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General related inequalities to Carlson-type inequality for the Sugeno integral
Applied Mathematics and Computation 305 (2017) 323–329
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General related inequalities to Carlson-type inequality for the Sugeno integral Bayaz Daraby a,∗, Hassan Ghazanfary Asll b, Ildar Sadeqi b a b
Department of Mathematics, University of Maragheh, P.O. Box 55181–83111, Maragheh, Iran Department of Mathematics, Sahand University of Technology, Tabriz, Iran
a r t i c l e
i n f o
a b s t r a c t
MSC: 35A23 45G99 28E10 45N05
In this paper, the Carlson-type inequality for the Sugeno integral is generalized on an abstract fuzzy measure space (X, , μ). Moreover, several examples are given to illustrate the validity of main results. © 2017 Elsevier Inc. All rights reserved.
Keywords: Carlson-type inequality Sugeno integral Fuzzy measure Comonotone functions
1. Introduction The integral inequalities are useful tools in several theoretical and applied fields. For instance, integral inequalities play a role in the development of a time scales calculus [12]. For more information on classical inequalities, we refer the reader to the recent monograph [7]. The study of inequalities for Sugeno integral was initiated by Flores-Franulicˇ and Román-Flores [6], Román-Flores et al. [15,16] Román-Flores and Chalco-Cano [17,18] and then followed by the authors in [1,2,8,9,11]. In [16] Román-Flores et al. studied some properties of Sugeno integral for strictly monotone real functions, they also provided some Yong-type inequalities. Based on these results, Flores-Franulicˇ and Román-Flores [6] provided some Chebyshev-type inequalities for Sugeno integral of continuous and strictly monotone real functions based on Lebesgue measure. Some other classical inequalities have also been generalized to Sugeno integral by them (see, for example [15,18]). Also some other classical inequalities have been generalized by others [1,2,4]. Later on, Ouyang and Fang [9] generalized the main results of Román-Flores et al. [16] to the case of monotone real functions. Based on these results, Ouyang et al. further generalized the fuzzy Chebyshev type inequalities to the case of arbitrary fuzzy measure-based Sugeno integrals [8,11]. In fact, they proved the following result. Theorem 1.1. Let f, g ∈ F μ (X ) and μ be an arbitrary fuzzy measure such that both −A f dμ and −A gdμ are finite. And let : [0, ∞)2 → [0, ∞) be continuous and non-decreasing in both arguments and bounded from above by minimum. If f, g are comonotone, then the inequality
− f gdμ ≥ − f dμ − gdμ A
A
A
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (B. Daraby), [email protected] (H. Ghazanfary Asll), [email protected] (I. Sadeqi). http://dx.doi.org/10.1016/j.amc.2017.02.005 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.
(1.1)
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B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
holds. In view of the fact that
− f gdμ ≤ − f dμ − gdμ A
A
(1.2)
A
holds for comonotone functions f, g ∈ F μ (X ) wherever ≥ max, it is of great interest to determine the operator such that
− f gdμ = − f dμ − gdμ A
A
(1.3)
A
holds for any comonotone functions f and g and for a fuzzy measure μ and any measurable set A. Ouyang et al. [10] proved that, there are only 18 operators such that (1.3) holds, including that four well-known operators: min, max, PF and PL, where PF is defined as the operator such that x y = x for each pair (x, y) and PL is defined as the operator such that x y = y for each pair (x, y). Recently, Daraby and Arabi [5] proved two related inequalities to Fritz Carlson-type inequality for the Sugeno integral, which are following. Theorem 1.2. Let f: [0, 1] → [0, ∞) be an increasing function and μ be the Lebesgue measure on R. And let : [0, ∞)2 → [0, ∞) be continuous and non-decreasing in both arguments and bounded from below by maximum. Then, the inequality
1 12 1 12 1 1 − f ( x ) f ( x )d μ ( x ) ≤ − x2 f 2 ( x )d μ ( x ) − f 2 ( x )d μ ( x ) 2 0 0 0
(1.4)
holds. Theorem 1.3. Let f: [0, 1] → [0, ∞) be an increasing function and μ be the Lebesgue measure on R. And let be a binary operation such that the Sugeno integral possesses comonotonic--property. Then, the inequality
1 2
1
− f ( x ) f ( x )d μ ( x ) ≤ 0
1
− x2 f 2 ( x )d μ ( x ) 0
12
1
− f 2 ( x )d μ ( x )
12 (1.5)
0
holds. In the present paper, we intend to prove Carlson-type inequality for the Sugeno integral on an abstract space. Generally, any integral inequality can be a very strong tool for applications. For example, when we think of an integral operator as a predictive tool, then an integral inequality can be very important in measuring and dimensioning such process. This paper is organized as follows: in Section 2, we present some basic concepts and previous results on related topics. In Section 3, we establish the general related inequalities to Carlson-type inequality for the Sugeno integral on an abstract fuzzy measure space. Finally, we close this paper by a conclusion. 2. Preliminaries As usual, we denote by R, the set of all real numbers. Let X be a non-empty set and be a σ -algebra of subsets of X. Throughout this paper, all considered subsets are supposed to be in . Definition 2.1 (Ralescu and Adams [14]). A set function μ : → [0, +∞] is called a fuzzy measure if the following properties are satisfied: (FM1) (FM2) (FM3) (FM4)
μ ( ∅ ) = 0, A ⊂ B ⇒ μ(A) ≤ μ(B), A1 ⊂ A2 ⊂ · · · ⇒ μ ( ∞ Ai ), i=1 Ai ) = lim μ ( A1 ⊃ A2 ⊃ · · · and μ(A1 ) < ∞ ⇒ μ( ∞ i=1 Ai ) = lim μ (Ai ).
