Soft Comput DOI 10.1007/s00500-015-1909-9
FOUNDATIONS
On Carlson’s inequality for Sugeno and Choquet integrals Michał Boczek1 · Marek Kaluszka1
© Springer-Verlag Berlin Heidelberg 2015
Abstract We present a Carlson type inequality for the Sugeno integral and a much wider class of functions than the comonotone functions. We also provide three Carlson type inequalities for the Choquet integral. Our inequalities generalize many known results. Keywords Choquet integral · Sugeno integral · Capacity · Semicopula · Carlson inequality
1 Introduction The pioneering concept of the fuzzy integral was introduced by Sugeno (1974) as a tool for modeling nondeterministic problems. Theoretical investigations of the integral and its generalizations have been pursued by many researchers. Wang and Klir (2009) presented an excellent general overview on fuzzy integration theory. On the other hand, fuzzy integrals have also been successfully applied to various fields (see, e.g., Hu 2007; Narukawa and Torra 2007). The study of inequalities for Sugeno integral was initiated by Román-Flores et al. (2007). Since then, the fuzzy integral counterparts of several classical inequalities, including Chebyshev’s, Jensen’s, Minkowski’s and Hölder’s inequalities, are given by Flores-Franuliˇet al. (2007), Agahi et al. (2012), Wu et al. (2010), Mesiar and Ouyang (2009) and others. Furthermore, many researchers started to study inequalities for the seminormed Sugeno integral (Agahi and Communicated by A. Di Nola.
B 1
Yaghoobi 2010; Kaluszka et al. 2014a, b; Ouyang and Mesiar 2009). The integral Carlson inequality is of the form ⎛
∞ f (x) dx ≤ 0
⎞1 ⎛ ∞ ⎞1 4 4 2 2 2 ⎠ ⎠ ⎝ f (x) dx x f (x) dx
0
0
(1) with f being any nonnegative, measurable function such that the integrals on the right-hand side converge (Carlson 1934). α The equality in (1) is attained iff f (x) = β+x 2 for some constants α ≥ 0, β > 0. The modified versions of the Carlson inequality can be found in Barza et al. (1998) and Mitrinovi´c et al. (1991), Chapter 8. Caballero and Sadarangani (2010) showed that the Carlson inequality (1) is not true for the Sugeno integral defined as f dμ = sup{α ∧ μ(A ∩ { f ≥ α})},
(S) A
(2)
α≥0
where f : X → [0, ∞), μ is a monotone measure on X, A is a measurable subset of X and { f ≥ α} denotes {x ∈ X : f (x) ≥ α} (see Wang and Klir 2009). Hereafter, a ∧ b = min(a, b). They also proved that if f : [0, 1] → [0, ∞) is a non-decreasing function and μ is the Lebesgue measure, then the following Carlson type inequality holds
Michał Boczek
[email protected] Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland
√ π⎝
∞
(S) [0,1]
⎛ √ ⎜ f dμ ≤ 2 ⎝(S)
[0,1]
⎞1 ⎛ 4
⎟ ⎜ f dμ⎠ ⎝(S)
2
⎞1 4
⎟ x f dμ⎠ . (3) 2
2
[0,1]
123
M. Boczek, M. Kaluszka
The assumption of monotonicity of f cannot be omitted. Xu and Ouyang (2012) presented a generalization of inequality (3) of the form (S) A
⎛ 1 f 1 dμ ≤ √ ⎝(S) C ⎛ × ⎝(S)
⎞
1 2p
2 Carlson’s type inequalities for Sugeno integral
