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Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2011, Article ID 761430, 9 pages doi:10.1155/2011/761430 Research Article General Fritz Carlson’s Type Inequ

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,czbai8@sohu.com

Received 18 August 2010; Revised 23 November 2010; Accepted 20 January 2011

Academic Editor: L´aszl ´o Losonczi

Copyrightq 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 integral1 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

to1,12

Suppose thatΣ is a σ-algebra of subsets of X, and let μ : Σ → 0, ∞ be a nonnegative, extended real-valued set function We say that μ is a fuzzy measure if it satisfies

1 μ∅  0,

2 E, F ∈ Σ and E ⊂ F imply μE ≤ μF monotonicity;

3 {E n } ⊂ Σ, E1⊂ E2⊂ · · · imply limn→ ∞μ E n   μ∞n1E n continuity from below,

4 {E n } ⊂ Σ, E1 ⊃ E2 ⊃ · · · , μE1 < ∞, imply lim n→ ∞μ E n   μ∞

n1E n continuity from above

If f is a nonnegative real-valued function defined on X, we will denote by L α f  {x ∈

X : f x ≥ α}  {f ≥ α} the α-level of f for α > 0, and L0f  {x ∈ : fx > 0}  supp f is the support of f Note that if α ≤ β, then {f ≥ β} ⊂ {f ≥ α}.

LetX, Σ, μ be a fuzzy measure space; by F μ

X we denote the set of all nonnegative

μ-measurable functions with respect toΣ

Definition 1.1see 1 Let X, Σ, μ be a fuzzy measure space, with f ∈ F μ

X, and A ∈ Σ,

then the Sugeno integralor fuzzy integral of f on A with respect to the fuzzy measure μ is

defined by

A

fdμ

α≥0



where∨ and ∧ denote the operations sup and inf on 0, ∞, respectively.

It is well known that the Sugeno integral is a type of nonlinear integral; that is, for general cases,



does not hold

The following properties of the fuzzy integral are well known and can be found in12

Proposition 1.2 Let X, Σ, μ be a fuzzy measure space, with A, B ∈ Σ and f, g ∈ F μ

X; then

1

A fdμ ≤ μA,

2

A kdμ  k ∧ μA, for k a nonnegative constant,

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 f dμ ≤ α,

7

A fdμ < α ⇔ there exists γ < α such that μA ∩ {f ≥ γ} < α,

8

A fdμ > α ⇔ there exists γ > α such that μA ∩ {f ≥ γ} > α.

Remark 1.3 Let F be the distribution function associated with f on A, that is, F α  μA ∩ {f ≥ α} By 5 and 6 ofProposition 1.2

F α  α ⇒

A

Thus, from a numerical point of view, the Sugeno integral can be calculated by solving the

equation Fα  α.

Fritz Carlson’s integral inequality states13,14 that



∞ 0

f xdx ≤√π



∞ 0

f2xdx

1/4

·



∞ 0

x2f2xdx

1/4

Recently, Caballero and Sadarangani8 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 xdμx ≤√2

1 0

x2f2xdμx

1/4

0

f2xdμx

1/4

In this paper, our purpose is to give a generalization of the above Fritz Carlson’s inequality for fuzzy integrals Moreover, we will give many interesting corollaries of our main results

2 Main Results

This section provides a generalization of Fritz Carlson’s type inequality for Sugeno integrals Before stating our main results, we need the following lemmas

Lemma 2.1 see 11 Let X, Σ, μ be a fuzzy measure space, f ∈ F μ

X, A ∈ Σ,

A fdμ ≤ 1, and

s ≥ 1 Then

A

f s dμ≥

A

fdμ

s

If the fuzzy measure μ inLemma 2.1is the Lebesgue measure, then



1

0 fdμ ≤ 1 is satisfied readily Thus, byLemma 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 xdμx ≥ 1

0

f xdμx

s

Definition 2.3 Two functions f, g : X → R are said to be comonotone if for all x, y ∈ X2,



f x − fy 

g x − gy

An important property of comonotone functions is that for any real numbers p, q, either {f ≥ p} ⊂ {g ≥ q} or {g ≥ q} ⊂ {f ≥ p}.

Note that two monotone functionsin the same sense are comonotone

Theorem 2.4 Let X, Σ, μ be a fuzzy measure space, f, g ∈ Fμ X and f and g comonotone

functions, A ∈ Σ with

A fdμ ≤ 1, and

A gdμ ≤ 1 Then

A

f · gdμ ≥

A

fdμ

·

A

gdμ

Proof If



A fdμ 0 or

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 by8 ofProposition 1.2, there exist 1 > γ α > α and 1 > γ β > β such that

μ

A∩f ≥ γ α



> α, μ

A∩g ≥ γ β



> β. 2.6

As f and g are comonotone functions, then either {f ≥ γ α } ⊂ {g ≥ γ β } or {g ≥ γ β } ⊂ {f ≥ γ α} Suppose that{f ≥ γ α } ⊂ {g ≥ γ β} In this case, we have the following:

μ

A∩fg ≥ γ α γ β

≥ μA∩f ≥ γ α



∩A∩g ≥ γ β



 μA∩f ≥ γ α



> α ≥ αβ.

