In this paper, related inequalities to Jensen type inequality for the seminormed fuzzy integrals are studied. Several examples are given to illustrate the validity of theorems. Some results on Jensen type inequalities are obtained.Publisher's Versio

Daraby, Bayaz

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.8, N.1, 2018, pp. 1-7GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZYINTEGRALSB. DARABY1, §Abstract. In this paper, related inequalities to Jensen type inequality for the semi-normed fuzzy integrals are studied. Several examples are given to illustrate the validityof theorems. Some results on Jensen type inequalities are obtained.Keywords: Fuzzy measure, Jensen’s inequality, Seminormed fuzzy integral, Semiconormedfuzzy integral.AMS Subject Classification: 03E72, 26E50, 28E101. IntroductionThe properties and applications of the Sugeno integral have been studied by manyauthors, including Ralescu and Adams [11], Román-Flores et al. [12, 13] and Wang andKlir [17].In connection with the topics that will be discussed in this article, there are severalother theoretical and applied papers related to fuzzy measure theory on metric spaces,such as Li et al. [8, 9] and Narukawa et al. [10]. Many authors generalized the Sugenointegral by using some other operators to replace the special operator(s) ∧ and/or ∨ (see,e.g., [1, 2, 3, 4, 6, 8, 18]). Suárez and Gil Álvarez presented two families of fuzzy integrals,the so-called seminormed fuzzy integrals and semiconormed fuzzy integrals [15].This paper is organized as follows: in Section 2, some preliminaries and summarizationof some previous known results are briefly recalled. Section 3 deals with a general relatedJensen type inequality for the fuzzy integrals.2. PreliminariesIn this section we recall some basic definitions and previous results that will be used inthe next section.Let X be a non-empty set, Σ be a σ-algebra of subsets of X. Throughout this paper, allconsidered subsets are supposed to belong to Σ.Definition 2.1 (Sugeno [16]). A set function µ : Σ → [0, 1] is called a fuzzy measure ifthe following properties are satisfied:(FM1) µ(∅) = 0 and µ(X) = 1;(FM2) A ⊂ B implies µ(A) ≤ µ(B);(FM3) An → A implies µ(An)→ µ(A).1 Department of Mathematics, University of Maragheh, P. O. Box 55181-83111, Maragheh, Iran.e-mail: bdaraby@maragheh.ac.ir; ORCID: https://orcid.org/0000-0001-6872-8661.§ Manuscript received: September 1, 2016; accepted: February 15, 2017.TWMS Journal of Applied and Engineering Mathematics, Vol.8, No.1 c© Işık University, Departmentof Mathematics, 2018; all rights reserved.12 TWMS J. APP. ENG. MATH. V.8, N.1, 2018When µ is a fuzzy measure, the triple (X,Σ, µ) is called a fuzzy measure space.Let (X,Σ, µ) be a fuzzy measure space, and F+(X) = {f |f : X → [0, 1] is measurablewith respect to Σ}. In what follows, all considered functions belong to F+(X). For anyα ∈ [0, 1], we will denote the set {x ∈ X|f(x) ≥ α} by Fα and {x ∈ X|f(x) > α} by Fᾱ.Clearly, both Fα and Fᾱ are non-increasing with respect to α, i.e., α ≤ β implies Fα k Fβand Fᾱ k Fβ̄.Definition 2.2 (Sugeno [16]). Let (X,Σ, µ) be a fuzzy measure space, and A ∈ Σ, theSugeno integral of f over A, with respect to the fuzzy measure µ, is defined by−∫Afdµ =∨α∈[0,1](α ∧ µ(A ∩ Fα)).When A=X, then−∫Xfdµ = −∫fdµ =∨α∈[0,1](α ∧ µ(Fα)).Note in the above definition, ∧ is just the prototypical t-norm minimum and ∨ theprototypical t-conorm maximum.Definition 2.3 ([7]). A t-norm is a function T : [0,1]×[0,1] → [0,1] satisfying the fol-lowing conditions:(A) T (x, 1) = T (1, x) = x ∀x ∈ [0, 1].