p p f 1 f 2 dμ⎠
A
⎞
1 2q
q q f 1 f 3 dμ⎠
,
(4)
A
where C = (S) A f 2 dμ (S) A f 3 dμ , μ is a monotone measure on X , A is a measurable subset of X , p, q ≥ 1, and f i : X → [0, ∞) are measurable functions such that any two out of three functions f i , i = 1, 2, 3, are comonotone and (S) A f i dμ ≤ 1 for all i. A similar result was obtained by Wang and Bai (2011). They proved that if f, g, h are measurable functions from X to [0, ∞), both pairs f, h and f, g are comonotone set such
functions, and A ⊂ X is a measurable that (S) A f dμ ≤ 1, (S) A g dμ ≤ 1, (S) A h dμ ≤ 1, (S) A f g dμ ≤ 1 and (S) A f h dμ ≤ 1, then (S)
⎞ 1 ⎛ p+q 1 ⎝ p p ⎠ f dμ ≤ f g dμ (S) K
A
A
⎛ × ⎝(S)
⎞
1 p+q
f q h q dμ⎠
,
(5)
A
p q where K = (S) A g dμ p+q (S) A h dμ p+q . Some related Carlson type inequalities for the Sugeno integral were also given in Daraby and Arabi (2013). A counterpart of the Carlson inequality for the Choquet integral was provided in Tang and Ouyang (2012) (see Sect. 3 below). The purpose of this paper is to study the Carlson type inequalities of the form Iμ ( f ) ≤ H (Iμ (φ( f, g)), Iμ (ψ( f, h))),
(6)
where Iμ ( f ) stands for the Sugeno, the generalized Sugeno or the Choquet integrals with respect to μ and H, φ, ψ are real valued functions. In Sect. 2, we provide a generalization of inequalities (3)–(5) for the Sugeno as well as the generalized Sugeno integral. The results are obtained for a rich class of functions, including the comonotone functions as a special case. In Sect. 3, we present the corresponding results for the Choquet integral. First, we show that (6) does not hold if we do not impose some extra conditions either on functions f, g, h or monotone measure μ. Next, we obtain the Carlson type inequalities for comonotone functions and an arbitrary
123
monotone measure. Finally, we present inequalities for any non-negative measurable functions f, g, h and a subadditive or submodular monotone measure μ.
Let (X, F) be a measurable space and μ : F → Y be a monotone measure, i.e., μ(∅) = 0, μ(X ) > 0 and μ(A) ≤ μ(B) whenever A ⊂ B. Throughout the paper Y = [0, 1] or Y = [0, ∞]. We say that ◦ : Y × Y → Y is a non-decreasing operator if a ◦ c ≥ b ◦ d for a ≥ b and c ≥ d. Let f, g : X → Y be measurable functions and A, B ∈ F. The functions f | A and g| B are positively dependent with respect to μ and an operator : Y × Y → Y if for any a, b ∈ Y μ({ f | A ≥ a}∩{g| B ≥ b}) ≥ μ({ f | A ≥ a}) μ({g| B ≥ b}), (7) where h |C denotes the restriction of the function h : X → Y to a set C ⊂ X (Kaluszka et al. 2014a). Obviously, {h |C ≥ a} = {x ∈ C : h(x) ≥ a} = C ∩ {h ≥ a}. Taking a b = a ∧ b and a b = ab, we recover two important examples of positively dependent functions, namely comonotone functions and independent random variables. Recall that f and g are comonotone if ( f (x) − f (y))(g(x) − g(y)) ≥ 0 for all x, y ∈ X . More examples of positively dependent functions can be found in Kaluszka et