FURTHER DEVELOPMENT OF CHEBYSHEV TYPE INEQUALITIES FOR SUGENO INTEGRALS AND T-(S-)EVALUATORS

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1 K Y BERNETIK VOLUM E 46 21, NUMBER 1, P GES FURTHER DEVELOPMENT OF CHEBYSHEV TYPE INEQULITIES FOR SUGENO INTEGRLS ND T-S-EVLUTORS Hamzeh gahi, Radko Mesiar and Yao Ouyang In this paper further development of Chebyshev type inequalities for Sugeno integrals based on an aggregation function H and a scale transformation ϕ is given. Consequences for T-S-evaluators are established. Keywords: Sugeno integral, fuzzy measure, comonotone functions, Chebyshev s inequality, t-norm; t-conorm, T-S-evaluators Classification: 3E72, INTRODUCTION In 1974, Sugeno [25] initiated researches on fuzzy measures and fuzzy integrals. Fuzzy measure is a generalization of the notion of measure in mathematical analysis see, e. g., [18, 26]. Sugeno integral is analogous to Lebesgue integral which has been studied by many authors, including Pap [18], Ralescu and dams [19], Román Flores et al. [6, 2, 21, 22, 23] and, Wang and Klir [26], among others. The difference between Sugeno integral and Lebesgue integral is that addition and multiplication in the definition of Lebesgue integral are replaced respectively by the operations max and min when Sugeno integral is considered. Román Flores et al. [6, 2, 21, 22, 23], started the studies of inequalities for Sugeno integral, and then followed by the authors [1, 2, 11, 12, 13]. In [11], Mesiar and Ouyang proved the following Chebyshev type inequalities for Sugeno integrals: Theorem 1.1. Let f, g F + X and µ be an arbitrary fuzzy measure such that f dµ and g dµ are finite. Let : [, 2 [, be continuous and nondecreasing in both arguments and bounded from above by minimum. If f, g are comonotone, then the inequality holds. f g dµ f dµ g dµ 1

2 84 H. GHI, R. MESIR ND Y. OUYNG It is known that f g dµ f dµ g dµ where f, g are comonotone functions whenever max for a similar result, see [15], it is of great interest to determine the operator such that f g dµ = f dµ g dµ 3 holds for any comonotone functions f, g, and for any fuzzy measure µ and any measurable set. Ouyang et al. [16, 17] proved that there are only 18 operators such that 3 holds, including the four well-known operators: minimum, maximum, PF called the first projection, PF for short, if x y = x for each pair x, y and PL called the last projection, PL for short, if x y = y for each pair x, y. Recently, gahi and Yaghoobi [1] proved a Minkowski type inequality for monotone functions and arbitrary fuzzy measure-based Sugeno integrals on real line, and then gahi et al. [2] further generalized it to comonotone functions and arbitrary fuzzy measure-base Sugeno integrals on an arbitrary measurable space. For 1, gahi, Mesiar and Ouyang [3, 14] gave some strengthened versions and extensions. In particular, in [14], Ouyang et al presented a generalization of Chebyshev inequality for Sugeno integral given in Theorem 1.2 below. Theorem 1.2. Let f, g F + X and µ be an arbitrary fuzzy measure such that both fs dµ and gs dµ are finite. nd let : [, 2 [, be continuous and nondecreasing in both arguments and bounded from above by minimum. If f, g are comonotone, then the inequality f g s s dµ f s s dµ g s s dµ holds for all < s <. 2 4 The aim of this paper is a deep generalization of some of the above results based on an aggregation function H and a scale transformation ϕ. The paper is arranged as follows. For convenience of the reader, in the next section, we review some preliminaries and summarization of some previous known results. In Section 3, we construct further development of Chebyshev type inequalities for Sugeno integrals and relate them to T-evaluators and S-evaluators. Finally, a conclusion is given. 2. PRELIMINRIES In this section, we are going to review some well known results from the theory of fuzzy measures, Sugeno integrals and T-S-evaluators. For details, we refer to [9, 19, 25, 26] and [5]. For the convenience of the reader, we provide in this section

