CUBO A Mathematical Journal Vol.21, No¯ 03, (29–38). December 2019 http: // dx. doi. org/ 10. 4067/ S0719-06462019000300029 Ostrowski-Sugeno fuzzy inequalities George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. ganastss@memphis.edu ABSTRACT We present Ostrowski-Sugeno fuzzy type inequalities. These are Ostrowski-like inequal- ities in the context of Sugeno fuzzy integral and its special properties are investigated. Tight upper bounds to the deviation of a function from its Sugeno-fuzzy averages are given. This work is greatly inspired by [3] and [1]. RESUMEN Presentamos desigualdades de Ostrowski-Sugeno de tipo fuzzy. Estas son desigualdades de tipo Ostrowski en el contexto de integrales fuzzy de Sugeno y se investigan sus propiedades especiales. Se entregan cotas superiores ajustadas para la desviación de una función de sus promedios fuzzy de Sugeno. Este trabajo está inspirado principalmente por [3] y [1]. Keywords and Phrases: Sugeno fuzzy, integral, function fuzzy average, deviation from fuzzy mean, fuzzy Ostrowski inequality. 2010 AMS Mathematics Subject Classification: Primary: 26D07, 26D10, 26D15, 41A44, Secondary: 26A24, 26D20, 28A25. http://dx.doi.org/10.4067/S0719-06462019000300029 30 George A. Anastassiou CUBO 21, 3 (2019) 1 Introduction The famous Ostrowski ([3]) inequality motivates this work and has as follows: ∣ ∣ ∣ ∣ ∣ 1 b − a ∫b a f (y) dy − f (x) ∣ ∣ ∣ ∣ ∣ ≤ ( 1 4 + ( x − a+b 2 )2 (b − a) 2 ) (b − a) ‖f′‖ ∞ , where f ∈ C′ ([a, b]), x ∈ [a, b], and it is a sharp inequality. One can easily notice that ( 1 4 + ( x − a+b 2 )2 (b − a) 2 ) (b − a) = (x − a) 2 + (b − x) 2 2 (b − a) . Another motivation is author’s article [1]. First we give a survey about Sugeno fuzzy integral and its basic properties. Then we derive a series of Ostrowski-like inequalities to all directions in the context of Sugeno integral and its basic important particular properties. We also give applications to special cases of our problem we deal with. 2 Background In this section, some definitions and basic important properties of the Sugeno integral which will be used in the next section are presented. Definition 2.1. (Fuzzy measure [5, 7]) Let Σ be a σ-algebra of subsets of X, and let µ : Σ → [0, +∞] be a non-negative extended real-valued set function. We say that µ is a fuzzy measure iff: (1) µ (∅) = 0, (2) E, F ∈ Σ : E ⊆ F imply µ (E) ≤ µ (F) (monotonicity), (3) En ∈ Σ (n ∈ N), E1 ⊂ E2 ⊂ ..., imply lim n→∞ µ (En) = µ (∪ ∞ n=1En) (continuity from below); (4) En ∈ Σ (n ∈ N), E1 ⊃ E2 ⊃ ..., µ (E1) < ∞, imply lim n→∞ µ (En) = µ (∩ ∞ n=1En) (continuity from above). Let (X, Σ, µ) be a fuzzy measure space and f be a non-negative real-valued function on X. We denote by F+ the set of all non-negative real valued measurable functions, and by Lαf the set: Lαf := {x ∈ X : f (x) ≥ α}, the α-level of f for α ≥ 0. Definition 2.2. Let (X, Σ, µ) be a fuzzy measure space. If f ∈ F+ and A ∈ Σ, then the Sugeno integral (fuzzy integral) [6] of f on A with respect to the fuzzy measure µ is defined by (S) ∫ A fdµ := ∨α≥0 (α ∧ µ (A ∩ Lαf)) , (1) where ∨ and ∧ denote the sup and inf on [0, ∞], respectively. CUBO 21, 3 (2019) Ostrowski-Sugeno fuzzy inequalities 31 The basic properties of Sugeno integral follow: Theorem 2.3. ([4, 7]) Let (X, Σ, µ) be a fuzzy measure space with A, B ∈ Σ and f, g ∈ F+. Then 1) (S) ∫ A fdµ ≤ µ (A) ; 2) (S) ∫ A kdµ = k ∧ µ (A) for a non-negative constant k; 3) if f ≤ g on A, then (S) ∫ A fdµ ≤ (S) ∫ A gdµ; 4) if A ⊂ B, then (S) ∫ A fdµ ≤ (S) ∫ B fdµ; 5) µ (A ∩ Lαf) ≤ α ⇒ (S) ∫ A fdµ ≤ α; 6) if µ (A) < ∞, then µ (A ∩ Lαf) ≥ α ⇔ (S) ∫ A fdµ ≥ α; 7) when A = X, (S) ∫ A fdµ = ∨α≥0 (α ∧ µ (Lαf)) ; 8) if α ≤ β, then Lβf ⊆ Lαf; 9) (S) ∫ A fdµ ≥ 0. Theorem 2.4. ([7, p. 135]) Let f ∈ F+, the class of all finite nonnegative measurable functions on (X, Σ, µ). Then 1) if µ (A) = 0, then (S) ∫ A fdµ = 0, for any f ∈ F+; 2) if (S) ∫ A fdµ = 0, then µ (A ∩ {x|f (x) > 0}) = 0; 3) (S) ∫ A fdµ = (S) ∫ A f · χAdµ, where χA is the characteristic function of A; 4) (S) ∫ A (f + a) dµ ≤ (S) ∫ A fdµ + (S) ∫ A adµ, for any constant a ∈ [0, ∞). Corollary 2.5. ([7, p. 136]) Let f, f1, f2 ∈ F+. Then 1) (S) ∫ A (f1 ∨ f2) dµ ≥ (S) ∫ A f1dµ ∨ (S) ∫ A f2dµ; 2) (S) ∫ A (f1 ∧ f2) dµ ≤ (S) ∫ A f1dµ ∧ (S) ∫ A f2dµ; 3) (S) ∫ A∪B fdµ ≥ (S) ∫ A fdµ ∨ (S) ∫ B fdµ; 4) (S) ∫ A∩B fdµ ≤ (S) ∫ A fdµ ∧ (S) ∫ B fdµ. In general we have (S) ∫ A (f1 + f2) dµ 6= (S) ∫ A f1dµ + (S) ∫ A f2dµ, and (S) ∫ A afdµ 6= a (S) ∫ A fdµ, where a ∈ R, see [7, p. 137]. Lemma 2.6. ([7, p. 138]) (S) ∫ A fdµ = ∞ if and only if µ (A ∩ Lαf) = ∞ for any α ∈ [0, ∞). We need 32 George A. Anastassiou CUBO 21, 3 (2019) Definition 2.7. ([2]) A fuzzy measure µ is subadditive iff µ (A ∪ B) ≤ µ (A) + µ (B), for all A, B ∈ Σ. We mention the following result Theorem 2.8. ([2]) If µ is subadditive, then (S) ∫ X (f + g) dµ ≤ (S) ∫ X fdµ + (S) ∫ X gdµ, (2) for all measurable functions f, g : X → [0, ∞). Moreover, if (2) holds for all measurable functions f, g : X → [0, ∞) and µ (X) < ∞, then µ is subadditive. Notice here in (1) we have that α ∈ [0, ∞). We have the following corollary. Corollary 2.9. If µ is aubadditive, n ∈ N, and f : X → [0, ∞) is a measurable function, then (S) ∫ X nfdµ ≤ n (S) ∫ X fdµ, (3) in particular it holds (S) ∫ A nfdµ ≤ n (S) ∫ A fdµ, (4) for any A ∈ Σ. Proof. By inequality (2). A very important property of Sugeno integral follows. Theorem 2.10. If µ is subadditive measure, and f : X → [0, ∞) is a measurable function, and c > 0, then (S) ∫ A cfdµ ≤ (c + 1) (S) ∫ A fdµ, (5) for any A ∈ Σ. Proof. Let the ceiling ⌈c⌉ = m ∈ N, then by Theorem 2.3 (3) and (4) we get (S) ∫ A cfdµ ≤ (S) ∫ A mfdµ ≤ m (S) ∫ A fdµ ≤ (c + 1) (S) ∫ A fdµ, proving (5). CUBO 21, 3 (2019) Ostrowski-Sugeno fuzzy inequalities 33 3 Main Results From now on in this article we work on the fuzzy measure space ([a, b] , B, µ), where [a, b] ⊂ R, B is the Borel σ-algebra on [a, b], and µ is a finite fuzzy measure on B. Typically we take it to be subadditive. The functions f we deal with here are continuous from [a, b] into R+. We make the following remark Remark 3.1. Let f ∈ C1 ([a, b] , R+), and µ is a subadditive fuzzy measure such that µ ([a, b]) > 0, x ∈ [a, b]. We will estimate E := ∣ ∣ ∣ ∣ ∣ (S) ∫ [a,b] f (x) dµ (t) − µ ([a, b]) ∧ f (x) ∣ ∣ ∣ ∣ ∣ (6) (by Theorem 2.3 (2)) = ∣ ∣ ∣ ∣ ∣ (S) ∫ [a,b] f (t) dµ (t) − (S) ∫ [a,b] f (x) dµ (t) ∣ ∣ ∣ ∣ ∣ . We notice that f (t) = f (t) − f (x) + f (x) ≤ |f (t) − f (x)| + f (x) , then (by Theorem 2.3 (3) and Theorem 2.4 (4)) (S) ∫ [a,b] f (t) dµ (t) ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) + (S) ∫ [a,b] f (x) dµ (t) , (7) that is (S) ∫ [a,b] f (t) dµ (t) − (S) ∫ [a,b] f (x) dµ (t) ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) . (8) Similarly, we have f (x) = f (x) − f (t) + f (t) ≤ |f (t) − f (x)| + f (t) , then (by Theorem 2.3 (3) and Theorem 2.8) (S) ∫ [a,b] f (x) dµ (t) ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) + (S) ∫ [a,b] f (t) dµ (t) , that is (S) ∫ [a,b] f (x) dµ (t) − (S) ∫ [a,b] f (t) dµ (t) ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) . (9) By (8) and (9) we derive that ∣ ∣ ∣ ∣ ∣ (S) ∫ [a,b] f (t) dµ (t) − (S) ∫ [a,b] f (x) dµ (t) ∣ ∣ ∣ ∣ ∣ ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) . (10) 34 George A. Anastassiou CUBO 21, 3 (2019) Consequently it holds E (by (6), (10)) ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) (and by |f (t) − f (x)| ≤ ‖f′‖ ∞ |t − x|) ≤ (S) ∫ [a,b] ‖f′‖ ∞ |t − x| dµ (t) (by (5)) ≤ (‖f′‖ ∞ + 1) (S) ∫ [a,b] |t − x| dµ (t) . (11) We have proved the following Ostrowski-like inequality ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − µ ([a, b] ∧ f (x)) µ ([a, b]) ∣ ∣ ∣ ∣ ∣ ≤ (12) (‖f′‖ ∞ + 1) µ ([a, b]) (S) ∫ [a,b] |t − x| dµ (t) . The last inequality can be better written as follows: ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (‖f′‖ ∞ + 1) µ ([a, b]) (S) ∫ [a,b] |t − x| dµ (t) . (13) Notice here that ( 1 ∧ f(x) µ([a,b]) ) ≤ 1, and 1 µ([a,b]) (S) ∫ [a,b] f (t) dµ (t) ≤ µ([a,b]) µ([a,b]) = 1, where (S) ∫ [a,b] f (t) dµ (t) ≥ 0. I.e. If f : [a, b] → R+ is a Lipschitz function of order 0 < α ≤ 1, i.e. |f (x) − f (y)| ≤ K |x − y| α , ∀ x, y ∈ [a, b], where K > 0, denoted by f ∈ Lipα,K ([a, b] , R+), then we get similarly the following Ostrowski-like inequality: ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (K + 1) µ ([a, b]) (S) ∫ [a,b] |t − x| α dµ (t) . (14) We have proved the following Ostrowski-Sugeno inequalities: Theorem 3.2. Suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b] . 1) Let f ∈ C1 ([a, b] , R+), then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (‖f′‖ ∞ + 1) µ ([a, b]) (S) ∫ [a,b] |t − x| dµ (t) . (15) CUBO 21, 3 (2019) Ostrowski-Sugeno fuzzy inequalities 35 2) Let f ∈ Lipα,K ([a, b] , R+), 0 < α ≤ 1, then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (K + 1) µ ([a, b]) (S) ∫ [a,b] |t − x| α dµ (t) . (16) We make the following remark Remark 3.3. Let f ∈ C1 ([a, b] , R+) and g ∈ C 1 ([a, b]), by Cauchy’s mean value theorem we get that (f (t) − f (x)) g′ (c) = (g (t) − g (x)) f′ (c) , for some c between t and x; for any t, x ∈ [a, b]. If g′ (c) 6= 0, we have (f (t) − f (x)) = ( f′ (c) g′ (c) ) (g (t) − g (x)) . Here we assume that g′ (t) 6= 0, ∀ t ∈ [a, b]. Hence it holds |f (t) − f (x)| ≤ ∥ ∥ ∥ ∥ f′ g′ ∥ ∥ ∥ ∥ ∞ |g (t) − g (x)| , (17) for all t, x ∈ [a, b] . We have again as before (see (11)) E ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) (by (17)) ≤ (S) ∫ [a,b] ∥ ∥ ∥ ∥ f′ g′ ∥ ∥ ∥ ∥ ∞ |g (t) − g (x)| dµ (t) (by (5)) ≤ ( ∥ ∥ ∥ ∥ f′ g′ ∥ ∥ ∥ ∥ ∞ + 1 ) (S) ∫ [a,b] |g (t) − g (x)| dµ (t) . (18) We have established the following general Ostrowski-Sugeno inequality: Theorem 3.4. Suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b]. Let f ∈ C1 ([a, b] , R+) and g ∈ C 1 ([a, b]) with g′ (t) 6= 0, ∀ t ∈ [a, b] . Then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ∥ ∥ ∥ f ′ g′ ∥ ∥ ∥ ∞ + 1 ) µ ([a, b]) (S) ∫ [a,b] |g (t) − g (x)| dµ (t) . (19) 36 