al. (2014a). First, we present a Carlson type inequality for the Sugeno integral (2). Suppose : Y × Y → Y is non-decreasing and left-continuous operator, i.e., limn→∞ (xn yn ) = x y for all xn x and yn y, where an a means that limn→∞ an = a and an < an+1 for all n. Theorem 2.1 Assume that p, q ≥ 1, r, s > 0 and : Y × Y → Y is an arbitrary operator. Then, for any pairs of positively dependent functions f |A , g|B and f |A , h |B such that (S) A∩B f ψ dμ ≤ 1 for ψ = g, h, the following inequality ⎛⎛ ⎝⎝(S)
⎞ ⎛ f dμ⎠ ⎝(S)
A
⎝⎝(S)
≤ ⎝(S)
⎞⎞r g dμ⎠⎠
B
⎛⎛
⎛
A
A∩B
⎞ ⎛ f dμ⎠ ⎝(S) ⎞r
p
( f g) p dμ⎠
B
⎞⎞s h dμ⎠⎠ ⎛
⎝(S)
⎞s
q
( f h)q dμ⎠
A∩B
(8)
On Carlson’s inequality for Sugeno and Choquet…
is satisfied if (a b) ∧ (c d) ≥ (a ∧ c) (b ∧ d) for all a, b, c, d ≥ 0. Proof We recall that the Jensen type inequality
z f z dμ ≥ (S) f dμ
(S) A
(9)
A
holds, where z ≥ 1 and f : X → Y is any measurable function such that (S) A f dμ ≤ 1 (see Xu and Ouyang 2012, Lemma 3.1; or Kaluszka et al. 2014a, Theorem 3.1). From (9), we have ⎛
f g dμ ≤ ⎝(S)
(S) A∩B
f h dμ ≤ ⎝(S)
(S)
p
( f g) p dμ⎠ ,
(10)
A∩B
⎛
⎞1
A∩B
q
( f h)q dμ⎠ ,
(11)
A∩B
⎞r
⎝(S)
⎛
( f g) dμ⎠ ⎝(S)
A∩B
⎛
≤ ⎝(S)
⎞r
p
( f g) p dμ⎠
⎞s ( f h) dμ⎠
A∩B
⎛
⎝(S)
A∩B
⎞s
q
( f h)q dμ⎠ .
From the Chebyshev type inequality for positively dependent functions f |A , g|B and f |A , h |B (see Theorem 2.1 in Kaluszka et al. 2014a), we get f ψ dμ A∩B
⎛
≥ ⎝(S)
A
⎞ ⎛ f dμ⎠ ⎝(S)
f dμ = sup {α ◦ μ(A ∩ { f ≥ α})}, α∈Y
(14)
where μ is a monotone measure and ◦ : Y × Y → Y is a non-decreasing operator (see Kaluszka et al. 2014b). The functional in (14) is the universal integral in the sense of Definition 2.5 in Klement et al. (2010) if ◦ is the pseudomultiplication function (see Klement et al. 2010, Definition 2.3). An example of non-decreasing operator is a t-seminorm, also called a semicopula (Durante et al. 2005; Ouyang and Mesiar 2009). There are three important t-seminorms: ∧, and ◦L , where (a, b) = ab and ◦L (a, b) = (a + b − 1) ∨ 0 usually called the Łukasiewicz t-norm (Klement et al. 2000). From now on, a ∨ b = max(a, b). For ◦ = ∧, we get the Sugeno integral
A∩B
(12)
(S)
◦, A
⎞1
since (S) A∩B f ψ dμ ≤ 1 for ψ = g, h. The operator is non-decreasing, so by (10) and (11), ⎛
Sadarangani.
If μ is the Lebesgue measure, then (3) holds since (S) [0,1] x dμ = 0.5 and if f and g are comonotone, then f |A and g|A are positively dependent with respect to the operator ∧ (see Example 2.1 in Kaluszka et al. 2014a). Setting r = s = 1 and = · in Theorem 2.1, we obtain the inequality (4) of Xu and Ouyang. Taking r = p/( p + q) and s = 1−r , we get (5) (see also Wang and Bai 2011). Combining the above results with other inequalities for comonotone functions, one can also derive (similarly as in Daraby and Arabi 2013) some related Carlson type inequalities for the Sugeno integral. Next, we define the generalized Sugeno integral of a measurable function f : X → Y on a set A ∈ F as
⎞ ψ dμ⎠ for ψ = g, h.