When μ is a fuzzy measure, the triple (X, , μ) is called a fuzzy measure space. Let (X, , μ) be a fuzzy measure space. We denote the set of all non-negative μ-measurable functions with respect to , by F μ (X ). Let f be a non-negative real-valued function defined on X. We will denote the set {x ∈ X|f(x) ≥ α } by Fα for α ≥ 0. Clearly, Fα is non-increasing with respect to α , i.e. α ≤ β implies that Fα ⊃Fβ . Definition 2.2 (Pap [13], Wang and Klir [19]). Let μ be a fuzzy measure on (X, ). If f ∈ F μ (X ) and A ∈ , then the Sugeno integral (or fuzzy integral) of f on A, with respect to the fuzzy measure μ, is defined as
− f dμ = (α ∧ μ(A ∩ Fα )), A
α ≥0
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where ∨ , ∧ denotes the operation sup and inf on [0, ∞) respectively. In particular, if A = X, then
− f dμ = − f dμ = (α ∧ μ(Fα )). X
α ≥0
The following properties of the Sugeno integral can be found in [13,19]. Theorem 2.3 (Pap [13], Wang and Klir [19]). Let (X, , μ) be a fuzzy measure space, then (1) μ(A ∩ Fα ) ≥ α ⇒ −A f dμ ≥ α , (2) μ(A ∩ Fα ) ≤ α ⇒ −A f dμ ≤ α , (3) −A f dμ = α ⇔ μ(A ∩ Fα ) ≥ α ≥ μ(A ∩ Fα + ), where μ(A ∩ Fα + ) = limε→0 μ(A ∩ Fα +ε ), (4) −A f dμ < α ⇔ there exists γ < α such that μ(A ∩ Fγ ) < α , (5) −A f dμ > α ⇔ there exists γ > α such that μ(A ∩ Fγ ) > α , (6) f ≤ g on A, then −A f dμ ≤ −A gdμ, (7) −A kdμ = k ∧ μ(A ), for k non-negative constant, (8) −A f dμ ≤ μ(A ). Remark 2.4. Let F be a distribution function associated to f on A, that is,
F (α ) = μ(A ∩ { f ≥ α} ). Then, by Theorem 2.3, (1) and (2), we have
F (α ) = α ⇒ − f d μ = α . A
Thus, from a numerical point of view, the Sugeno integral can be calculated by solving the equation F (α ) = α . Recall that two functions f, g : X → R are said comonotone, if for all (x, y) ∈ X2 , ( f (x ) − f (y ))(g(x ) − g(y )) ≥ 0. Clearly, if f and g are comonotone, then, for any real numbers p and q, either Fp ⊂ Gq or Gq ⊂ Fp . Definition 2.5 (Ouyang et al. [10]). Let : [0, ∞)2 → [0, ∞) be a binary operation. We say that the Sugeno integral possesses comonotonic--property, if (1.3) holds for any fuzzy measure space (X, , μ), and any measurable set A and for any comonotone functions f, g: X → [0, ∞). In [20], Xu and Ouyang proved the following lemma. Lemma 2.6 (Xu and Ouyang [20]). Let (X, , μ) be a fuzzy measure space, let A ∈ and let f : X → R be a measurable function such that −A f dμ ≤ 1. Then, for any s ≥ 1, we have
s − f s dμ ≥ − f dμ . A
(2.1)
A
Caballero and Sadarangani [3] studied Carlson’s inequality for the Sugeno integral, which is following. Theorem 2.7 (Caballero and Sadarangani [3]). Let f: [0, 1] → [0, ∞] be a non-decreasing function and μ be the Lebesgue measure on R, Then
1 14 1 14 √ 1 − f ( x )d μ ( x ) ≤ 2 − x2 f 2 ( x )d μ ( x ) − f 2 ( x )d μ ( x ) . 0
0
(2.2)
0
In [20], Xu and Ouyang stated a general version of Carlson’s inequality for the Sugeno integral as follows. Theorem 2.8 (Xu and Ouyang [20]). Let (X, , μ) be a fuzzy measure space, let fi : X → R, i = 1, 2, 3, be measurable functions such that −A fi dμ ≤ 1. If any two functions of fi , i = 1, 2, 3, are comonotone, then for any p, q ≥ 1, we have
21p 21q 1 − f1 dμ ≤ √ − f1p f2p dμ − f1q f3q dμ , C A A A
(2.3)
where, C = (−A f2 dμ )(−A f3 dμ ). 3. Main results
In this section, as an main result, we study general related inequalities to Carlson-type inequality for the Sugeno integral. Theorem 3.1. Let (X, , μ) be a fuzzy measure space, and f i : X → R, i = 1, 2, 3, be measurable functions such that −A fi dμ ≤ 1. 2 Let : [0, ∞) → [0, ∞) be continuous and non-decreasing in both arguments and bounded from below by maximum. If any two functions of fi , i = 1, 2, 3, are comonotone, then for any p, q ≥ 1 we have
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
A
(3.1)
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where, C = (−A f2 dμ )(−A f3 dμ ). (1 ) Proof. From −A f1 dμ ≤ 1 and −A f2 dμ ≤ 1 we get −A f1 f2 dμ ≤ 1. In fact, by (3) of Theorem 2.3, we have μ(A ∩ F1+ ) ≤ 1 and (2 ) (i ) μ(A ∩ F1+ ) ≤ 1, where μ(A ∩ F1+ ) = limε→0 μ(A ∩ {x | fi (x ) ≥ 1 + ε} ). Thus, the comonotonicity of f1 and f2 implies that
(1 ) (2 ) μ A (F1+ ∪ F1+ ) ≤ 1.
By the fact {x|f1 (x)f2 (x) > 1} ⊂ {x|f1 (x) > 1} ∪ {x|f2 (x) > 1}, we have
μ(A ∩ {x | f1 (x ) f2 (x ) ≥ 1 + ε} ) ≤ 1,
for any ε > 0. Again, by (3) of Theorem 2.3 , we conclude that −A f1 f2 dμ ≤ 1 (see [20]). Now, by Lemma 2.6, for p, q ≥ 1, we have
− f1 f2 dμ
p
≤ − f1p f2p dμ,
A
and
− f1 f3 dμ
(3.2)
A
q
≤ − f1q f3q dμ.
A
(3.3)
A
Since is non-decreasing, we have
− f1 f2 dμ
− f1 f3 dμ ≤
A
A
− f1p f2p dμ
1p
A
− f1q f3q dμ
1q
A
.
(3.4)
Since the usual product is bounded from above by minimum on [0, 1]2 , by (1.1) we have
− f1 f2 dμ ≥ − f1 dμ − f2 dμ ,
(3.5)
− f1 f3 dμ ≥ − f1 dμ − f3 dμ .
(3.6)
A
and
A
A
A
A
A
Since ≥ max, by (1.2) we get
− f1 f1 dμ ≤ − f1 dμ − f1 dμ . A
Thus,
A
(3.7)
A
− f2 dμ − f3 dμ − f1 f1 dμ ≤ − f2 dμ − f3 dμ − f1 dμ − f1 dμ , A
A
A
A
A
A
(3.8)
A
from Inequalities (3.5) and (3.6) we have
− f f dμ − f f dμ 1 3 A 1 2 − f2 dμ − f3 dμ − f1 dμ − f1 dμ ≤ − f2 dμ − f3 dμ A . − − A A A A A A A f2 dμ A f3 dμ
(3.9)
Note that if a, b ≥ 0, then αβ (ab) ≤ α aβ b for all α , β ∈ R, where 0 ≤ α , β ≤ 1. So
− f f dμ − f f dμ 1 3 A 1 2 − f2 dμ − f3 dμ A ≤ − f 1 f 2 d μ − f 1 f 3 d μ, − − A A A A A f2 dμ A f3 dμ
(3.10)
and by Inequalities (3.8)–(3.10) we obtain
− f2 dμ − f3 dμ − f 1 f 1 d μ ≤ − f 1 f 2 d μ − f 1 f 3 d μ. A
A
A
A
(3.11)
A
Form Inequalities (3.4) and (3.11) we conclude that
1p 1q − f2 dμ − f3 dμ − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ . A
A
A
A
A
If we denote (−A f2 dμ )(−A f3 dμ ) by C, then we get the desired result.