George A. Anastassiou CUBO 21, 3 (2019) We give for g (t) = et the next result Corollary 3.5. Suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b]. Let f ∈ C1 ([a, b] , R+), then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ∥ ∥ ∥ f ′ et ∥ ∥ ∥ ∞ + 1 ) µ ([a, b]) (S) ∫ [a,b] ∣ ∣et − ex ∣ ∣dµ (t) . (20) When g (t) = ln t we get the following corollary. Corollary 3.6. Suppose that µ is a fuzzy subadditive measure with µ ([a, b]) > 0, x ∈ [a, b] and a > 0. Let f ∈ C1 ([a, b] , R+) . Then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ (‖tf′ (t)‖ ∞ + 1) µ ([a, b]) (S) ∫ [a,b] ∣ ∣ ∣ ∣ ln t x ∣ ∣ ∣ ∣ dµ (t) . (21) Many other applications of Theorem 3.4 could follow but we stop it here. We make the following remark. Remark 3.7. Let f ∈ [ C ([a, b] , R+) ∩ C n+1 ([a, b]) ] , n ∈ N, x ∈ [a, b]. Then by Taylor’s theorem we get f (y) − f (x) = n∑ k=1 f(k) (x) k! (y − x) k + Rn (x, y) , (22) where the remainder Rn (x, y) := ∫y x ( f(n) (t) − f(n) (x) ) (y − t) n−1 (n − 1) ! dt; (23) here y can be ≥ x or ≤ x. By [1] we get that |Rn (x, y)| ≤ ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! |y − x| n+1 , for all x, y ∈ [a, b] . (24) Here we assume f(k) (x) = 0, for all k = 1, ..., n. Therefore it holds |f (t) − f (x)| ≤ ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! |t − x| n+1 , for all t, x ∈ [a, b] . (25) CUBO 21, 3 (2019) Ostrowski-Sugeno fuzzy inequalities 37 Here we have again E ≤ (S) ∫ [a,b] |f (t) − f (x)| dµ (t) (by Theorem 2.3 (3) and (25)) ≤ (S) ∫ [a,b] ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! |t − x| n+1 dµ (t) (by (5)) ≤ ( ∥ ∥f(n+1) ∥ ∥ ∞ (n + 1) ! + 1 ) (S) ∫ [a,b] |t − x| n+1 dµ (t) . (26) We have derived the following high order Ostrowski-Sugeno inequality: Theorem 3.8. Let f ∈ [ C ([a, b] , R+) ∩ C n+1 ([a, b]) ] , n ∈ N, x ∈ [a, b]. We assume that f(k) (x) = 0, all k = 1, ..., n. Here µ is subadditive with µ ([a, b]) > 0. Then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f (x) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ‖f(n+1)‖ ∞ (n+1)! + 1 ) µ ([a, b]) (S) ∫ [a,b] |t − x| n+1 dµ (t) , (27) which generalizes (15). When x = a+b 2 we get the following corollary Corollary 3.9. Let f ∈ [ C ([a, b] , R+) ∩ C n+1 ([a, b]) ] , n ∈ N. Assume that f(k) ( a+b 2 ) = 0, k = 1, ..., n. Here µ is subadditive with µ ([a, b]) > 0. Then ∣ ∣ ∣ ∣ ∣ 1 µ ([a, b]) (S) ∫ [a,b] f (t) dµ (t) − ( 1 ∧ f ( a+b 2 ) µ ([a, b]) ) ∣ ∣ ∣ ∣ ∣ ≤ ( ‖f(n+1)‖ ∞ (n+1)! + 1 ) µ ([a, b]) (S) ∫ [a,b] ∣ ∣ ∣ ∣ t − a + b 2 ∣ ∣ ∣ ∣ n+1 dµ (t) . (28) 38 George A. Anastassiou CUBO 21, 3 (2019) References [1] G.A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc. 123(1995), 3775-3781. [2] M. Boczek, M. Kaluszka, On the Minkowaki-Hölder type inequalities for generalized Sugeno integrals with an application, Kybernetica, 52(3) (2016), 329-347. [3] A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Inte- gralmittelwert, Comment. Math. Helv., 10 (1938), 226-227. [4] E. Pap, Null-Additive Set functions, Kluwer Academic, Dordrecht, 1995. [5] D. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75 (1980), 562-570. [6] M. Sugeno, Theory of fuzzy integrals and its applications, PhD thesis, Tokyo Institute of Tech- nology (1974). [7] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum, New York, 1992. Introduction Background Main Results