3 Further Development of Chebyshev Type Inequalities for Sugeno Integrals a summary of the mathematical notations and definitions used in this paper see [2, 16, 17]. s usual we denote by R the set of real numbers. Let X be a non-empty set, F be a σ-algebra of subsets of X and m be the Lebesgue measure on R. Let N denote the set of all positive integers and R + denote [, + ]. Throughout this paper, we fix the measurable space X, F, and all considered subsets are supposed to belong to F. Definition 2.1. Ralescu and dams [19] set function µ : F R + is called a fuzzy measure if the following properties are satisfied: FM1 µ = ; FM2 B implies µ µb; FM3 1 2 implies µ n=1 n = lim n µ n ; FM4 1 2, and µ 1 < + imply µ n=1 n = lim n µ n. When µ is a fuzzy measure, the triple X, F, µ then is called a fuzzy measure space. Let X, F, µ be a fuzzy measure space, by F + X we denote the set of all nonnegative measurable functions f : X [, with respect to F. In what follows, all considered functions belong to F + X. Let f be a nonnegative realvalued function defined on X, we will denote the set {x X fx α} by F α for α. Clearly, F α is nonincreasing with respect to α, i. e., α β implies F α F β. Moreover, for any fixed k in, denote by F k X the set of all measurable functions f : X [, k]. Observe that the system F k X is strictly increasing and F k X F + X. Definition 2.2. Pap [18], Sugeno [25], Wang and Klir [26] Let X, F, µ be a fuzzy measure space and F, the Sugeno integral of f on, with respect to the fuzzy measure µ, is defined as f dµ = α µ F α. α When = X, then f dµ = X f dµ = α α µf α. It is well known that Sugeno integral is a type of nonlinear integral [1]. I. e., for general case, af + bgdµ = a f dµ + b g dµ does not hold. Some basic properties of Sugeno integral are summarized in [18, 26], we cite some of them in the next theorem.

4 86 H. GHI, R. MESIR ND Y. OUYNG Theorem 2.3. Pap [18], Wang and Klir [26] Let X, F, µ be a fuzzy measure space, then i µ F α α = f dµ α; ii µ F α α = f dµ α; iii f dµ < α there exists γ < α such that µ F γ < α; iv f dµ > α there exists γ > α such that µ F γ > α; v If µ <, then µ F α α f dµ α; vi If f g, then f dµ g dµ. In [12], Ouyang and Fang proved the following result which generalized the corresponding one in [22]. Lemma 2.4. Let f : [, [, be a nonincreasing function. If a f dm = p, then fp a f dm = p for all a, where fp = lim x p fx. Moreover, if p < a and f is continuous at p, then fp = fp = p. Notice that if f is nonincreasing, then fp p implies a f dm p for any a p. In fact, the monotonicity of f and the fact fp p imply that [, p F p. Thus, m[, a] F p m[, a] [, p = m[, p = p. Now, by Theorem 2.3i, we have a f dm p. Based on Lemma 2.4, Ouyang et al. proved some Chebyshev type inequalities [13] and their united form [11] i. e., Theorem 1.1 in this paper. Notice that when proving these theorems, the following lemma, which is derived from the transformation theorem for Sugeno integrals see [26], plays a fundamental role. Lemma 2.5. Let f dµ = p. Then r p, f dµ = r µ F αdm. In this contribution, we will prove further development of Chebyshev type inequalities for Sugeno integrals and T-S-evaluators of comonotone functions. Recall that two functions f, g: X R are said to be comonotone if for all x, y X 2, fx fygx gy. Clearly, if f and g are comonotone, then for all non-negative real numbers p, q either F p G q or G q F p. Indeed, if this assertion does not hold, then there are x F p \G q and y G q \F p. That is, fx p, gx < q and fy < p, gy q, and hence fx fygx gy <, contradicts with the comonotonicity of f and g. Notice that comonotone functions can be defined on any abstract space. Now, we give the following definitions which will be used later.