The following example shows the validity of Theorem 3.1. √ Example 3.2. Suppose that f1 (x ) = 1, f2 (x ) = x, f3 (x ) = x, A = [0, 1], p = 2, q = 1 and μ is the Lebesgue measure, then simple calculation shows that 1 1 1 1√ 1 1 −0 f1 (x )dμ = −0 1dμ = 1, −0 f2 (x )dμ = −0 xdμ ≈ 0.618, −0 f3 (x )dμ = −0 xdμ = 0.5, 1 1 C = (−0 f2 dμ )(−0 f3 dμ ) ≈ 0.618 × 0.5 ≈ 0.309,
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
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2 1 1 2 1 2 ( x )d μ 2 = − f ( x ) f − xd μ = (0.5 ) 2 ≈ 0.707, 0 1 0 2 1 1 − 0 f 1 ( x ) f 3 ( x )d μ = − 0 xd μ = 0.5. 1
1
Now, if we let = ∨, then we have
1 1 1 12 1 C − f1 ∨ f1 dμ = C − f1 dμ ≈ 0.309 ≤ 0.707 ≈ − f12 f22 dμ ∨ − f1 f3 dμ . 0
0
0
0
Remark 3.3. Note that, in Theorem 3.1, letting f2 (x ) = x, f3 (x ) = 1, for all x ∈ X and p = q = 2, we conclude Theorem 1.2. In the following example, we show that comonotonicity of fi , i = 1, 2, 3, in Theorem 3.1 is necessary. Example 3.4. Let f1 (x ) = x, f2 (x ) = f3 (x ) = 2 − x, p = q = 1, = ∨, A = [0, 2] and μ be the Lebesgue measure. Then 2 2 2 2 2 − f1 dμ = −0 f2 dμ = −0 f3 dμ = 1, C = (−0 f2 dμ )(−0 f3 dμ ) = 2, √ 02 2 2 −0 f1 f2 dμ = −0 f1 f3 dμ = −0 x(2 − x )dμ = 2 2 − 2. Thus
2
2 2 √ C − f1 ∨ f1 dμ = 2 > 2 2 − 2 = − f1 f2 dμ ∨ − f1 f3 dμ . 0
0
0
The following example shows that, if we omit the condition −A fi dμ ≤ 1, i = 1, 2, 3, in Theorem 3.1, then Inequality (3.1) may not hold. Example 3.5. Let A = [0, 3], fi (x ) = x, i = 1, 2, 3, and p = q = 1, = ∨ and μ be the Lebesgue measure. Then 3 3 3 3 3 3 − f1 dμ = −0 f2 dμ = −0 f3 dμ = −0 xdμ = 32 , C = (−0 f2 dμ )(−0 f3 dμ ) = ( 32 )2 , √ 03 3 3 2 −0 f1 f2 dμ = −0 f1 f3 dμ = −0 x dμ = 7−2 13 . Thus,
3
3 √ 3 3 3 7 − 13 C − f1 ∨ f1 dμ = = 3.375 > 1.697 ≈ = − f1 f2 dμ ∨ − f1 f3 dμ . 2 2 0 0 0
Theorem 3.6. Let (X, , μ) be a fuzzy measure space, and f i : X → R, i = 1, 2, 3, be measurable functions such that −A fi dμ ≤ 1. Let : [0, ∞)2 → [0, ∞) be a non-decreasing binary operation such that the Sugeno integral possesses comonotonic--property. Then we have
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
(3.12)
A
where, C = (−A f2 dμ )(−A f3 dμ ). Proof. Since −A f1 dμ ≤ 1 and −A f2 dμ ≤ 1, from (1.3) we get −A f1 f2 dμ = −A f1 dμ −A f2 dμ ≤ 1. Similarly −A f1 f3 dμ ≤ 1. Therefore, by Lemma 2.6, for p, q ≥ 1 we have
− f1 f2 dμ
p
≤ − f1p f2p dμ,
A
and
− f1 f3 dμ
(3.13)
A
q
≤ − f1q f3q dμ.
A
(3.14)
A
Since is non-decreasing, we have
− f1 f2 dμ A
− f1 f3 dμ ≤
− f1p f2p dμ
A
A
1p
− f1q f3q dμ
1q
A
.
(3.15)
Thus, by using (1.3), we have
− f2 dμ − f3 dμ − f1 f1 dμ = − f2 dμ − f3 dμ − f1 dμ − f1 dμ .
A
A
A
A
A
A
(3.16)
A
Since a, b ≥ 0, then αβ (ab) ≤ α aβ b for all α , β ∈ R, where 0 ≤ α , β ≤ 1. So
− f2 dμ − f3 dμ − f1 dμ − f1 dμ ≤ − f1 dμ − f2 dμ − f1 dμ − f3 dμ . A
A
A
A
A
A
A
Note that
− f i d μ ≤ 1 , − f j d μ ≤ 1 ⇒ − f i d μ − f j d μ ≤ − f i d μ − f j d μ, A
A
A
A
A
A
A
(3.17)
328
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
hence,
− f1 dμ − f2 dμ − f1 dμ − f3 dμ ≤ − f1 dμ − f2 dμ − f1 dμ − f3 dμ , A
A
A
A
A
A
A
(3.18)
A
again by using (1.3) we get
− f1 dμ − f2 dμ − f1 dμ − f3 dμ = − f1 f2 dμ − f1 f3 dμ . A
A
A
A
A
(3.19)
A
Using Inequalities (3.15)–(3.19), we obtain
1p 1q − f2 dμ − f3 dμ − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ .
A
A
A
A
A
If we denote (−A f2 dμ )(−A f3 dμ ) by C, then we get the desired result.