B
(13) To complete the proof, it is enough to apply (13) to (12). Theorem 2.1 extends all known (obtained by different methods) Carlson type inequalities for the Sugeno integral. To see this, we first put A = B, = ∧ and = · in Theorem 2.1. Putting further g = 1, h = x, p = q = 2, r = s = 1 and A = [0, 1] yields the inequality (3) of Caballero and
f dμ = sup {α ∧ μ(A ∩ { f ≥ α})},
(S) A
α∈Y
(see Sugeno 1974 for Y = [0, 1]). If ◦ = , then (14) is called the Shilkret integral (see Shilkret 1971 for Y = [0, ∞]; Ouyang and Mesiar 2009
for Y = [0, 1]). We denote the Shilkret integral as (N ) A f dμ. Moreover, we obtain the seminormed fuzzy integral if Y = [0, 1] and ◦ is a semicopula (Suárez García and Gil Álvarez 1986). Suppose , : Y × Y → Y are non-decreasing operators. Let ♦ : Y × Y → Y be a non-decreasing and left-continuous operator. Assume : Y × Y → Y is an arbitrary operator and f, g, h : X → Y, are measurable functions. Now, we are ready to derive a Carlson inequality for the generalized Sugeno integral. Theorem 2.2 Suppose p, q ≥ 1 and r, s > 0. Then, for arbitrary pairs of positively dependent functions f |A , g|B and f |A , h |B , the following inequality
123
M. Boczek, M. Kaluszka
f dμ ♦
◦, A
♦
◦, B
◦, B
r g dμ
◦, A
s h dμ ≤
6. = ♦ , is any operator, a ◦ b = a for all a, b ∈ Y and Y = [0, 1] or Y = [0, ∞].
r p ( f g) dμ p
◦, A∩B
s q
◦, A∩B
f dμ
( f h)q dμ
Example 2.1 The following inequality for the Shilkret integral of a non-decreasing function f is valid (15)
(N )
is satisfied if for all a, b, c, d ∈ Y and z > 1,
A
⎛ 1 f dμ ≤ √ · ⎝(N ) K ⎛
a ◦ b ≥ (a ◦ b) , (a b) ◦ (c d) ≥ (a ◦ c) ♦ (b ◦ d). (16) z
z
Proof The proof is similar to that of Theorem 2.1. Observe that all integrals in (15) are elements of Y . The Jensen type inequality ◦, A
z u dμ ≤
◦, A
u z dμ
◦, A∩B
r ( f g) dμ
≤
◦, A∩B
◦, A∩B
s ( f h) dμ
r p p ( f g) dμ
◦, A∩B
◦, A∩B
( f ψ) dμ ≥
for ψ = g, h.
◦, A
A
⎞1 4
x 2 f 2 dμ⎠ ,
A
where K = μ(A) · (N ) A x dμ ; to see this put = ∧ or = · , g = 1, h = x, ♦ = = = ◦ = ·, p = q = 2, r = s = 1 and A = B in Theorem 2.2.
f dμ ♦
◦, B
Example 2.2 Let (X, F, P) be a probability space. Put Y = [0, 1], r = s = 1, g = 1, A = B = X, f = φ U ) and h = 1 − ψ(U ), where U has the uniform distribution on [0, 1] and φ, ψ : [0, 1] → [0, 1] are increasing functions. The functions f and h are not comonotone but
= P({ f ≥ a}) ◦L P({h ≥ b}),
s q q ( f h) dμ .