5 Further Development of Chebyshev Type Inequalities for Sugeno Integrals Definition 2.6. Bodjanova and Kalina [5] For a complete lattice X,,, with the least and the greatest elements and, respectively, a function ϕ : X [, 1] is said to be an evaluator on X iff it satisfies the following properties: 1 ϕ =, ϕ = 1. 2 for all a, b X, if a b then ϕa ϕb. Definition 2.7. Klement et al. [9] binary operation T : [, 1] [, 1] [, 1] is said to be a t-norm iff it satisfies the following properties: i for each y [, 1] T1, y = y, ii for all x, y [, 1] Tx, y = Ty, x, iii for all x, y 1, y 2 [, 1] if y 1 y 2 then Tx, y 1 Tx, y 2, iv for all x, y, z [, 1] Tx, Ty, z = TTx, y, z. The four basic t-norms are: the minimum t-norm, T M x, y = min{x, y}, the product t-norm, T P x, y = x.y, the Lukasiewicz t-norm, T L x, y = max{, x + y 1}, the drastic product, { T D x, y = if max{x, y} < 1, min{x, y} if max{x, y} = 1. function S : [, 1] [, 1] [, 1] is called a t-conorm [9], if there is a t-norm T such that Sx, y = 1 T1 x, 1 y. Evidently, a t-conorm S satisfies: i Sx, = S, x = x, x [, 1] as well as conditions ii, iii and iv. The basic t-conorms dual of four basic t-norms are: the maximum t-conorm, S M x, y = max{x, y}, the probabilistic sum, S P x, y = x + y xy, the Lukasiewicz t-conorm, S L x, y = min{1, x + y}, the drastic sum, { S D x, y = 1 if min{x, y} >, max{x, y} if min{x, y} =. Definition 2.8. Bodjanova and Kalina [5] Consider a complete lattice X,,,, a t-norm T and a t-conorm S. n evaluator on X is called a T- evaluator iff for all a, b X and it is called an S-evaluator iff T ϕa,ϕb ϕa b, S ϕa,ϕb ϕa b.

6 88 H. GHI, R. MESIR ND Y. OUYNG 3. MIN RESULTS The aim of this paper is to give the following results. Theorem 3.1. Let k, be fixed. For any continuous and non-decreasing ϕ : [, k] [, k] satisfying ϕx x for all x [, k] and any non-decreasing n- place function H : [, n [, such that H is continuous and bounded from above by minimum and any comonotone system f 1, f 2,...,f n from F k X and any fuzzy measure µ it holds H ϕf 1,ϕf 2...,ϕf n dµ H ϕ f 1 dµ, ϕ f 2 dµ...,ϕ f n dµ. 5 P roof. Let f i dµ = p i, i = 1, 2,..., n. Theorem 2.3v implies that f i dµ = p i = µ {x f i x p i } p i. Then µ {x ϕf i x ϕp i } p i. Since ϕx x for all x [, k] and H : [, n [, is continuous and bounded from above by minimum there holds H ϕp 1,ϕp 2...,ϕp n min ϕp 1, ϕp 2,...,ϕp n min p 1, p 2...,p n. Therefore µ {x H ϕf 1,ϕf 2...,ϕf n H ϕp 1,ϕp 2...,ϕp n } {x ϕf µ 1 x ϕp 1 } {x ϕf 2 x ϕp 2 }... {x ϕf n x ϕp n } = min µ {x ϕf 1 x ϕp 1 },...,µ {x ϕf n x ϕp n } min p 1, p 2,..., p n H ϕp 1,ϕp 2,...,ϕp n, and, consequently, from Theorem 2.3i we obtain: H ϕf 1,...,ϕf n dµ H ϕ f 1 dµ,..., ϕ f n dµ. This completes the proof. Remark 3.2. If Hk,..., k = k, then the function H required in Theorem 3.1 is a conjunctive continuous aggregation function on [, k], compare [7]. Typical examples of such functions on [, 1] interval, i. e., if k = 1, are continuous t-norms, copulas, quasi-copulas, etc. Note also that the function ϕ required in Theorem 3.1 can be seen as a contracting transformation of the scale [, k].

7 Further Development of Chebyshev Type Inequalities for Sugeno Integrals Corollary 3.3. Let X, F, µ be a fuzzy measure space and f, g: X [, 1] two comonotone measurable functions. If T is a continuous t-norm and ϕ is continuous T-evaluator on X such that ϕx x, then the inequality holds for any F. T ϕf,ϕg dµ T ϕ f dµ, ϕ g dµ 6 The following example shows that ϕx x for all x [, k] in Theorem 3.1 is inevitable. Example 3.4. Let X [, 1 2 ], f 1x = x, f 2 x = 1 2, ϕx = x and Hx, y = x.y. Then ϕf 1 ϕf 2 dµ =. 39 2, ϕ f 1 dµ = 1 2 and but = ϕ H ϕf 1,ϕf 2 dµ < H 1 f 2 dµ = 2, ϕ f 1 dµ, ϕ f 2 dµ = , which violates Theorem 3.1. The following example shows that the comonotonicity of f 1, f 2,..., f n in Theorem 3.1 cannot be omitted. Example 3.5. Let X = [, 1], f 1 x = x, f 2 x = 1 x, Hx, y = min {x, y}, ϕx = x 2 and the fuzzy measure µ be defined as µ = m. Then f 1 dµ = f 2 dµ = [ α 1 α 2 ] = , α [,1] and Hence f1 2 f2dµ 2 = α 1 2α = α [, 1 2 ] ϕf 1 ϕf 2 dµ = < ϕ = ϕ f 1 dµ f 2 dµ, which violates Theorem 3.1.