2.7 Therefore, by applying8 ofProposition 1.2again, we find that

A

Since the values of α, β > 0 are arbitrary, we obtain the desired inequality Similarly, for the

case{g ≥ γ β } ⊂ {f ≥ γ α} we can get the desired inequality too

FromTheorem 2.4, we get the following

Corollary 2.5 see 15 Let μ be an arbitrary fuzzy measure on 0, a and f, g : 0, a → be two real-valued measurable functions such that



a

0 fdμ ≤ 1 and

a

0 gdμ ≤ 1 If f and g are increasing (or

decreasing) functions, then the inequality

a

0

f · gdμ ≥

0

fdμ

·

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≤ 1 and

a

0 gdμ≤ 1 are satisfied readily Thus, byCorollary 2.5, we obtain

Corollary 2.6 see 2 Let f, g : 0, 1 → be two real-valued functions, and let μ be the Lebesgue measure on  If f, g are both continuous and strictly increasing (decreasing) functions, then the inequality

1 0

0

fdμ



0

gdμ



2.10

holds.

The following result presents a fuzzy version of generalized Carlson’s inequality

Journal of Inequalities and Applications 5

Theorem 2.7 Let X, Σ, μ be a fuzzy measure space, f, g, h ∈ F μ

X, f and g, and f and h are

comonotone functions, respectively, A ∈ Σ with

A fdμ ≤ 1,

A gdμ ≤ 1, ≤

A hdμ ≤ 1,

A fgdμ≤

1, and



A fhdμ ≤ 1 Then

A

f xdμx ≤ 1

K

A

f p xg p xdμx

1/pq

·

A

f q xh q xdμx

1/pq

, 2.11

where K 

A g xdμx p/ pq· 

A h xdμx q/ pq Proof ByLemma 2.1, for p, q≥ 1, we have the following:

A

f x · gxdμx

p

≤

A

f p xg p xdμx,

A

f x · hxdμx

q

≤

A

f q xh q xdμx.

2.12

Multiplying these inequalities, we get that

A

f x · gxdμx

p

·

A

f x · hxdμx

q

≤

A

f p xg p xdμx

·

A

f q xh q xdμx

.

2.13

ByTheorem 2.4

A

f · gdμ ≥

A

fdμ

·

A

gdμ

,

A

f · hdμ ≥

A

fdμ

·

A

hdμ

Substitutes2.14 into 2.13, we obtain

A

f xdμx

p q

·

A

g xdμx

p

·

A

h xdμx

q

≤

A

f p xg p xdμx

·

A

f q x · h q xdμx

.

2.15

This inequality implies that2.11 holds

ByTheorem 2.7, we have the following

Corollary 2.8 Assume that p, q ≥ 1 Let f, g, h : 0, 1 → 0, ∞ are increasing (or decreasing)

functions and μ the Lebesgue measure on Then be

1

0

f xdμx ≤ 1

K

1 0

f p xg p xdμx

1/pq

0

f q xh q xdμx

1/pq

, 2.16

where K 

1

0 g xdμx p/ pq· 

1

0 h xdμx q/ pq

Theorem 2.9 Let g : 0, 1 → 0, ∞ be a μ-measurable function with μ the Lebesgue measure If

g s (s ≥ 1) is a convex function such that, g0 / g1, then

1 0

g xdμx ≤ min

 max

g 0, g1



1g s 1 − g s0 1/s , 1



Proof Firstly, we consider the case of g s 0 < g s 1 As g sis a convex function, we have by Theorem 1 of Caballero and Sadarangani7 that

1 0

g s xdμx ≤ min

1 g s 1 − g s0, 1



ByCorollary 2.2and2.18, we get

1 0

g xdμx

s

≤ min



g s1

1 g s 1 − g s0, 1



which implies that2.17 holds Similarly, we can obtain 2.17 by of 7, Theorem 2 for the

case of g s 0 > g s1

FromTheorem 2.9andCorollary 2.8, we have the following

Theorem 2.10 Assume that p, q ≥ 1 Let f, g, h : 0, 1 → 0, ∞ be increasing (or decreasing)

functions and μ the Lebesgue measure on If g s ( s ≥ 1) or h r ( r ≥ 1) is a convex function such that