From Theorem 2.1 in Kaluszka et al. (2014a) we get
4
f 2 dμ⎠
P({ f ≥ a, h ≥ b}) = (ψ −1 (1 − b) − φ −1 (a))+
(18)
⎞1
(17)
z holds if a z ◦ b ≥ a ◦ b for all a, b, c, d ∈ Y , where z > 1 and u : X → Y is any measurable function (see Kaluszka et al. 2014b, Theorem 2.1). The operator is non-decreasing, so from (17) it follows that
× ⎝(N )
ψ dμ
so f and h are positively dependent with respect to P and ◦L . The conditions (16) are satisfied for = = ♦ = ◦L and = ◦ = · (see Kaluszka et al. 2014a, formula (40)), thus the corresponding Carlson inequality takes the form 1
(N ( f ) ◦L 1) · (N ( f ) ◦L N (h)) ≤ (N ( f p )) p 1
× (N (( f ◦L h)q )) q ,
(19)
To complete the proof, it is enough to apply (19) to (18).
where N ( f ) = (N )
From Theorem 2.2, one can obtain many other Carlson type inequalities since the conditions (16) are fulfilled by many systems of operators. Examples are:
3 Carlson’s type inequality for Choquet integral
1. = ∧ and = ♦ = ◦, where ◦ is any t-norm satisfying the condition (a s ◦ b) ≥ (a ◦ b)s for s ≥ 1 since a ◦ b ≤ a ∧ b and any t-norm is an associative and commutative operator (Klement et al. 2000); 2. = = ◦ = ♦ = · on Y = [0, 1]; 3. = = ♦ = · and ◦ = ∧ with Y = [0, 1]; 4. = = ♦ , ◦ = ∧ and Y = [0, 1]; 5. = = ♦ = ◦, where ◦ is any t-norm satysfying the condition (a s ◦ b) ≥ (a ◦ b)s for s ≥ 1, e.g., the Dombi t-norm a ◦ b = ab/(a + b − ab);
123
X
f dP.
In this section, μ : F → [0, ∞] is a monotone measure. Denote by M the set of monotone measures on (X, F). The Choquet integral of f : X → [0, ∞) on A ∈ F is defined as ∞
f dμ = A
μ(A ∩ { f ≥ t}) dt, 0
where the integral on the right-hand side is the improper Riemann integral. A function f is said to be integrable on
a measurable set A if A f dμ < ∞. The importance of
On Carlson’s inequality for Sugeno and Choquet…
the Choquet integral still increases due to many applications in mathematics and economics, see for instance (Cerdà et al. 2011; Denneberg 1994; Grabisch and Labreuche 2010; Heilpern 2003; Kaluszka and Krzeszowiec 2012). If μ is a measure, the Choquet integral coincides with the Lebesgue integral (Wang and Klir 2009). It was Tang and Ouyang (2012) who first presented the following Carlson type inequality for the Choquet integral
− r
− s
r +s r +s and d = 2 − g dμ h dμ where K = A A 1 r s r +s p + q . Proof Without loss of generality, we assume that 0 < μ(A) < ∞. Put m(B) = μ(A ∩ B)/μ(A) for B ∈ F. For a given c ≥ 1, the following Jensen type inequality
c
≤
f dm ⎛ ⎝
⎞2
1 1 3− p + q
μ(A)
f dμ⎠ ≤
A
h dμ
⎛ ⎝
⎛ ×⎝
is satisfied (Girotto and Holzer 2012;
Mesiar et al. 2010;
Zhao et al. 2011). Hereafter, we write f dm instead of A f dm. From (23), we have
⎞1 q
f q h q dμ⎠ ,
(20)
r
f dμ ≤ c(μ) ⎝
X
⎞1 ⎛ ⎞1 4 4 2 2 g f dμ⎠ ⎝ h f dμ⎠
X
⎛ ⎞ r ⎛ ⎞ s p(r +s) q(r +s) p p q q ⎝ f h dμ⎠ f dμ ≤ K (μ(A)) ⎝ f g dμ⎠ , d
A
A
(22)
p
f p g p dm
.