8 9 H. GHI, R. MESIR ND Y. OUYNG The following results are easy to obtain. Corollary 3.6. Let X, F, µ be a fuzzy measure space and f, g: X [, 1] two comonotone measurable functions. Let : [, 1] 2 [, 1] be continuous and nondecreasing in both arguments and bounded from above by minimum. Then for any F. f g dµ f dµ g dµ Corollary 3.7. Let f 1, f 2,...,f n be such that any two of them are comonotone and be as in Corollary 3.3. Then ϕt... T T f 1, f 2,f 3...,f n dµ T... T ϕ T f 1 dµ, ϕ f 2 dµ,......, ϕ f n dµ. 7 Now, by using the Lemma 2.4 and 2.5 we obtain the following result. Theorem 3.8. Let k, be fixed. For any continuous and non-decreasing ϕ : [, k] [, k] satisfying ϕx x for all x [, k] and any non-decreasing n- place function H : [, n [, such that H is continuous and bounded from below by maximum and any comonotone system f 1, f 2,...,f n from F k X and any fuzzy measure µ it holds H ϕf 1,ϕf 2...,ϕf n dµ H ϕ f 1 dµ, ϕ f 2 dµ...,ϕ f n dµ. 8 P roof. Let H ϕf 1,ϕf 2...,ϕf n dµ = r v <. Theorem 2.3v implies that: µ {x H ϕf 1,ϕf 2...,ϕf n r} r. Denote 1 α = µ {x f 1 x α}, 2 α = µ {x f 2 x α},..., n α = µ {x f 2 x α} and Cα = µ {x H ϕf 1,ϕf 2...,ϕf n α. By Lemma 2.5 we have v H ϕf 1,ϕf 2...,ϕf n dµ = Cαm dα. Therefore, it is suffices to prove v ϕ Cαm dα H v 1 αm dα, ϕ...,ϕ v v n αm dα 2 αm dα,.

9 Further Development of Chebyshev Type Inequalities for Sugeno Integrals Let p 1 = v 1αm dα, p 2 = v 2αm dα,...,p n = v nαm dα. Without loss of generality, let p 1, p 2,...,p n < v. Since 1 α, 2 α,..., n α are non- increasing with respect to α, by Lemma 2.4 moreover part, we have the following equalities: 1 p 1 = p 1, 2 p 2 = p 2,..., n p n = p n. Since ϕx x for all x [, k] and H : [, n [, is continuous and bounded from below by maximum there holds H ϕp 1,ϕp 2,...,ϕp n max ϕp 1,ϕp 2,...,ϕp n max p 1, p 2...,p n. Now, on the contrary suppose Then r > H ϕp 1,ϕp 2,...,ϕp n. 9 µ {x H ϕf 1,ϕf 2,...,ϕf n r} µ {x H ϕf 1,ϕf 2,...,ϕf n > H ϕp 1,ϕp 2,...,ϕp n } {x ϕf1 x > ϕp µ 1 } {x ϕf 2 x > ϕp 2 }... {x ϕf n x > ϕp n } µ {x f 1 x > p 1 } {x f 2 x > p 2 }... {x f n x > p n }. Therefore for sufficiently small ε >, we have µ {x H ϕf 1,ϕf 2,...,ϕf n r} {x f1 x p µ 1 ε} {x f 2 x p 2 ε}... {x f n x p n ε} Letting ε, then we have r lim µ {x H ϕf 1,ϕf 2,...,ϕf n r} ε lim max 1 p 1 ε, 2 p 2 ε,..., n p n ε ε = maxp 1, p 2..., p n H ϕp 1,ϕp 2...,ϕp n,. which is a contradiction to 9. Hence r H ϕp 1,ϕp 2,...,ϕp n and the proof is completed. Observe that if H,.., for a function H required in Theorem 3.8 holds, then H is a disjunctive continuous aggregation function on [, k], see [7]. Typical examples in the case k = 1 of such aggregation functions are continuous t-conorms, cocopulas, etc. Corollary 3.9. Let X, F, µ be a fuzzy measure space and f, g: X [, 1] two comonotone measurable functions. If S is a continuous t-conorm and ϕ is continuous S -evaluator on X such that ϕx x, then the inequality holds for any F. S ϕf,ϕg dµ S ϕ f dµ, ϕ g dµ 1