(B) ∀x1, x2, y1, y2 in [0, 1], if x1 ≤ x2, y1 ≤ y2 then T (x1, y1) ≤ T (x2,y2).(C) T (x, y) = T (y, x).(D) T (T (x, y), z) = T (x, T (y, z)).Definition 2.4 ([7]). A function S : [0, 1] × [0, 1] → [0, 1] is called a t-conorm, if thereis a t-norm T such that S(x, y) = 1 − T (1 − x, 1 − y). Evidently, a t-conorm S satisfies:(A′) S(x, 0) = S(0, x) = x, ∀x ∈ [0, 1] as well as conditions (B), (C) and (D).A binary operator T (S) on [0,1] is called a t-seminorm (t-semiconorm), if it satisfiesthe above conditions (A) and (B) ((A′) and (B))[15]. Notice that in the literature, at-seminorm is also called a semicopula. By using the concepts of t-seminorm and t-semiconorm, Suárez and Gil (1986) proposed two families of fuzzy integrals:Definition 2.5. Let T be a t-seminorm, then the seminormed fuzzy integral of f over Awith respect to T and the fuzzy measure µ is defined by∫T, Afdµ =∨α∈[0,1]T (α, µ(A ∩ Fα)). (1)Definition 2.6. Let S be a t-semiconorm, then the semiconormed fuzzy integral of f overA with respect to S and the fuzzy measure µ is defined by∫S, Afdµ =∧α∈[0,1]S(α, µ(A ∩ Fᾱ)).Remark 2.1. Maybe one can define the so-called conormed-seminormed fuzzy integral∫TS, A fdµ by using an arbitrary t-conorm S to replace the special t- conorm“∨” in (1).However, as Suárez and Gil Álvarez [15] pointed out that the conormed-seminormed fuzzyintegral satisfies the essential property∫TS, Xadµ = a, ∀a ∈ [0, 1]B. DARABY,: GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS 3if and only if S =∨. The case of semiconormed fuzzy integral is similar. It is easy tosee that the Sugeno integral is a special seminormed fuzzy integral. Moreover, Kandel andByatt in (see [5]) showed another expression of the Sugeno integral as follows:−∫Afdµ =∧α∈[0,1](α ∨ µ(A ∩ Fᾱ)).So the semiconormed fuzzy integrals also generalize the concept of the Sugeno integral.Notice that the seminormed fuzzy integral is just the family of the weakest universal fuzzyintegrals. Note that if∫T,A fdµ = a, then T (α, µ(A ∩ Fα)) ≤ a for all α ∈ [0, 1] and, forany ε > o there exists αε such that T (αε, µ(A ∩ Fαε)) ≥ a − ε. Also, if∫S,A fdµ = a,then S(α, µ(A ∩ Fᾱ)) ≥ a for all α ∈ [0, 1] and, for any ε > o there exists αε such thatS(αε, µ(A ∩ Fᾱε)) ≤ a + ε. The well-known Jensen inequality is a part of the classicalmathematical analysis, (see [14]).Theorem 2.1. Let h : [0, 1] → (a, b) be real and integrable function and Φ a convexfunction on (a,b). ThenΦ(∫ 10h(x)dx)≤∫ 10Φ(h(x))dx. (2)In [3], Daraby et al. have proved general version of the Jensen type inequality as follows:Theorem 2.2. Let (X,Σ, µ) be a fuzzy measure space and f : X → [0, 1] be a measurablefunction. Let T be a seminorm such that∫T,A fdµ = a for any A ∈ Σ. Let Φ : [0, 1]→ [0, 1]be a continuous and strictly increasing function such that Φ(x) ≤ x, for every x ∈ [0, a],then:Φ(∫T,Afdµ)≤∫T,AΦ(f)dµ. (3)3. Main results3.1. Some results on Jensen type seminormed fuzzy integrals. In this section, wewill show some results on general Jensen type inequality. At the first, we show that con-ditions “Φ strictly increasing” and “Φ(x) ≤ x, for every x ∈ [0,∫T,A fdµ]” on the functionΦ in Theorem 2.2 cannot be avoided, as we will illustrate in the following examples.Example 3.1. We observe