(24)
Since f, g are comonotone functions, the following Chebyshev type inequality
f g dm ≥
f dm
g dm
holds (see Girotto and Holzer 2011). The functions f, h are also comonotone, so from (24) we get
r +s
≤
r g dm
s h dm
r p p
p
s
q
q q
f g dm
f h dm
X
Theorem 3.1 Let p, q ≥ 1 and r, s > 0. Suppose f, g : X → [0, ∞) and f, h : X → [0, ∞) are pairs of comonotone functions. If f is integrable on A, then
A
q
f h dm
(21)
r
≤
s q q
f dm
is satisfied provided inf x∈X (g(x)h(x)) = 0. In (1), we have g(x) = 1 and h(x) = x 2 for x ∈ X = [0, ∞), so this assumption is fulfilled. Indeed, put μ(A) = 1 for all A = ∅, f (x) = 1 for x = t and f (x)
= 0 otherwise, where t is any fixed point of X. Since X ψ dμ = supx∈X ψ(x), from (21) we get 1 ≤ c(μ)(g(t)h(t))1/4 , a contradiction with inf x∈X (g(x)h(x)) = 0 and c(μ) < ∞. Therefore, some extra conditions should be imposed on f, g, h or μ. First, we present the Carlson inequality under the assumption of comonotonicity.
×
s f h dm
f g dm
where p, q > 1 and f, h are comonotone functions. This result was also obtained by Ouyang (2015) as a consequence of Hölder’s inequality and Chebyshev’s inequality for the Choquet integral. The assumption of comonotonicity is essential for inequality (20) (see Girotto and Holzer 2011). Unfortunately, Eq. (20) is not a generalization of the classical Carlson inequality (1). Our aim is to give such an extension. We begin by showing that it does not exist a functional c : M → [0, ∞) such that for any monotone measure μ and any integrable function f the following Carlson type inequality ⎛
(23)
p
f p dμ⎠
A
f c dm,
⎞1
A
A
Combining this with the equality A φ dμ completes the proof.
.
φ dm = (μ(A))−1
The inequality (22) is sharp. In fact, if μ(B) = 1 for B =
∅, then A φ dμ = s(φ), where s(φ) denotes the supremum of φ on A, so the inequality (22) takes the form r
s
r
s
s( f ) ≤ s(g)− r +s s(h)− r +s (s( f g)) r +s (s( f h)) r +s .
(25)
Since s(φψ) = s(φ)s(ψ) for comonotone functions φ, ψ (see Murofushi and Sugeno 1991), the equality in (25) is attained. Putting g = 1 and r = s in Theorem 3.1, we obtain inequality (20) as A 1 dμ = μ(A). Now, we provide the Carlson inequality for the Choquet integral with respect to a submodular monotone measure μ and any integrable functions f, g, h from X to [0, ∞). Recall that μ is submodular if μ(A ∩ B) + μ(A ∪ B) ≤ μ(A) + μ(B)
123
M. Boczek, M. Kaluszka
for A, B ∈ F. The Choquet integral is subadditive for all measurable functions if and only if μ is submodular (see Pap 1995, Theorem 7.7). Define ⎛ 1 (ab) p
H pq (a, b) =
⎝
A
q
1 dμ⎠ . (bg + ah)q−1
(26)
⎞ ⎛ f dμ ≤ 21/ p H pq ⎝ g f p dμ, h f p dμ⎠ ,
A
A
(bg f p + ah f p ) dμ = A
(27)
A
A
Proof Since μ is submodular, the following Hölder inequality ⎛ φψ dμ ≤ ⎝
A
⎞1 ⎛ ⎞1 p q p q ⎠ ⎠ ⎝ φ dμ ψ dμ
A
(28)
A
f dμ =
A
1 1 2 √ +√ p q ⎞ ⎛ × H pq ⎝ g f p dμ, h f p dμ⎠ ,
A
A
⎛ ≤ ⎝b
A
g f p dμ + a
A
⎛ ×⎝ A
⎞1 1 dμ⎠ (bg + ah)q/ p
q
A
A
⎞1
A
where H pq (a, b) is given by (26), p > 1 and 1/ p +1/q = 1. Proof The proof is similar to that of Theorem 3.2, but we use the following inequalities (see Shirali 2008) ⎞1 ⎛ ⎞1 ⎛
2 p q 1 1 p q ⎠ ⎠ ⎝ ⎝ f g dμ ≤ √ + √ f dμ g dμ , p q A A A ⎛ ⎞ ( f + g) dμ ≤ 2 ⎝ f dμ + g dμ⎠ .