For the validity of Theorem 3.6 we present two examples. √ Example 3.7. Suppose that f1 (x ) = 1, f2 (x ) = x, f3 (x ) = x, A = [0, 1], p = 2 and q = 1. Let μ be the Lebesgue measure and ∈ {∨, ∧, PF, PL}, then simple calculation shows that 1 1 1 1√ 1 1 −0 f1 (x )dμ = −0 1dμ = 1, −0 f2 (x )dμ = −0 xdμ ≈ 0.618, −0 f3 (x )dμ = −0 xdμ = 0.5, 1 1 C = (−0 f2 dμ )(−0 f3 dμ ) ≈ 0.618 × 0.5 = 0.309, 12 12 12 1 1 1 1 −0 f12 (x )dμ = −0 1dμ = −0 1dμ = ( 1 ) 2 = 1,
1
1
−0 f22 (x )dμ
12
=
−0 xdμ
−0 f12 (x ) ∨ f22 (x )dμ
1
12
1
= (0.5 ) 2 ≈ 0.707,
−0 1 ∨ xdμ 1
12
=
−0 1dμ 1
12
1
= ( 1 ) 2 = 1,
2 2 1 1 1 −0 1 ∧ xdμ = −0 xdμ = (0.5 ) 2 ≈ 0.707, 1 1 1 − f1 (x ) ∨ f3 (x )dμ = −0 1 ∨ xdμ = −0 1dμ = 1, 01 1 1 −0 f1 (x ) ∧ f3 (x )dμ = −0 1 ∧ xdμ = −0 xdμ = 0.5. Now, we consider four cases:
−0 f12 (x ) ∧ f22 (x )dμ 1
12
=
12
=
1
1
(1) If = ∨, then 12 1 1 1 2 1 2 C− − ∨ − 0 f1 ∨ f1 dμ = C − 0 f 1 d μ ≈ 0.309 ≤ 1 = 0 f1 ∨ f2 dμ 0 f1 ∨ f3 dμ . (2) If = ∧, then 12 1 1 1 2 1 2 C− − ∧ − 0 f1 ∧ f1 dμ = C − 0 f 1 d μ ≈ 0.309 ≤ 0.5 = 0 f1 ∧ f2 dμ 0 f1 ∧ f3 dμ . (3) If = P F , then
12 12 1 1 1 1 1 2 2 C− −0 f12 dμ = PF − , −0 P F ( f1 , f3 )dμ . 0 P F ( f 1 , f 1 )d μ = C − 0 f 1 d μ ≈ 0.309 ≤ 1 = 0 P F ( f 1 , f 2 )d μ (4) If = P L, then
1 1 1 1 1 2 , f 2 )d μ 2 , −1 P L ( f , f )d μ C− P L ( f , f ) d μ = C − f d μ ≈ 0 . 309 ≤ 0 . 5 = − f d μ = P L − P F ( f . 1 1 1 3 1 3 0 0 0 0 0 1 2 √ Example 3.8. Suppose that f1 (x ) = x, f2 (x ) = x2 , f3 (x ) = x, A = [0, 1] and p = q = 1. Let μ be the Lebesgue measure and ∈ {∨, ∧, PF, PL}, then simple calculation shows that 1 1 1 1 1 1√ −0 f1 (x )dμ = −0 xdμ = 0.5, −0 f2 (x )dμ = −0 x2 dμ ≈ 0.118, −0 f3 (x )dμ = −0 xdμ ≈ 0.618, 1 1 C = (−0 f2 dμ )(−0 f3 dμ ) ≈ 0.118 × 0.618 ≈ 0.073, 1 1 1 − f1 (x ) ∨ f2 (x )dμ = −0 x ∨ x2 dμ = −0 xdμ = 0.5, 01 1 1 − f1 (x ) ∧ f2 (x )dμ = −0 x ∧ x2 dμ = −0 x2 dμ ≈ 0.118, 01 1 1√ √ − f1 (x ) ∨ f3 (x )dμ = −0 x ∨ xdμ = −0 xdμ ≈ 0.618, 01 1 1 √ −0 f1 (x ) ∧ f3 (x )dμ = −0 x ∧ xdμ = −0 xdμ = 0.5. Now, we consider four cases: (1) If = ∨, then 1 1 1 1 C− − 0 f1 ∨ f1 dμ = C − 0 f 1 d μ ≈ 0.036 ≤ 0.618 ≈ 0 f1 ∨ f2 dμ ∨ − 0 f1 ∨ f3 dμ . (2) If = ∧, then 1 1 1 1 C− − 0 f1 ∧ f1 dμ = C − 0 f 1 d μ ≈ 0.036 ≤ 0.118 ≈ 0 f1 ∧ f2 dμ ∧ − 0 f1 ∧ f3 dμ . (3) If = P F , then
1 1 1 1 1 C− P F ( f , f ) d μ = C − f d μ ≈ 0 . 036 ≤ 0 . 5 = − f d μ = P F − P F ( f , f ) d μ , − P F ( f , f ) d μ . 1 1 1 1 1 2 1 3 0 0 0 0 0
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(4) If = P L, then
1 1 1 1 1 C− − . 0 P L ( f 1 , f 1 )d μ = C − 0 f 1 d μ ≈ 0.036 ≤ 0.618 ≈ −0 f 3 d μ = P L 0 P L ( f 1 , f 2 )d μ , −0 P L ( f 1 , f 3 )d μ The following example shows that, if we omit the condition comonotonicity of f i , i = 1, 2, 3, then Inequality (3.12) does not hold true, in general. Example 3.9. Let A = [0, 2] and f1 (x ) = x, f2 (x ) = f3 (x ) = 2 − x, p = q = 1, = ∧ and μ be the Lebesgue measure, then 2 2 2 2 2 − f1 (x )dμ = −0 f2 (x )dμ = −0 f3 (x )dμ = 1, C = (−0 f2 dμ )(−0 f3 dμ ) = 1, 02 2 −0 f1 (x ) ∧ f2 (x )dμ = −0 f1 (x ) ∧ f3 (x )dμ = 23 . Thus,
2 2 2 2 C − f1 ∧ f1 dμ = 1 > = − f1 ∧ f2 dμ ∧ − f1 ∧ f3 dμ . 3 0 0 0
The following example shows that the condition −A fi dμ ≤ 1, i = 1, 2, 3, in Theorem 3.6 is necessary. Example 3.10. Consider A = [0, 5], fi (x ) = 5x, i = 1, 2, 3, p = q = 1, = ∨ and let μ be the Lebesgue measure, then 5 5 5 5 5 25 2 25 −0 fi (x )dμ = 25 6 , C = (−0 f 2 d μ )(−0 f 3 d μ ) = ( 6 ) , −0 f 1 (x ) ∨ f 2 (x )d μ = −0 f 1 (x ) ∨ f 3 (x )d μ = 6 . Thus,
5 5 25 3 25 5 C − f1 ∨ f1 dμ = > = − f1 ∨ f2 dμ ∨ − f1 ∨ f3 dμ . 6 6 0 0 0
4. Conclusion In this paper, the general related inequalities to Carlson-type inequality for the Sugeno integral on an abstract fuzzy measure space (X, , μ) are studied. It is also pointed out that for the measurable pairwise comonotone functions f i : X → R, i = 1, 2, 3, with −A fi dμ ≤ 1, one has
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
A
for which, : [0, ∞)2 → [0, ∞) be continuous and non-decreasing in both arguments and bounded from below by maximum, C = (−A f2 dμ )(−A f3 dμ ) and p, q ≥ 1. In addition, by the same assumptions, we get
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
A
where : [0, ∞)2 → [0, ∞) be a non-decreasing binary operation such that the Sugeno integral possesses comonotonic-property. Acknowledgments The authors would like to thank the reviewers for their valuable comments and suggestions to improve the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
H. Agahi, R. Mesiar, Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets Syst. 161 (2010) 708–715. H. Agahi, M.A. Yaghoobi, A Minkowski type inequality for fuzzy integrals, J. Uncertain. Syst. 4 (2010) 187–1940. J. Caballero, K. Sadarangani, Fritz Carlson’s inequality for fuzzy integrals, Comput. Math. Appl. 59 (2010) 2763–2767. B. Daraby, Investigation of a Stolarsky type inequality for integrals in pseudo-analysis, Fract. Calc. Appl. Anal. 13 (5) (2010) 467–473. B. Daraby, L. Arabi, Related Fritz Carlson type inequality for Sugeno integrals, Soft Comput. 17 (2013) 1745–1750. A. Flores-Franulicˇ , H. Román-Flores, A Chebyshev type inequality for fuzzy integrals, Appl. Math. Comput. 190 (2007) 1178–1184. J. Kuang, Applied Inequalities, third ed., Shandong Science and Technology Press, 2004. (in Chinese) R. Mesiar, Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets Syst. 160 (2009) 58–64. Y. Ouyang, J. Fang, Sugeno integral of monotone functions based on Lebesgue measure, Comput. Math. Appl. 56 (2008) 367–374. Y. Ouyang, R. Mesiar, J. Li, On the comonotonic--property for Sugeno integral, Appl. Math. Comput. 211 (2009) 450–458. Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, Int. J. Approx. Reason. 48 (2008) 829–835. U.M. Özkan, M.Z. Sarikaya, H. Yildirim, Extensions of certain integral inequalities on time scales, Appl. Math. Lett. 21 (2008) 993–1000. E. Pap, Null-Additive Set Functions, Kluwer, Dordrecht, 1995. D. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl. 75 (1980) 562–570. H. Román-Flores, A. Flores-Franulicˇ , Y. Chalco-Cano, A Jensen type inequality for fuzzy integrals, Inf. Sci. 177 (2007) 3192–3201. H. Román-Flores, A. Flores-Franulic, Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput. 185 (2007) 492– 498. H. Román-Flores, Y. Chalco-Cano, H-continuity of fuzzy measures and set defuzzification, Fuzzy Sets Syst. 157 (2006) 230–242. H. Román-Flores, Y. Chalco-Cano, Sugeno integral and geometric inequalities, Int. J. Uncertain. Fuzziness Knowl. Based Syst. 15 (2007) 1–11. Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. Q. Xu, Y. Ouyang, A note on a Carlson-type inequality for the Sugeno integral, Appl. Math. Lett. 25 (2012) 619–623.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
General related inequalities to Carlson-type inequality for the Sugeno integral Bayaz Daraby a,∗, Hassan Ghazanfary Asll b, Ildar Sadeqi b a b
Department of Mathematics, University of Maragheh, P.O. Box 55181–83111, Maragheh, Iran Department of Mathematics, Sahand University of Technology, Tabriz, Iran
a r t i c l e
i n f o
a b s t r a c t
MSC: 35A23 45G99 28E10 45N05
In this paper, the Carlson-type inequality for the Sugeno integral is generalized on an abstract fuzzy measure space (X, , μ). Moreover, several examples are given to illustrate the validity of main results. © 2017 Elsevier Inc. All rights reserved.