that “Φ strictly increasing” condition of function Φ in Theorem2.2 cannot be avoided. Let µ be the Lebesgue measure on R and T be defined asTD(x, y) ={0 if max{x, y} < 1,min{x, y} if max{x, y} = 1.and consider f(x) = xχ[0,1](x). If we define:Φ(x) ={x if 0 ≤ x < 12(1− x)χ[0,1](x) if x ≥ 12 .Thus:If max{x, y} < 1, then TD(x, y) = 0. So we have∫T,A fdm = 0.Consequently,Φ(∫T,Afdm)= 0.HenceΦ(∫T,Afdm)= 0 ≤∫T,AΦ(f)dm.4 TWMS J. APP. ENG. MATH. V.8, N.1, 2018If max{x, y} = 1, then ∫T,Afdm = −∫fdm=∨α∈[0,1](α ∧m(Fα))=∨α∈[0,1](α ∧ (1− α)) = 12.Consequently,Φ(∫T,Afdm)= Φ(12)=12. (4)In addition, because Φ(f)(x) = Φ(x)χ[0,1](x), then:m({Φ(f) > 12})= m({12})= 0. (5)But, from (4) and (5) we have:∫T,AΦ(f)dm ≥ 12⇒ m({Φ(f) ≥ 12})≥ 12in contradiction with (5), which implies that∫T,AΦ(f)dm <12.Consequently, inequality (3) is not verified.Example 3.2. The condition “Φ(x) ≤ x for all 0 ≤ x ≤∫T,A fdµ” in Theorem (2.9)cannot be avoided. In fact, consider f(x) = xχ[0, 12](x) and define Φ(x) =√x. Also T bedefined asTD(x, y) ={0 if max{x, y} < 1,min{x, y} if max{x, y} = 1.Thus: If max{x, y} < 1, then TD(x, y) = 0. So we have∫T,A fdm = 0. Consequently,Φ(∫T,Afdm)= 0.HenceΦ(∫T,Afdm)= 0 ≤∫T,AΦ(f)dm.If max{x, y} = 1, then ∫T,Afdµ = −∫fdµ=∨α∈[0, 12][α ∧ µ(Fα))=∨α∈[0, 12][α ∧ (12− α)] = 14.B. DARABY,: GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS 5Consequently,Φ(∫T,Afdµ)= Φ(14)=12.Furthermore, because Φ(f)(x) =√xχ[0, 12](x), then:∫T,AΦ(f)dµ = −∫Φ(f)dµ=∨α∈[0,√12][α ∧ µ(Fα)]=∨α∈[0,√12][α ∧ (12− α2)]=−1 +√32<12= Φ(∫T,Afdµ).Which implies that inequality (3) is not verified.Theorem 3.1. Let (X,Σ) be a fuzzy measurable space and f : X → [0, 1] be a measurablefunction. Let T be a seminorm such that F (x) =∫T,[0,x] f(t)dt. Let Φ : [0, 1] → [0, 1] bea continuous and strictly increasing function such that Φ(x) ≤ x, for every x ∈ [0, F (x)],then: ∨α∈[0,1]T(α,(∫T,A Φ(f)dxα))≥∫T,AΦ (F (x))xdx (6)holds for any A ∈ Σ.Proof. By Theorem 2.2 we haveK =∫T,AΦ(f)dx ≥ Φ(∫T,Afdµ).Since [0, x] ⊆ A, then we haveΦ(∫T,Afdµ)≥ Φ(∫T,[0,x]fdµ)thenK =∫T,AΦ(f)dx ≥ Φ(∫T,[0,x]fdµ)= Φ (F (x))thus ∨α∈[0,1]T(α,(∫T,A Φ(f)dxα))=∫T,AKxdx ≥∫T,AΦ ((F (x))xdx.This completes the proof. 6 TWMS J. APP. ENG. MATH. V.8, N.1, 20183.2. A related inequality. In this section, we prove a related inequality for semiconormedfuzzy integrals. The main result of this section is as follows.Theorem 3.2. Let (X,Σ, µ) be a fuzzy measure space and f : X → [0, 1] measurablefunction. Let S be a semiconorm such that∫S,A fdµ = a for any A ∈ Σ. Let Φ : [0, 1] →[0, 1] be a continuous and strictly increasing function such that Φ(x) ≤ x, for every x ∈[0, a], then:Φ(∫S, Afdµ)≥∫S, AΦ(f)dµ. (7)Proof. Let∫S,A fdµ = a. Then, for any ε > 0, there exist aε such that µ(A ∩ Fāε) = a0,where S(aε, a0) ≤ a+ ε, (Thus aε ≤ a+ ε).Denote ϕᾱ = {x|Φ(f(x)) > α}. Since Φ : [0, 1] → [0, 1] is a continuous and strictlyincreasing function, there holdsϕΦ(āε) ⊂ {x|f(x) > āε}.Thenµ(A ∩ ϕΦ(āε))≤ µ(A ∩ {x|f(x) > āε}) = a0.Hence ∫S, AΦ(f)dµ =∧a∈[0,1]S(a, µ(A ∩ ϕā))≤ S(Φ(aε), µ(A ∩ ϕΦ(āε)))≤ S (Φ(aε), a0)≤ (Φ(S(aε, a0)))≤ Φ(a+ ε).Thus ∫S,AΦ(f)dµ ≤ Φ(a), (8)follows from the arbitrariness of ε. Consequently from 8 we conclude:∫S, AΦ(f)dµ =∧a∈[0,1]S(a, µ(A ∩ ϕā))≤ Φ(a) = Φ(∫S, Afdµ).This completes the proof. As a direct consequence of Theorem (3.2.1) we haveCorollary 3.1. Let (X,Σ, µ) be a fuzzy measure space and f : X → [0, 1] be a measurablefunction. If S be a semiconorm. Let Φ : [0, 1] → [0, 1] be a continuous and strictlyincreasing function such that Φ(x) ≤ x, for every x ∈ [0, µ(x)], then:Φ(∫S, Afdµ)≥∫S, AΦ(f)dµ.Acknowledgement I am very grateful to the referee(s) for the careful reading of thepaper and for the useful comments.B. DARABY,: GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS 7References[1] Daraby, B., Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets and Sys-tems 194 (2012) 90-96.