A
A
√ √ instead of those in (29). Since (1/ p + 1/ q)2 ≤ 2, the bounds obtained are better than the bounds of Cerdà et al. (2011).
p
h f p dμ⎠
4
A
⎞1 q 1 ⎠ . dμ (bg + ah)q−1
(30)
Remark 1 The upper bound in (30) is worse than that in (27) (under the weaker assumption on μ) as 1/ p
1 1 √ +√ p q
2 > 21/ p
for all p, q > 1 such that 1/ p + 1/q = 1. (29)
Putting a = A g f p dμ and b = A h f p dμ, we obtain (27). Note that if μ is modular then from Theorem 7.7 of Pap (1995), it follows that
123
f dμ ≤ 41/ p
A
1 (bg + ah)1/ p f dμ (bg + ah)1/ p
⎛ ⎞1 ⎛ p p ⎝ ⎠ ⎝ ≤ (bg + ah) f dμ
A
Theorem 3.3 Suppose f : X → [0, ∞), A ⊂ [0, ∞] and μ ∈ M such that μ(A ∪ B) ≤ μ(A) + μ(B) for A, B ∈ F. Then,
is valid, where φ, ψ ≥ 0 (see Wang 2011, Theorem 3.5). The equality in (28) holds if αφ p = βψ q for α, β ≥ 0, α +β > 0 (Niculescu and Persson 2006). By (28) and the subadditivity and positively homogeneity of the Choquet integral, we get
ah f p dμ.
If μ is the Lebesgue measure, g(x) = 1, h(x) = x 2 , p = q = 2 and A = [0, ∞], we obtain the classical Carlson inequality (1). Finally, we present the Carlson type inequality for the Choquet integral with respect to a subadditive monotone measure μ.
where p > 1 and in (27)
1. The equality
1/ p + 1/q = q is attained if g A h f p dμ + h A g f p dμ f p = γ for some γ ≥ 0 provided μ is modular or g f p and h f p are comonotone functions.
bg f p dμ +
⎞1
Theorem 3.2 If μ is submodular, A ∈ F and f, g, h : X → [0, ∞), then
Acknowledgments The authors would like to thank the referees for their valuable comments which led to improvements in the paper. Compliance with ethical standards Conflict of interest The authors declare that have no conflict of interest.