Keywords: Carlson-type inequality Sugeno integral Fuzzy measure Comonotone functions
1. Introduction The integral inequalities are useful tools in several theoretical and applied fields. For instance, integral inequalities play a role in the development of a time scales calculus [12]. For more information on classical inequalities, we refer the reader to the recent monograph [7]. The study of inequalities for Sugeno integral was initiated by Flores-Franulicˇ and Román-Flores [6], Román-Flores et al. [15,16] Román-Flores and Chalco-Cano [17,18] and then followed by the authors in [1,2,8,9,11]. In [16] Román-Flores et al. studied some properties of Sugeno integral for strictly monotone real functions, they also provided some Yong-type inequalities. Based on these results, Flores-Franulicˇ and Román-Flores [6] provided some Chebyshev-type inequalities for Sugeno integral of continuous and strictly monotone real functions based on Lebesgue measure. Some other classical inequalities have also been generalized to Sugeno integral by them (see, for example [15,18]). Also some other classical inequalities have been generalized by others [1,2,4]. Later on, Ouyang and Fang [9] generalized the main results of Román-Flores et al. [16] to the case of monotone real functions. Based on these results, Ouyang et al. further generalized the fuzzy Chebyshev type inequalities to the case of arbitrary fuzzy measure-based Sugeno integrals [8,11]. In fact, they proved the following result. Theorem 1.1. Let f, g ∈ F μ (X ) and μ be an arbitrary fuzzy measure such that both −A f dμ and −A gdμ are finite. And let : [0, ∞)2 → [0, ∞) be continuous and non-decreasing in both arguments and bounded from above by minimum. If f, g are comonotone, then the inequality
− f gdμ ≥ − f dμ − gdμ A
A
A
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (B. Daraby), [email protected] (H. Ghazanfary Asll), [email protected] (I. Sadeqi). http://dx.doi.org/10.1016/j.amc.2017.02.005 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.
(1.1)
324
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
holds. In view of the fact that
− f gdμ ≤ − f dμ − gdμ A
A
(1.2)
A
holds for comonotone functions f, g ∈ F μ (X ) wherever ≥ max, it is of great interest to determine the operator such that
− f gdμ = − f dμ − gdμ A
A
(1.3)
A
holds for any comonotone functions f and g and for a fuzzy measure μ and any measurable set A. Ouyang et al. [10] proved that, there are only 18 operators such that (1.3) holds, including that four well-known operators: min, max, PF and PL, where PF is defined as the operator such that x y = x for each pair (x, y) and PL is defined as the operator such that x y = y for each pair (x, y). Recently, Daraby and Arabi [5] proved two related inequalities to Fritz Carlson-type inequality for the Sugeno integral, which are following. Theorem 1.2. Let f: [0, 1] → [0, ∞) be an increasing function and μ be the Lebesgue measure on R. And let : [0, ∞)2 → [0, ∞) be continuous and non-decreasing in both arguments and bounded from below by maximum. Then, the inequality
1 12 1 12 1 1 − f ( x ) f ( x )d μ ( x ) ≤ − x2 f 2 ( x )d μ ( x ) − f 2 ( x )d μ ( x ) 2 0 0 0
(1.4)
holds. Theorem 1.3. Let f: [0, 1] → [0, ∞) be an increasing function and μ be the Lebesgue measure on R. And let be a binary operation such that the Sugeno integral possesses comonotonic--property. Then, the inequality
1 2
1
− f ( x ) f ( x )d μ ( x ) ≤ 0
1
− x2 f 2 ( x )d μ ( x ) 0
12
1
− f 2 ( x )d μ ( x )
12 (1.5)
0
holds. In the present paper, we intend to prove Carlson-type inequality for the Sugeno integral on an abstract space. Generally, any integral inequality can be a very strong tool for applications. For example, when we think of an integral operator as a predictive tool, then an integral inequality can be very important in measuring and dimensioning such process. This paper is organized as follows: in Section 2, we present some basic concepts and previous results on related topics. In Section 3, we establish the general related inequalities to Carlson-type inequality for the Sugeno integral on an abstract fuzzy measure space. Finally, we close this paper by a conclusion. 2. Preliminaries As usual, we denote by R, the set of all real numbers. Let X be a non-empty set and be a σ -algebra of subsets of X. Throughout this paper, all considered subsets are supposed to be in . Definition 2.1 (Ralescu and Adams [14]). A set function μ : → [0, +∞] is called a fuzzy measure if the following properties are satisfied: (FM1) (FM2) (FM3) (FM4)
μ ( ∅ ) = 0, A ⊂ B ⇒ μ(A) ≤ μ(B), A1 ⊂ A2 ⊂ · · · ⇒ μ ( ∞ Ai ), i=1 Ai ) = lim μ ( A1 ⊃ A2 ⊃ · · · and μ(A1 ) < ∞ ⇒ μ( ∞ i=1 Ai ) = lim μ (Ai ).
