Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 761430, 9 pages
doi:10.1155/2011/761430
Research Article
General Fritz Carlson’s Type Inequality for
Sugeno Integrals
Xiaojing Wang and Chuanzhi Bai
Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China
Correspondence should be addressed to Chuanzhi Bai, [email protected]
Received 18 August 2010; Revised 23 November 2010; Accepted 20 January 2011
Academic Editor: L
´
aszl
´
o Losonczi
Copyright q 2011 X. Wang and C. Bai. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Fritz Carlson’s type inequality for fuzzy integrals is studied in a rather general form. The main
results of this paper generalize some previous results.
1. Introduction and Preliminaries
Recently, the study of fuzzy integral inequalities has gained much attention. The most popu-
lar method is using the Sugeno integral 1. The study of inequalities for Sugeno integral was
initiated by Rom
´
an-Flores et al. 2, 3 and then followed by the others 4 –11.
Now, we introduce some basic notation and properties. For details, we refer the reader
to 1, 12.
Suppose that Σ is a σ-algebra of subsets of X,andletμ : Σ → 0, ∞ be a nonnegative,
extended real-valued set function. We say that μ is a fuzzy measure if it satisfies
n
μ
∞
n1
E
n
continuity
from above.
If f is a nonnegative real-valued function defined on X, we will denote by L
α
f {x ∈
X : fx ≥ α} {f ≥ α} the α-level of f for α>0, and L
0
f {x ∈ : fx > 0} supp f is
the support of f.Notethatifα ≤ β,then{f ≥ β}⊂{f ≥ α}.
Let X, Σ,μ be a fuzzy measure space; by F
μ
X we denote the set of all nonnegative
μ-measurable functions with respect to Σ.
2 Journal of Inequalities and Applications
Definition 1.1 see 1.LetX, Σ,μ be a fuzzy measure space, with f ∈F
μ
X,andA ∈ Σ,
then the Sugeno integral or fuzzy integral of f on A with respect to the fuzzy measure μ is
defined by
A
fdμ
3 if f ≤ g on A then
A
fdμ ≤
A
gdμ,
4 if A ⊂ B then
A
fdμ ≤
A
fdμ,
5 μA ∩{f ≥ α} ≥ α ⇒
A
fdμ ≥ α,
6 μA ∩{f ≥ α} ≤ α ⇒
A
fdμ≤ α,
7
A
fdμ < α ⇔ there exists γ<αsuch that μA ∩{f ≥ γ} <α,
8
A
fdμ > α ⇔ there exists γ>αsuch that μA ∩{f ≥ γ} >α.
Remark 1.3. Let F be the distribution function associated with f on A,thatis,FαμA ∩
{f ≥ α}.By5 and 6 of Proposition 1.2
F
α
α ⇒
A
1/4
. 1.4
Recently, Caballero and Sadarangani 8 have shown that in general, the Carlson’s
integral inequality is not valid in the fuzzy context. And they presented a fuzzy version of
Fritz Carlson’s integral inequality as follows.
Journal of Inequalities and Applications 3
Theorem 1.4. Let f : 0, 1 → 0, ∞ be a nondecreasing function and μ the Lebesgue measure on
.Then,
1
0
f
x
dμ
x
≤
√
2
1
0
x
2
f
2
xdμx
A
fdμ
s
. 2.1
If the fuzzy measure μ in Lemma 2.1 is the Lebesgue measure, then
1
0
fdμ ≤ 1is
satisfied readily. Thus, by Lemma 2.1, we have the following.
Corollary 2.2 see 8. Let f : 0, 1 → 0, ∞ be a μ-measurable function with μ the Lebesgue
measure and s ≥ 1.Then
1
0
f
s
x
dμ
x
≥
1
0
fxdμx
s
A
fdμ ≤ 1,and
A
gdμ ≤ 1.Then
A
f ·gdμ ≥
A
fdμ
·
A
gdμ
. 2.4
4 Journal of Inequalities and Applications
Proof. If
A
fdμ 0or
A
gdμ 0 then the inequality is obvious. Now choose α, β such that
1 ≥
A
fdμ > α > 0, 1 ≥
A
gdμ > β > 0. 2.5
Then by 8 of Proposition 1.2,thereexist1>γ
α
>αand 1 >γ
}. In this case, we have the following:
μ
A ∩
fg ≥ γ
α
γ
β
≥ μ
A ∩
f ≥ γ
α
∩
A ∩
g ≥ γ
β
μ
A ∩
f ≥ γ
α
·
a
0
gdμ
2.9
holds.