[2] Daraby, B., Shafiloo, A. and Rahimi, A., Generalization of the Feng Qi type inequality for pesudo-integral, Gazi University Journal of Science 28 (4) (2015), 695-702.[3] Daraby, B. and Rahimi, A., Jensen type inequality for seminormed fuzzy integrals, Acta UniversitatisApulensis 46 (2016) 1-8.[4] Daraby, B. and Ghazanfary Asll, H. and Sadeqi, I., General related inequalities to Carlson-typeinequlity for the Sugeno integral, Applied Mathematics and Computation 305 (2017) 323-329.[5] Kandel, A., Byatt, W.J., Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proceedings of the IEEE 66(1978) 1619-1639.[6] Klement, E.P., Mesiar, R. and Pap, E., Integration with respect to decomposable measures, basedon a conditionally distributive semiring on the unit interval, International Journal of UncertaintyFuzziness Knowledge-Based Systems 8 (2000) 701-717.[7] Klement, E.P., Mesiar,R. and Pap, E., Triangular norms, in: Trends in Logic, in: Studia LogicaLibrary, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.[8] Mesiar, R., Choquet-like integrals, Journal of Mathematical Analysis and Applications 194 (1995)477-488.[9] Murofushi, T. and Sugeno, M., Fuzzy t-conorm integral with respect to fuzzy measures: Generalizationof Sugeno integral and Choquet integral, Fuzzy Sets and Systems 42 (1991) 57-71.[10] Narukawa, Y., Murofushi, T. and Sugeno, M., Regular fuzzy measure and representation of comono-tonically additive functionals, Fuzzy Sets and Systems 112 (2000) 177-186.[11] Ralescu, D. and Adams, G., The fuzzy integral, Journal of Mathematical Analysis and Application75 (1980) 562-570.[12] Román-Flores, H., Flores-Franulič, A. and Chalco-Cano, Y., A Jensen type inequality for fuzzy inte-grals, Information Sciences 177 (2007) 3192-3201.[13] Román-Flores, H., Flores-Franulič, A. and Chalco-Cano, Y., A Hardy type inequality for fuzzy inte-grals, Applied Mathematics and Computation 204 (2008) 178-183.[14] Royden, H.L., Real Analysis, Macmillan, New York, 1988.[15] Suárez Garćia, F. and Gil Álvarez, P., Two families of fuzzy integrals, Fuzzy Sets and Systems 18(1986) 67-81.[16] Sugeno, M., Theory of Fuzzy Integrals and Its Applications, Ph.D. Dissertation, Tokyo Institute ofTechnology, 1974.[17] Wang, Z. and Klir, G.J., Fuzzy Measure Theory, Plenum Press, New York, 1992.[18] Weber, S., Measures of fuzzy sets and measures of fuzziness, Fuzzy Sets and Systems 13 (1984) 247-271.Bayaz Daraby received his B.S. degree from University of Tabriz, M.S. degreefrom Tarbiat Modares University in Iran and Ph.D. degree from Puna University,Puna, India. He is an associate professor in Department of Mathematics, Universityof Maragheh, Iran. His area of interest includes Fuzzy Analysis and Fuzzy Topology.

General related jensen type inequalities for fuzzy integrals

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General related jensen type inequalities for fuzzy integrals

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