When μ is a fuzzy measure, the triple (X, , μ) is called a fuzzy measure space. Let (X, , μ) be a fuzzy measure space. We denote the set of all non-negative μ-measurable functions with respect to , by F μ (X ). Let f be a non-negative real-valued function defined on X. We will denote the set {x ∈ X|f(x) ≥ α } by Fα for α ≥ 0. Clearly, Fα is non-increasing with respect to α , i.e. α ≤ β implies that Fα ⊃Fβ . Definition 2.2 (Pap [13], Wang and Klir [19]). Let μ be a fuzzy measure on (X, ). If f ∈ F μ (X ) and A ∈ , then the Sugeno integral (or fuzzy integral) of f on A, with respect to the fuzzy measure μ, is defined as
− f dμ = (α ∧ μ(A ∩ Fα )), A
α ≥0
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
325
where ∨ , ∧ denotes the operation sup and inf on [0, ∞) respectively. In particular, if A = X, then
− f dμ = − f dμ = (α ∧ μ(Fα )). X
α ≥0
The following properties of the Sugeno integral can be found in [13,19]. Theorem 2.3 (Pap [13], Wang and Klir [19]). Let (X, , μ) be a fuzzy measure space, then (1) μ(A ∩ Fα ) ≥ α ⇒ −A f dμ ≥ α , (2) μ(A ∩ Fα ) ≤ α ⇒ −A f dμ ≤ α , (3) −A f dμ = α ⇔ μ(A ∩ Fα ) ≥ α ≥ μ(A ∩ Fα + ), where μ(A ∩ Fα + ) = limε→0 μ(A ∩ Fα +ε ), (4) −A f dμ < α ⇔ there exists γ < α such that μ(A ∩ Fγ ) < α , (5) −A f dμ > α ⇔ there exists γ > α such that μ(A ∩ Fγ ) > α , (6) f ≤ g on A, then −A f dμ ≤ −A gdμ, (7) −A kdμ = k ∧ μ(A ), for k non-negative constant, (8) −A f dμ ≤ μ(A ). Remark 2.4. Let F be a distribution function associated to f on A, that is,
F (α ) = μ(A ∩ { f ≥ α} ). Then, by Theorem 2.3, (1) and (2), we have
F (α ) = α ⇒ − f d μ = α . A
Thus, from a numerical point of view, the Sugeno integral can be calculated by solving the equation F (α ) = α . Recall that two functions f, g : X → R are said comonotone, if for all (x, y) ∈ X2 , ( f (x ) − f (y ))(g(x ) − g(y )) ≥ 0. Clearly, if f and g are comonotone, then, for any real numbers p and q, either Fp ⊂ Gq or Gq ⊂ Fp . Definition 2.5 (Ouyang et al. [10]). Let : [0, ∞)2 → [0, ∞) be a binary operation. We say that the Sugeno integral possesses comonotonic--property, if (1.3) holds for any fuzzy measure space (X, , μ), and any measurable set A and for any comonotone functions f, g: X → [0, ∞). In [20], Xu and Ouyang proved the following lemma. Lemma 2.6 (Xu and Ouyang [20]). Let (X, , μ) be a fuzzy measure space, let A ∈ and let f : X → R be a measurable function such that −A f dμ ≤ 1. Then, for any s ≥ 1, we have
s − f s dμ ≥ − f dμ . A
(2.1)
A
Caballero and Sadarangani [3] studied Carlson’s inequality for the Sugeno integral, which is following. Theorem 2.7 (Caballero and Sadarangani [3]). Let f: [0, 1] → [0, ∞] be a non-decreasing function and μ be the Lebesgue measure on R, Then
1 14 1 14 √ 1 − f ( x )d μ ( x ) ≤ 2 − x2 f 2 ( x )d μ ( x ) − f 2 ( x )d μ ( x ) . 0
0
(2.2)
0
In [20], Xu and Ouyang stated a general version of Carlson’s inequality for the Sugeno integral as follows. Theorem 2.8 (Xu and Ouyang [20]). Let (X, , μ) be a fuzzy measure space, let fi : X → R, i = 1, 2, 3, be measurable functions such that −A fi dμ ≤ 1. If any two functions of fi , i = 1, 2, 3, are comonotone, then for any p, q ≥ 1, we have
21p 21q 1 − f1 dμ ≤ √ − f1p f2p dμ − f1q f3q dμ , C A A A
(2.3)
where, C = (−A f2 dμ )(−A f3 dμ ). 3. Main results
In this section, as an main result, we study general related inequalities to Carlson-type inequality for the Sugeno integral. Theorem 3.1. Let (X, , μ) be a fuzzy measure space, and f i : X → R, i = 1, 2, 3, be measurable functions such that −A fi dμ ≤ 1. 2 Let : [0, ∞) → [0, ∞) be continuous and non-decreasing in both arguments and bounded from below by maximum. If any two functions of fi , i = 1, 2, 3, are comonotone, then for any p, q ≥ 1 we have
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
A
(3.1)
326
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
where, C = (−A f2 dμ )(−A f3 dμ ). (1 ) Proof. From −A f1 dμ ≤ 1 and −A f2 dμ ≤ 1 we get −A f1 f2 dμ ≤ 1. In fact, by (3) of Theorem 2.3, we have μ(A ∩ F1+ ) ≤ 1 and (2 ) (i ) μ(A ∩ F1+ ) ≤ 1, where μ(A ∩ F1+ ) = limε→0 μ(A ∩ {x | fi (x ) ≥ 1 + ε} ). Thus, the comonotonicity of f1 and f2 implies that
(1 ) (2 ) μ A (F1+ ∪ F1+ ) ≤ 1.
By the fact {x|f1 (x)f2 (x) > 1} ⊂ {x|f1 (x) > 1} ∪ {x|f2 (x) > 1}, we have
μ(A ∩ {x | f1 (x ) f2 (x ) ≥ 1 + ε} ) ≤ 1,
for any ε > 0. Again, by (3) of Theorem 2.3 , we conclude that −A f1 f2 dμ ≤ 1 (see [20]). Now, by Lemma 2.6, for p, q ≥ 1, we have
− f1 f2 dμ
p
≤ − f1p f2p dμ,
A
and
− f1 f3 dμ
(3.2)
A
q
≤ − f1q f3q dμ.
A
(3.3)
A
Since is non-decreasing, we have
− f1 f2 dμ
− f1 f3 dμ ≤
A
A
− f1p f2p dμ
1p
A
− f1q f3q dμ
1q
A
.
(3.4)
Since the usual product is bounded from above by minimum on [0, 1]2 , by (1.1) we have
− f1 f2 dμ ≥ − f1 dμ − f2 dμ ,
(3.5)
− f1 f3 dμ ≥ − f1 dμ − f3 dμ .
(3.6)
A
and
A
A
A
A
A
Since ≥ max, by (1.2) we get
− f1 f1 dμ ≤ − f1 dμ − f1 dμ . A
Thus,
A
(3.7)
A
− f2 dμ − f3 dμ − f1 f1 dμ ≤ − f2 dμ − f3 dμ − f1 dμ − f1 dμ , A
A
A
A
A
A
(3.8)
A
from Inequalities (3.5) and (3.6) we have
− f f dμ − f f dμ 1 3 A 1 2 − f2 dμ − f3 dμ − f1 dμ − f1 dμ ≤ − f2 dμ − f3 dμ A . − − A A A A A A A f2 dμ A f3 dμ
(3.9)
Note that if a, b ≥ 0, then αβ (ab) ≤ α aβ b for all α , β ∈ R, where 0 ≤ α , β ≤ 1. So
− f f dμ − f f dμ 1 3 A 1 2 − f2 dμ − f3 dμ A ≤ − f 1 f 2 d μ − f 1 f 3 d μ, − − A A A A A f2 dμ A f3 dμ
(3.10)
and by Inequalities (3.8)–(3.10) we obtain
− f2 dμ − f3 dμ − f 1 f 1 d μ ≤ − f 1 f 2 d μ − f 1 f 3 d μ. A
A
A
A
(3.11)
A
Form Inequalities (3.4) and (3.11) we conclude that
1p 1q − f2 dμ − f3 dμ − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ . A
A
A
A
A
If we denote (−A f2 dμ )(−A f3 dμ ) by C, then we get the desired result.
