TWMS J. App. Eng. Math. V.8, N., 08, pp. -7 GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS B. DARABY, Abstract. 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. Keywords: Fuzzy measure, Jensen s inequality, Seminormed fuzzy integral, Semiconormed fuzzy integral. AMS Subject Classification: 03E7, 6E50, 8E0. Introduction The properties and applications of the Sugeno integral have been studied by many authors, including Ralescu and Adams [], Román-Flores et al. [, 3] and Wang and Klir [7]. In connection with the topics that will be discussed in this article, there are several other theoretical and applied papers related to fuzzy measure theory on metric spaces, such as Li et al. [8, 9] and Narukawa et al. [0]. Many authors generalized the Sugeno integral by using some other operators to replace the special operator(s and/or (see, e.g., [,, 3, 4, 6, 8, 8]. Suárez and Gil Álvarez presented two families of fuzzy integrals, the so-called seminormed fuzzy integrals and semiconormed fuzzy integrals [5]. This paper is organized as follows: in Section, some preliminaries and summarization of some previous known results are briefly recalled. Section 3 deals with a general related Jensen type inequality for the fuzzy integrals.. Preliminaries In this section we recall some basic definitions and previous results that will be used in the next section. Let X be a non-empty set, Σ be a σ-algebra of subsets of X. Throughout this paper, all considered subsets are supposed to belong to Σ. Definition. (Sugeno [6]. A set function µ : Σ [0, ] is called a fuzzy measure if the following properties are satisfied: (FM µ( = 0 and µ(x = ; (FM A B implies µ(a µ(b; (FM3 A n A implies µ(a n µ(a. Department of Mathematics, University of Maragheh, P. O. Box 558-83, Maragheh, Iran. e-mail: bdaraby@maragheh.ac.ir; ORCID: https://orcid.org/0000-000-687-866. Manuscript received: September, 06; accepted: February 5, 07. TWMS Journal of Applied and Engineering Mathematics, Vol.8, No. c Işık University, Department of Mathematics, 08; all rights reserved.
TWMS J. APP. ENG. MATH. V.8, N., 08 When 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, ] is measurable with respect to Σ}. In what follows, all considered functions belong to F + (X. For any α [0, ], we will denote the set {x X f(x α} by F α and {x X f(x > α} by Fᾱ. Clearly, both F α and Fᾱ are non-increasing with respect to α, i.e., α β implies F α F β and Fᾱ F β. Definition. (Sugeno [6]. Let (X, Σ, µ be a fuzzy measure space, and A Σ, the Sugeno integral of f over A, with respect to the fuzzy measure µ, is defined by = (α µ(a F α. When A=X, then A = = X (α µ(f α. Note in the above definition, is just the prototypical t-norm minimum and the prototypical t-conorm maximum. Definition.3 ([7]. A t-norm is a function T : [0,] [0,] [0,] satisfying the following conditions: (A T (x, = T (, x = x x [0, ]. (B x, x, y, y in [0, ], if x x, y y then T (x, y T (x, y. (C T (x, y = T (y, x. (D T (T (x, y, z = T (x, T (y, z. Definition.4 ([7]. A function S : [0, ] [0, ] [0, ] is called a t-conorm, if there is a t-norm T such that S(x, y = T ( x, y. Evidently, a t-conorm S satisfies: (A S(x, 0 = S(0, x = x, x [0, ] as well as conditions (B, (C and (D. A binary operator T (S on [0,] is called a t-seminorm (t-semiconorm, if it satisfies the above conditions (A and (B ((A and (B[5]. Notice that in the literature, a t-seminorm is also called a semicopula. By using the concepts of t-seminorm and t- semiconorm, Suárez and Gil (986 proposed two families of fuzzy integrals: Definition.5. Let T be a t-seminorm, then the seminormed fuzzy integral of f over A with respect to T and the fuzzy measure µ is defined by = T (α, µ(a F α. ( T, A Definition.6. Let S be a t-semiconorm, then the semiconormed fuzzy integral of f over A with respect to S and the fuzzy measure µ is defined by = S(α, µ(a Fᾱ. Remark.. Maybe one can define the so-called conormed-seminormed fuzzy integral T by using an arbitrary t-conorm S to replace the special t- conorm in (. However, as Suárez and Gil Álvarez [5] pointed out that the conormed-seminormed fuzzy integral satisfies the essential property adµ = a, a [0, ] T S, X