10 92 H. GHI, R. MESIR ND Y. OUYNG The following example shows that ϕx x for all x [, k] in Theorem 3.8 is inevitable. Example 3.1. Let X = [, 1], f 1 x = f 2 x = x, ϕx = x 2 and Hx, y = min{1, x + y} and the fuzzy measure µ be defined as µ = m 2. Then 1 H ϕf 1,ϕf 2 dµ = α [,1] 2α α 1 = = but ϕ 1 f 1 dµ = ϕ f 2 dµ = = Then 1 H ϕf 1,ϕf 2 dµ = > H ϕ = , 1 f 1 dµ, ϕ 1 f 2 dµ which violates Theorem 3.8. The following example shows that the comonotonicity of f 1, f 2,..., f n in Theorem 3.8 cannot be omitted. Example Let X = [, 1], f 1 x = x, f 2 x = 1 x, Hx, y = min{1, x + y}, ϕx = x and the fuzzy measure µ be defined as µ = m 2. Then f 1 dµ = f 2 dµ = α [,1] [ α 1 α 2] = and H x, 1 x dµ = 1. Hence H ϕf 1,ϕf 2 dµ = 1 > H ϕ f 1 dµ, ϕ f 2 dµ = , which violates Theorem 3.8. The following results are easy to obtain.

11 Further Development of Chebyshev Type Inequalities for Sugeno Integrals Corollary Let X, F, µ be a fuzzy measure space and f, g: X [, 1] two comonotone measurable functions. Let : [, 1] 2 [, 1] be continuous and nondecreasing in both arguments and bounded from below by maximum. Then for any F. f g dµ f dµ g dµ Corollary Let f 1, f 2,..., f n be such that any two of them are comonotone and be as in Corollary 3.9. Then ϕs...s S f 1, f 2,f 3...,f n dµ S... S ϕ S f 1 dµ, ϕ f 2 dµ,......, ϕ f n dµ. 4. CONCLUSION In this paper, we have investigated further development of Chebyshev type inequalities for Sugeno integrals and we have related them to T-evaluators and S-evaluators. s an interesting open problem for further investigation we pose the generalization of equality 3 for n-ary case. To be more precise, it is worth studying the case when the inequalities 5 and/or 8 became equalities, independently of incoming functions f 1,..., f n. CKNOWLEDGEMENT The work on this paper was supported by grants PVV-12-7, G ČR 42/8/618, and bilateral project SK-CN-19-7 between Slovakia and China and the first author was partially supported by the Fuzzy Systems and pplications Center of Excellence, Shahid Bahonar University of Kerman, Kerman, Iran. Our thanks go to anonymous referees who helped to improve the original version of our paper. 11 Received June 26, 28 R EFERENCES [1] H. gahi and M.. Yaghoobi: Minkowski type inequality for fuzzy integrals. J. Uncertain Systems, in press. [2] H. gahi, R. Mesiar, and Y. Ouyang: General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets and Systems , [3] H. gahi, R. Mesiar, and Y. Ouyang: New general extensions of Chebyshev type inequalities for Sugeno integrals, International Journal of pproximate Reasoning 51 29, [4] P. Benvenuti, R. Mesiar, and D. Vivona: Monotone set functions-based integrals. In: Handbook of Measure Theory E. Pap, ed., Vol II, Elsevier, 22, pp

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13 Further Development of Chebyshev Type Inequalities for Sugeno Integrals Hamzeh gahi, Department of Statistics, Faculty of Mathematics and Computer Science, Shahid Bahonar University of Kerman, Kerman, Iran Radko Mesiar, Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, SK Bratislava, Slovakia and Institute of Information Theory and utomation, Czech cademy of Sciences, CZ Prague, Czech Republic mesiar@math.sk Yao Ouyang, Faculty of Science, Huzhou Teacher s College, Huzhou, Zhejiang 313, People s Republic of China oyy@hutc.zj.cn