g 0 / g1 or h0 / h1, then

1

0

M1p/p q K q/p2 q

1 0

f p xg p xdμx

1/pq

0

f q xh q xdμx

1/pq

,

2.20

where

M1 min

 max

g 0, g1



1g s 1 − g s0 1/s , 1



, K2  1

0

or

1

0

K1p/p q M q/p2 q

1 0

f p xg p xdμx

1/pq

0

f q xh q xdμx

1/pq

,

2.22

Journal of Inequalities and Applications 7

where

K1 1

0

g xdμx, M2 min

 max{h0, h1}

1  |h r 1 − h r0|1/r , 1



Theorem 2.11 Assume that p, q ≥ 1 Let f, g, h : 0, 1 → 0, ∞ be increasing (or decreasing)

functions and μ the Lebesgue measure on If g s s ≥ 1 and h r r ≥ 1 are two convex functions such

that g 0 / g1 and h0 / h1, then,

1

0

M p/p1 q M q/p2 q

1 0

f p xg p xdμx

1/pq

0

f q xh q xdμx

1/pq

,

2.24

where M1 and M2 are as in2.21 and 2.23, respectively.

Straightforward calculus shows that

1 0

x2dμ x  3−

√ 5

1 0

xdμ x  1

2,

1

If p  q  2, gx  x and hx  1, gx  x2and hx  x, gx  x2, and hx  1,

respectively, thenCorollary 2.8reduces toTheorem 1.4, and the following Corollaries2.12

and2.13

Corollary 2.12 Let f : 0, 1 → 0, ∞ be a nondecreasing function and μ the Lebesgue measure on

 Then,

1

0

f xdμx ≤



3√5

1 0

x4f2xdμx

1/4

0

x2f2xdμx

1/4

Corollary 2.13 Let f : 0, 1 → 0, ∞ be a nondecreasing function and μ the Lebesgue measure on

 Then,

1

0

f xdμx ≤



6 2√5 2

1 0

x4f2xdμx

1/4

0

f2xdμx

1/4

Remark 2.14. Corollary 2.8is a generalization of the main result in8, Theorem 1

If p  q  1, gx  hx  x2, thenCorollary 2.8reduces to the following corollary

Corollary 2.15 Let f : 0, 1 → 0, ∞ be a nondecreasing function and μ the Lebesgue measure on

 Then

1 0

f xdμx ≤3

√ 5 2

1 0

Consider gx  e−√x1 on 0, 1 This function is nonincreasing gx 

−1/2√x  1e−√x1< 0 , nonnegative and convex gx  1/4x1e√x11/√x 11 ≥ 0

Let p  q  1, gx  hx  e−√x1, and s  r  1 As g0  1/e > 1/e√2  g1 and

h 0 > h1, we have the following

M1  M2 e

√ 2−1

Thus, byTheorem 2.11we can get the following corollary

Corollary 2.16 Let f : 0, 1 → 0, ∞ be a nonincreasing function and μ the Lebesgue measure on

 Then,

1 0

f xdμx ≤ e

√

2 e√2−1− 1

e√2−1

1 0

Consider gx  x − lnx  1 and hx  x − arc tan x on 0, 1 Obviously, g and h are

nonnegative, nondecreasing and convex on the interval0, 1 Let s  r  1, then, we have

the following:

M1 min

 max

g 0, g1



1g s 1 − g s0 1/s , 1



 1− ln 2

2− ln 2,

M2 min

 max{h0, h1}

1  |h r 1 − h r0|1/r , 1



 4− π

8− π .

2.31

Thus, byTheorem 2.11set p  q  1 we can get the following corollary.

Corollary 2.17 Let f : 0, 1 → 0, ∞ be a nondecreasing function and μ the Lebesgue measure on

 Then,

1

0

f xdμx ≤



2 − ln 28 − π

1 − ln 24 − π

1 0

x − lnx  1fxdμx

1/2

0

x − arctanx  1fxdμx

1/2

2.32

Consider gx x2 x  1/8 on 0, 1 Obviously, this function is nonnegative,

non-decreasinggx  2x  1/2x2 x  1/8 −1/2 ≥ 0, and nonconvex gx  −1/8x2

x  1/8 −3/2 ≤ 0 But g2x  x2 x  1/8 is convex Set s  2, then we obtain

M1



17/8



117/8−1/82  2

√ 34

√

Journal of Inequalities and Applications 9 Thus, byTheorem 2.10set g x2 x  1/8, hx  x, 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

0

√

8√17− 12

17

0



x2 x  1/8fxdμx

1/3

0

x2f2xdμx

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 Chinano 10771212

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Journal of Inequalities and...

2 Main Results

This section provides a generalization of Fritz Carlson’s type inequality for Sugeno integrals Before stating our main results, we need the following lemmas... Jensen type inequality for fuzzy

integrals,” Information Sciences, vol 177, no 15, pp 3192–3201, 2007.

4 R Mesiar and Y Ouyang, ? ?General Chebyshev type inequalities for Sugeno