If the fuzzy measure μ in Corollary 2.5 is the Lebesgue measure and a 1, then
a
0
fdμ ≤ 1and
a
0
gdμ ≤ 1 are satisfied readily. Thus, by Corollary 2.5,weobtain
Corollary 2.6 see 2. Let f, g : 0, 1 →
be two real-valued functions, and let μ be the Lebesgue
measure on
.Iff, g are both continuous and strictly increasing (decreasing) functions, then the
inequality
1
0
f ·gdμ ≥
1
0
fdμ
·
dμ
x
≤
1
K
A
f
p
x
g
p
xdμx
1/pq
·
A
f
q
xh
q
xdμx
1/pq
, 2.11
,
A
f
x
· h
x
dμ
x
q
≤
A
f
q
x
h
q
x
dμ
x
·
A
f
q
x
h
q
x
dμ
x
.
2.13
By Theorem 2.4
A
f ·gdμ ≥
gxdμx
p
·
A
hxdμx
q
≤
A
f
p
x
g
p
x
dμ
x
·
A
x
≤
1
K
1
0
f
p
xg
p
xdμx
1/pq
·
1
0
f
q
xh
q
xdμx
1/pq
, 2.16
where K
1
0
0
,g
1
1
g
s
1
− g
s
0
1/s
, 1
. 2.17
Proof. Firstly, we consider the case of g
− g
s
0
, 1
. 2.18
By Corollary 2.2 and 2.18,weget
1
0
gxdμx
s
≤ min
g
s
1
1 g
s
1
− g
s
dμ
x
≤
1
M
p/pq
1
K
q/pq
2
1
0
f
p
xg
p
xdμx
1/pq
·
1
0
f
q
x
1
g
s
1
− g
s
0
1/s
, 1
,K
2
1
0
h
x
dμ
xdμx
1/pq
·
1
0
f
q
xh
q
xdμx
1/pq
,
2.22
Journal of Inequalities and Applications 7
where
K
1
1
0
g
x
dμ
x
. 2.23
Theorem 2.11. Assume that p, q ≥ 1.Letf, g, h : 0, 1 → 0 , ∞ be increasing (or decreasing)
functions and μ the Lebesgue measure on
.Ifg
s
s ≥ 1 and h
r
r ≥ 1 are two convex functions such
that g0
/
g1 and h0
/
h1,then,
1
0
f
x
dμ
x
≤
1
M
p/pq
1
M
1/pq
,
2.24
where M
1
and M
2
are as in 2.21 and 2.23, respectively .
Straightforward calculus shows that
1
0
x
2
dμ
x
3 −
√
5
2
,
1
0
xdμ
x
≤
3
√
5
1
0
x
4
f
2
x
dμ
x
1/4
·
1
0
x
2
f
2
x
4
f
2
x
dμ
x
1/4
·
1
0
f
2
x
dμ
x
1/4
. 2.27
Remark 2.14. Corollary 2.8 is a generalization of the main result in 8,Theorem1.
x
.
2.28
8 Journal of Inequalities and Applications
Consider gxe
−
√
x1
on 0, 1. This function is nonincreasing g
x
−1/2
√
x 1e
−
√
x1
< 0, nonnegative and convex g
x1/4x 1e
√
x1
1/
√
x 1 1 ≥
0.
Let p q 1, gxhxe
−
√
dμ
x
≤
e
√
2
e
√
2−1
− 1
e
√
2−1
1
0
e
−
√
x1
f
x
dμ
x
0
1/s
, 1
1 −ln 2
2 −ln 2
,
M
2
min
max
{
h
0
,h
1
}
1
|
≤
2 −ln 2
8 −π
1 −ln 2
4 −π
1
0
x − ln
x 1
fxdμx
1/2
×
1
0
x − arctanx 1fxdμx
17/8
1
17/8 −
1/8
2
2
√
34
√
8
√
17 − 1
2
. 2.33
Journal of Inequalities and Applications 9
Thus, by Theorem 2.10 set g
x
2
x 1/8, hxx, s 2, p 1,q 2 we can get
the following corollary.
Corollary 2.18. Let f : 0, 1 → 0, ∞ be a nondecreasing function and μ the Lebesgue measure on
.Then
1/8
fxdμx
1/3
×
1
0
x
2
f
2
xdμx
2/3
.
2.34
Acknowledgments
The authors would like to thank the referees for reading this work carefully, providing
valuable suggestions and comments. This work is supported by the National Natural Science
Foundation of China no. 10771212.
References
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Technology, 1974.
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ˇ
candH.Rom
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an-Flores, “A Chebyshev type inequality for fuzzy integrals,” Applied
ˇ
c, and Y. Chalco-Cano, “A convolution type inequality for fuzzy
integrals,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 94–99, 2008.
11 J. Caballero and K. Sadarangani, “A Cauchy-Schwarz type inequality for fuzzy integrals,” Nonlinear
Analysis. Theory, Methods and Applications. Series A, vol. 73, no. 10, pp. 3329–3335, 2010.
12 Z. Wang and G. Klir,
Fuzzy Measure Theory, Plenum Pr ess, New York, NY, USA, 1992.
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¨
or Matematik, vol. 25, pp. 1–5, 1934.
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