The following example shows the validity of Theorem 3.1. √ Example 3.2. Suppose that f1 (x ) = 1, f2 (x ) = x, f3 (x ) = x, A = [0, 1], p = 2, q = 1 and μ is the Lebesgue measure, then simple calculation shows that 1 1 1 1√ 1 1 −0 f1 (x )dμ = −0 1dμ = 1, −0 f2 (x )dμ = −0 xdμ ≈ 0.618, −0 f3 (x )dμ = −0 xdμ = 0.5, 1 1 C = (−0 f2 dμ )(−0 f3 dμ ) ≈ 0.618 × 0.5 ≈ 0.309,
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
327
2 1 1 2 1 2 ( x )d μ 2 = − f ( x ) f − xd μ = (0.5 ) 2 ≈ 0.707, 0 1 0 2 1 1 − 0 f 1 ( x ) f 3 ( x )d μ = − 0 xd μ = 0.5. 1
1
Now, if we let = ∨, then we have
1 1 1 12 1 C − f1 ∨ f1 dμ = C − f1 dμ ≈ 0.309 ≤ 0.707 ≈ − f12 f22 dμ ∨ − f1 f3 dμ . 0
0
0
0
Remark 3.3. Note that, in Theorem 3.1, letting f2 (x ) = x, f3 (x ) = 1, for all x ∈ X and p = q = 2, we conclude Theorem 1.2. In the following example, we show that comonotonicity of fi , i = 1, 2, 3, in Theorem 3.1 is necessary. Example 3.4. Let f1 (x ) = x, f2 (x ) = f3 (x ) = 2 − x, p = q = 1, = ∨, A = [0, 2] and μ be the Lebesgue measure. Then 2 2 2 2 2 − f1 dμ = −0 f2 dμ = −0 f3 dμ = 1, C = (−0 f2 dμ )(−0 f3 dμ ) = 2, √ 02 2 2 −0 f1 f2 dμ = −0 f1 f3 dμ = −0 x(2 − x )dμ = 2 2 − 2. Thus
2
2 2 √ C − f1 ∨ f1 dμ = 2 > 2 2 − 2 = − f1 f2 dμ ∨ − f1 f3 dμ . 0
0
0
The following example shows that, if we omit the condition −A fi dμ ≤ 1, i = 1, 2, 3, in Theorem 3.1, then Inequality (3.1) may not hold. Example 3.5. Let A = [0, 3], fi (x ) = x, i = 1, 2, 3, and p = q = 1, = ∨ and μ be the Lebesgue measure. Then 3 3 3 3 3 3 − f1 dμ = −0 f2 dμ = −0 f3 dμ = −0 xdμ = 32 , C = (−0 f2 dμ )(−0 f3 dμ ) = ( 32 )2 , √ 03 3 3 2 −0 f1 f2 dμ = −0 f1 f3 dμ = −0 x dμ = 7−2 13 . Thus,
3
3 √ 3 3 3 7 − 13 C − f1 ∨ f1 dμ = = 3.375 > 1.697 ≈ = − f1 f2 dμ ∨ − f1 f3 dμ . 2 2 0 0 0
Theorem 3.6. Let (X, , μ) be a fuzzy measure space, and f i : X → R, i = 1, 2, 3, be measurable functions such that −A fi dμ ≤ 1. Let : [0, ∞)2 → [0, ∞) be a non-decreasing binary operation such that the Sugeno integral possesses comonotonic--property. Then we have
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
(3.12)
A
where, C = (−A f2 dμ )(−A f3 dμ ). Proof. Since −A f1 dμ ≤ 1 and −A f2 dμ ≤ 1, from (1.3) we get −A f1 f2 dμ = −A f1 dμ −A f2 dμ ≤ 1. Similarly −A f1 f3 dμ ≤ 1. Therefore, by Lemma 2.6, for p, q ≥ 1 we have
− f1 f2 dμ
p
≤ − f1p f2p dμ,
A
and
− f1 f3 dμ
(3.13)
A
q
≤ − f1q f3q dμ.
A
(3.14)
A
Since is non-decreasing, we have
− f1 f2 dμ A
− f1 f3 dμ ≤
− f1p f2p dμ
A
A
1p
− f1q f3q dμ
1q
A
.
(3.15)
Thus, by using (1.3), we have
− f2 dμ − f3 dμ − f1 f1 dμ = − f2 dμ − f3 dμ − f1 dμ − f1 dμ .
A
A
A
A
A
A
(3.16)
A
Since a, b ≥ 0, then αβ (ab) ≤ α aβ b for all α , β ∈ R, where 0 ≤ α , β ≤ 1. So
− f2 dμ − f3 dμ − f1 dμ − f1 dμ ≤ − f1 dμ − f2 dμ − f1 dμ − f3 dμ . A
A
A
A
A
A
A
Note that
− f i d μ ≤ 1 , − f j d μ ≤ 1 ⇒ − f i d μ − f j d μ ≤ − f i d μ − f j d μ, A
A
A
A
A
A
A
(3.17)
328
B. Daraby et al. / Applied Mathematics and Computation 305 (2017) 323–329
hence,
− f1 dμ − f2 dμ − f1 dμ − f3 dμ ≤ − f1 dμ − f2 dμ − f1 dμ − f3 dμ , A
A
A
A
A
A
A
(3.18)
A
again by using (1.3) we get
− f1 dμ − f2 dμ − f1 dμ − f3 dμ = − f1 f2 dμ − f1 f3 dμ . A
A
A
A
A
(3.19)
A
Using Inequalities (3.15)–(3.19), we obtain
1p 1q − f2 dμ − f3 dμ − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ .
A
A
A
A
A
If we denote (−A f2 dμ )(−A f3 dμ ) by C, then we get the desired result.