On Carlson’s inequality for Sugeno and Choquet…
References Agahi H, Yaghoobi MA (2010) General Hardy type inequality for seminormed fuzzy integrals. Appl Math Comput 216:1972–1977 Agahi H, Mesiar R, Ouyang Y (2012) On some advanced type inequalities for Sugeno integral and T-(S-)evaluators. Inf Sci 190:64–75 Barza S, Peoari J, Persson L-E (1998) Carlson type inequalities. J Inequal Appl 2:121–135 Caballero J, Sadarangani K (2010) Fritz Carlson’s inequality for fuzzy integrals. Comput Math Appl 59:2763–2767 Carlson F (1934) Une inegalité. Arkiv för Matematik Astronomi och Fysik 25:1–5 Cerdà J, Martín J, Silvestre P (2011) Capacitary function spaces. Collect Math 62:95–118 Daraby B, Arabi L (2013) Related Fritz Carlson type inequalities for Sugeno integrals. Soft Comput 17:1745–1750 Denneberg D (1994) Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht Durante F, Sempi C (2005) Semicopulae. Kybernetika 41:315–328 Flores-Franuliˇc A, Román-Flores H (2007) A Chebyshev type inequality for fuzzy integrals. Appl Math Comput 190:1178–1184 Girotto B, Holzer S (2011) Chebyshev type inequality for Choquet integral and comonotonicity. Int J Approx Reason 52:1118–1123 Girotto B, Holzer S (2012) Chebyshev and Jensen inequalities for Choquet integral. Math Pannonica 23:267–275 Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175(1):247–286 Heilpern S (2003) A rank-dependent generalization of zero utility principle. Insur Math Econ 33:67–73 Hu Y (2007) Fuzzy integral-based perceptron for two-class pattern classification problems. Inf Sci 177:1673–1686 Kaluszka M, Krzeszowiec M (2012) Pricing insurance contracts under cumulative prospect theory. Insur Math Econ 50:159–166 Kaluszka M, Okolewski A, Boczek M (2014a) On Chebyshev type inequalities for generalized Sugeno integrals. Fuzzy Sets Syst 244:51–62 Kaluszka M, Okolewski A, Boczek M (2014b) On the Jensen type inequality for generalized Sugeno integral. Inf Sci 266:140–147 Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht Klement EP, Mesiar R, Pap E (2010) A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans Fuzzy Syst 18:178–187 Mesiar R, Ouyang Y (2009) General Chebyshev type inequalities for Sugeno integrals. Fuzzy Sets Syst 160:58–64
Mesiar R, Li J, Pap E (2010) The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46:1098–1107 Mitrinovi´c DS, Peˇcari´c JE, Fink AM (1991) Inequalities involving functions and their integrals and derivatives. Kluwer, Dordrecht Murofushi T, Sugeno M (1991) A theory of fuzzy measures: representations, the Choquet integral and null sets. J Math Anal Appl 159:532–549 Narukawa Y, Torra V (2007) Fuzzy measures and integrals in evaluation of strategies. Inf Sci 177:4686–4695 Niculescu C, Persson L-E (2006) Convex functions and their applications. A contemporary approach. Springer, New York Ouyang Y (2015) On Carlson inequality for the Choquet integral (personal communication) Ouyang Y, Mesiar R (2009) On the Chebyshev type inequality for seminormed fuzzy integral. Appl Math Lett 22:1810–1815 Pap E (1995) Null-additive set functions. Kluwer, Dordrecht Román-Flores H, Flores-Franuliˇc A, Chalco-Cano Y (2007) A Jensen type inequality for fuzzy integrals. Inf Sci 177:3192–3201 Shilkret N (1971) Maxitive measure and integration. Indag Math 33:109–116 Shirali S (2008) Analog of L p with a subadditive measure. Ric di Mat 57:43–54 Suárez García F, Gil Álvarez P (1986) Two families of fuzzy integrals. Fuzzy Sets Syst 18:67–81 Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. dissertation, Tokyo Institute of Technology Tang YL, Ouyang Y (2012) On the Carlson inequality for the Choquet integral. J Huzhou Teach Coll 2:21–25 (in Chinese) Wang R-S (2011) Some inequalities and convergence theorems for Choquet integrals. J Appl Math Comput 35:305–321 Wang Z, Klir G (2009) Generalized measure theory. Springer, New York Wang X, Bai C (2011) General Fritz Carlson’s type inequality for Sugeno integrals. Hindawi Publ Corp J Inequal Appl Article ID 761430, 9 pages Wu L, Sun J, Ye X, Zhu L (2010) Hölder type inequality for Sugeno integral. Fuzzy Sets Syst 161:2337–2347 Xu Q, Ouyang Y (2012) A note on a Carlson-type inequality for the Sugeno integral. Appl Math Lett 25:619–623 Zhao X, Zhang Q (2011) Hölder type inequality and Jensen type inequality for Choquet integral. In: Wang Y, Li T (eds) Knowledge engineering and management. AISC 123, pp 219–224
123
