [(X + Y ) ] s .
s (X + Y )
(M + 1)
M +1
(4.9)
STOLARSKY’S INEQUALITY FOR CHOQUET-LIKE EXPECTATION
By (4.8) and (4.9), the following result holds
1
⊗ s 1s ⊗ s 1s
s s
.
Sµ [X ] + Sµ [Y ] K S⊗
µ [(X + Y ) ]
This completes the Proof.
Notice that if ⊗ is minimum (i.e., for Sugeno integral) in Theorem 4.3, then the following result
holds.
Let X, Y be two non-negative measurable functions such that
0<m
X(ω)
M
Y (ω)
for all
Then for the Sugeno integral, the inequality
1
O
PY
4.4
1
ω ∈ Ω.
1
(Suµ [X s ]) s + (Suµ [Y s ]) s K (Suµ [(X + Y )s ]) s
holds where K =
m(M+1)+(m+1)
(m+1)(M+1) ,
s > 0.
(4.10)
C
Remark 4.5 If X(ω) = αY (ω), α > 0 in Corollary 4.4, then m = M = α, K = 1 and (4.10)
reduces on the following inequality
1
1
1
(Suµ [X s ]) s + (Suµ [Y s ]) s (Suµ [(X + Y )s ]) s .
R
The following example proves that the inequality of Corollary 4.4 is sharp.
O
Example 4.6 Let Ω = [0, 2], X(ω) = Y (ω) ≡ 1. Let µ(A) = λ(A) where λ is the Lebesgue
measure on R. Clearly X(ω)
Y (ω) = 1. Then m = M = 1. So, by Corollary 4.4, K = 1 and
Suµ [X + Y ] = 2,
U
TH
Therefore,
Suµ [X] = Suµ [Y ] = 1.
K Suµ [X + Y ] = 2 = Suµ [X] + Suµ [Y ] .
Notice that when working on [0, 1] in Theorem 4.3, then ⊗ = is semicopula (t-seminorm) and
the following result holds.
4.7
Let X, Y : Ω → [0, 1] be two non-negative measurable functions such that
X(ω)
M
Y (ω)
for all
ω ∈ Ω.
A
0<m
Let s > 0. If semicopula satisfying
a
(m + 1)s ms a
s
b , (M + 1)
(a b) min
s b
ms
(m + 1)s
(M + 1)
(4.11)
then for the S
µ integral (2.3), the inequality
1
s 1s s 1s
s s
Sµ [X ] + Sµ [Y ] K S
µ [(X + Y ) ]
holds where K =
m(M+1)+(m+1)
(m+1)(M+1) .
The following example shows that the condition (4.11) in Corollary 4.7 cannot be omitted.
1245
HAMZEH AGAHI — RADKO MESIAR
Example 4.8 Let Ω = [0, 1], X(ω) = Y (ω) = ω+1
ukasiewicz t-norm
2 . Let semicopula be the L
λ(A)
TL (i.e., TL (x, y) = max {(x + y − 1) , 0}) and µ(A) = 1+λ(A) where λ is the Lebesgue measure
on R. Clearly
X(ω)
Y (ω)
= 1. Then m = M = 1. So, by Corollary 4.7, K = 1 and
STµL [X + Y ] =
TL (α, µ({X + Y ≥ α})) =
α∈[0,1]
[X] =
STµL
[Y ] =
TL (α, µ({X ≥ α}))
α∈[0,1]
=
α∈[ 12 ,1]
Therefore,
2α − 2
3 √
TL α,
= − 2.
2α − 3
2
O
PY
STµL
1
,
2
√
1
= K S
S
µ [X] + Sµ [Y ] = 3 − 2 2 µ [X + Y ] .
2
4.9
C
Note that if is the usual product (i.e., for Shilkret integral) in Corollary 4.7, then we have
the following result.
Let X, Y : Ω → [0, 1] be two non-negative measurable functions such that
Then the inequality
1
X(ω)
M
Y (ω)
for all
ω ∈ Ω.
R
0<m
1
s
1
holds where K =
O
(Shµ [X s ]) s + (Shµ [Y s ]) s K (Shµ [(X + Y ) ]) s
m(M+1)+(m+1)
(m+1)(M+1) ,
s > 0.
U
TH
Finally, if is minimum (i.e., for original Sugeno integral) in Corollary 4.7, then the following
result holds.
4.10
Let X, Y : Ω → [0, 1] be two non-negative measurable functions such that
0<m
X(ω)
M
Y (ω)
for all
ω ∈ Ω.
A
Then the inequality
holds where K =
1
1
1
(Suµ [X s ]) s + (Suµ [Y s ]) s K (Suµ [(X + Y )s ]) s
m(M+1)+(m+1)
(m+1)(M+1) ,
s > 0.
1
Example 4.11 Let Ω = [0, 1], X(ω) = ω + 100
and Y (ω) = 1 − ω2 . Let µ(A) = λ2 (A) where λ
X(ω)
1
1
101
is the Lebesgue measure on R. Clearly 100
Y (ω) 101
50 . Then m = 100 and M = 50 . So, by
Corollary 4.10, K = 0.341,
Suµ [X + Y ] = 1,
Suµ [X] = 0.3875 and
Suµ [Y ] = 0.6096.
Therefore,
K Suµ [X + Y ] = 0.341 0.9971 = Suµ [X] + Suµ [Y ] .
1246
STOLARSKY’S INEQUALITY FOR CHOQUET-LIKE EXPECTATION
5. Conclusions
O
PY
This paper has successfully proved the Stolarsky inequality for Choquet-like expectation, which
generalizes the pervious result of this inequality on Choquet expectation. Moreover, a new
Minkowski’s inequality without the comonotonicity condition for Choquet-like integrals is introduced. At first, we thoroughly described two classes of Choquet-like integrals. Then, we prepared
extensions of these inequalities from the Choquet expectation and the Sugeno integral to the two
classes of Choquet-like integrals. Observe that similarly some Chebyshev-type inequalities shown
for the Choquet and Sugeno integrals in [3, 14] were generalized for the Choquet-like integrals in
[5, 19].
Acknowledgement The authors are very grateful to the anonymous reviewers for their insightful
and constructive comments and suggestions that have led to an improved version of this paper.
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Received 24. 3. 2014
Accepted 13. 9. 2014
1248
* Corresponding author:
Department of Mathematics
Faculty of Basic Sciences
Babol Noshirvani University of Technology
Shariati Av., Babol
IRAN
Post Code: 47148-71167
E-mail : [email protected]
h [email protected]
** Department of Mathematics
and Descriptive Geometry
Faculty of Civil Engineering
Slovak University of Technology
SK–810 05 Bratislava
SLOVAKIA
Institute of Information Theory and Automation
Academy of Sciences of the Czech Republic
Pod vodarenskou vezi 4
CZ–182 08 Praha 8
CZECH REPUBLIC
E-mail : [email protected]