For the validity of Theorem 3.6 we present two examples. √ Example 3.7. Suppose that f1 (x ) = 1, f2 (x ) = x, f3 (x ) = x, A = [0, 1], p = 2 and q = 1. Let μ be the Lebesgue measure and ∈ {∨, ∧, PF, PL}, then simple calculation shows that 1 1 1 1√ 1 1 −0 f1 (x )dμ = −0 1dμ = 1, −0 f2 (x )dμ = −0 xdμ ≈ 0.618, −0 f3 (x )dμ = −0 xdμ = 0.5, 1 1 C = (−0 f2 dμ )(−0 f3 dμ ) ≈ 0.618 × 0.5 = 0.309, 12 12 12 1 1 1 1 −0 f12 (x )dμ = −0 1dμ = −0 1dμ = ( 1 ) 2 = 1,
1
1
−0 f22 (x )dμ
12
=
−0 xdμ
−0 f12 (x ) ∨ f22 (x )dμ
1
12
1
= (0.5 ) 2 ≈ 0.707,
−0 1 ∨ xdμ 1
12
=
−0 1dμ 1
12
1
= ( 1 ) 2 = 1,
2 2 1 1 1 −0 1 ∧ xdμ = −0 xdμ = (0.5 ) 2 ≈ 0.707, 1 1 1 − f1 (x ) ∨ f3 (x )dμ = −0 1 ∨ xdμ = −0 1dμ = 1, 01 1 1 −0 f1 (x ) ∧ f3 (x )dμ = −0 1 ∧ xdμ = −0 xdμ = 0.5. Now, we consider four cases:
−0 f12 (x ) ∧ f22 (x )dμ 1
12
=
12
=
1
1
(1) If = ∨, then 12 1 1 1 2 1 2 C− − ∨ − 0 f1 ∨ f1 dμ = C − 0 f 1 d μ ≈ 0.309 ≤ 1 = 0 f1 ∨ f2 dμ 0 f1 ∨ f3 dμ . (2) If = ∧, then 12 1 1 1 2 1 2 C− − ∧ − 0 f1 ∧ f1 dμ = C − 0 f 1 d μ ≈ 0.309 ≤ 0.5 = 0 f1 ∧ f2 dμ 0 f1 ∧ f3 dμ . (3) If = P F , then
12 12 1 1 1 1 1 2 2 C− −0 f12 dμ = PF − , −0 P F ( f1 , f3 )dμ . 0 P F ( f 1 , f 1 )d μ = C − 0 f 1 d μ ≈ 0.309 ≤ 1 = 0 P F ( f 1 , f 2 )d μ (4) If = P L, then
1 1 1 1 1 2 , f 2 )d μ 2 , −1 P L ( f , f )d μ C− P L ( f , f ) d μ = C − f d μ ≈ 0 . 309 ≤ 0 . 5 = − f d μ = P L − P F ( f . 1 1 1 3 1 3 0 0 0 0 0 1 2 √ Example 3.8. Suppose that f1 (x ) = x, f2 (x ) = x2 , f3 (x ) = x, A = [0, 1] and p = q = 1. Let μ be the Lebesgue measure and ∈ {∨, ∧, PF, PL}, then simple calculation shows that 1 1 1 1 1 1√ −0 f1 (x )dμ = −0 xdμ = 0.5, −0 f2 (x )dμ = −0 x2 dμ ≈ 0.118, −0 f3 (x )dμ = −0 xdμ ≈ 0.618, 1 1 C = (−0 f2 dμ )(−0 f3 dμ ) ≈ 0.118 × 0.618 ≈ 0.073, 1 1 1 − f1 (x ) ∨ f2 (x )dμ = −0 x ∨ x2 dμ = −0 xdμ = 0.5, 01 1 1 − f1 (x ) ∧ f2 (x )dμ = −0 x ∧ x2 dμ = −0 x2 dμ ≈ 0.118, 01 1 1√ √ − f1 (x ) ∨ f3 (x )dμ = −0 x ∨ xdμ = −0 xdμ ≈ 0.618, 01 1 1 √ −0 f1 (x ) ∧ f3 (x )dμ = −0 x ∧ xdμ = −0 xdμ = 0.5. Now, we consider four cases: (1) If = ∨, then 1 1 1 1 C− − 0 f1 ∨ f1 dμ = C − 0 f 1 d μ ≈ 0.036 ≤ 0.618 ≈ 0 f1 ∨ f2 dμ ∨ − 0 f1 ∨ f3 dμ . (2) If = ∧, then 1 1 1 1 C− − 0 f1 ∧ f1 dμ = C − 0 f 1 d μ ≈ 0.036 ≤ 0.118 ≈ 0 f1 ∧ f2 dμ ∧ − 0 f1 ∧ f3 dμ . (3) If = P F , then
1 1 1 1 1 C− P F ( f , f ) d μ = C − f d μ ≈ 0 . 036 ≤ 0 . 5 = − f d μ = P F − P F ( f , f ) d μ , − P F ( f , f ) d μ . 1 1 1 1 1 2 1 3 0 0 0 0 0
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(4) If = P L, then
1 1 1 1 1 C− − . 0 P L ( f 1 , f 1 )d μ = C − 0 f 1 d μ ≈ 0.036 ≤ 0.618 ≈ −0 f 3 d μ = P L 0 P L ( f 1 , f 2 )d μ , −0 P L ( f 1 , f 3 )d μ The following example shows that, if we omit the condition comonotonicity of f i , i = 1, 2, 3, then Inequality (3.12) does not hold true, in general. Example 3.9. Let A = [0, 2] and f1 (x ) = x, f2 (x ) = f3 (x ) = 2 − x, p = q = 1, = ∧ and μ be the Lebesgue measure, then 2 2 2 2 2 − f1 (x )dμ = −0 f2 (x )dμ = −0 f3 (x )dμ = 1, C = (−0 f2 dμ )(−0 f3 dμ ) = 1, 02 2 −0 f1 (x ) ∧ f2 (x )dμ = −0 f1 (x ) ∧ f3 (x )dμ = 23 . Thus,
2 2 2 2 C − f1 ∧ f1 dμ = 1 > = − f1 ∧ f2 dμ ∧ − f1 ∧ f3 dμ . 3 0 0 0
The following example shows that the condition −A fi dμ ≤ 1, i = 1, 2, 3, in Theorem 3.6 is necessary. Example 3.10. Consider A = [0, 5], fi (x ) = 5x, i = 1, 2, 3, p = q = 1, = ∨ and let μ be the Lebesgue measure, then 5 5 5 5 5 25 2 25 −0 fi (x )dμ = 25 6 , C = (−0 f 2 d μ )(−0 f 3 d μ ) = ( 6 ) , −0 f 1 (x ) ∨ f 2 (x )d μ = −0 f 1 (x ) ∨ f 3 (x )d μ = 6 . Thus,
5 5 25 3 25 5 C − f1 ∨ f1 dμ = > = − f1 ∨ f2 dμ ∨ − f1 ∨ f3 dμ . 6 6 0 0 0
4. Conclusion In this paper, the general related inequalities to Carlson-type inequality for the Sugeno integral on an abstract fuzzy measure space (X, , μ) are studied. It is also pointed out that for the measurable pairwise comonotone functions f i : X → R, i = 1, 2, 3, with −A fi dμ ≤ 1, one has
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
A
for which, : [0, ∞)2 → [0, ∞) be continuous and non-decreasing in both arguments and bounded from below by maximum, C = (−A f2 dμ )(−A f3 dμ ) and p, q ≥ 1. In addition, by the same assumptions, we get
1p 1q C − f1 f1 dμ ≤ − f1p f2p dμ − f1q f3q dμ , A
A
A
where : [0, ∞)2 → [0, ∞) be a non-decreasing binary operation such that the Sugeno integral possesses comonotonic-property. Acknowledgments The authors would like to thank the reviewers for their valuable comments and suggestions to improve the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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