B. DARABY,: GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS 3 if and only if S =. The case of semiconormed fuzzy integral is similar. It is easy to see that the Sugeno integral is a special seminormed fuzzy integral. Moreover, Kandel and Byatt in (see [5] showed another expression of the Sugeno integral as follows: = (α µ(a Fᾱ. A 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 fuzzy integrals. Note that if = a, then T (α, µ(a F α a for all α [0, ] and, for any ε > o there exists α ε such that T (α ε, µ(a F αε a ε. Also, if S,A = a, then S(α, µ(a Fᾱ a for all α [0, ] and, for any ε > o there exists α ε such that S(α ε, µ(a Fᾱε a + ε. The well-known Jensen inequality is a part of the classical mathematical analysis, (see [4]. Theorem.. Let h : [0, ] (a, b be real and integrable function and a convex function on (a,b. Then ( h(xdx (h(xdx. ( 0 0 In [3], Daraby et al. have proved general version of the Jensen type inequality as follows: Theorem.. Let (X, Σ, µ be a fuzzy measure space and f : X [0, ] be a measurable function. Let T be a seminorm such that = a for any A Σ. Let : [0, ] [0, ] be a continuous and strictly increasing function such that (x x, for every x [0, a], then: ( (fdµ. (3 3. Main results 3.. Some results on Jensen type seminormed fuzzy integrals. In this section, we will show some results on general Jensen type inequality. At the first, we show that conditions strictly increasing and (x x, for every x [0, ] on the function in Theorem. cannot be avoided, as we will illustrate in the following examples. Example 3.. We observe that strictly increasing condition of function in Theorem. cannot be avoided. Let µ be the Lebesgue measure on R and T be defined as { 0 if max{x, y} <, T D (x, y = min{x, y} if max{x, y} =. and consider f(x = xχ [0,] (x. If we define: { x if 0 x < (x = ( xχ [0,] (x if x. Thus: If max{x, y} <, then T D (x, y = 0. So we have fdm = 0. Consequently, ( fdm = 0. Hence ( fdm = 0 (fdm.
4 TWMS J. APP. ENG. MATH. V.8, N., 08 If max{x, y} =, then fdm = = fdm = (α m(f α (α ( α =. Consequently, ( ( fdm = =. (4 In addition, because (f(x = (xχ [0,] (x, then: m ({(f } = m ({ } = 0. (5 But, from (4 and (5 we have: (fdm in contradiction with (5, which implies that (fdm <. Consequently, inequality (3 is not verified. m ({(f } Example 3.. The condition (x x for all 0 x in Theorem (.9 cannot be avoided. In fact, consider f(x = xχ [0, ](x and define (x = x. Also T be defined as T D (x, y = { 0 if max{x, y} <, min{x, y} if max{x, y} =. Thus: If max{x, y} <, then T D (x, y = 0. So we have fdm = 0. Consequently, ( = 0. Hence If max{x, y} =, then fdm ( fdm = 0 (fdm. = = [α µ(f α α [0, ] = [α ( α] = 4. α [0, ]
B. DARABY,: GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS 5 Consequently, ( ( = = 4. Furthermore, because (f(x = xχ [0, ](x, then: (fdµ = = (fdµ α [0, = α [0, [α µ(f α ] ] [α ( α ] ] = + 3 < ( =. Which implies that inequality (3 is not verified. Theorem 3.. Let (X, Σ be a fuzzy measurable space and f : X [0, ] be a measurable function. Let T be a seminorm such that F (x = T,[0,x] f(tdt. Let : [0, ] [0, ] be a continuous and strictly increasing function such that (x x, for every x [0, F (x], then: ( ( T α, (fdx (F (x dx (6 α x holds for any A Σ. Proof. By Theorem. we have K = Since [0, x] A, then we have ( then thus K = (fdx ( (fdx. ( ( ( ( T α, (fdx = α T,[0,x] T,[0,x] = (F (x K x dx ((F (x dx. x This completes the proof.
6 TWMS J. APP. ENG. MATH. V.8, N., 08 3.. A related inequality. In this section, we prove a related inequality for semiconormed fuzzy integrals. The main result of this section is as follows. Theorem 3.. Let (X, Σ, µ be a fuzzy measure space and f : X [0, ] measurable function. Let S be a semiconorm such that S,A = a for any A Σ. Let : [0, ] [0, ] be a continuous and strictly increasing function such that (x x, for every x [0, a], then: ( (fdµ. (7 Proof. Let S,A = a. Then, for any ε > 0, there exist a ε such that µ(a Fāε = a 0, where S(a ε, a 0 a + ε, (Thus a ε a + ε. Denote ϕᾱ = {x (f(x > α}. Since : [0, ] [0, ] is a continuous and strictly increasing function, there holds Then Hence Thus ϕ (āε {x f(x > ā ε }. µ ( A ϕ (āε µ(a {x f(x > āε } = a 0. (fdµ = S,A a [0,] S(a, µ(a ϕā S ( (a ε, µ(a ϕ (āε S ((a ε, a 0 ((S(a ε, a 0 (a + ε. (fdµ (a, (8 follows from the arbitrariness of ε. Consequently from 8 we conclude: This completes the proof. (fdµ = a [0,] ( (a = S(a, µ(a ϕā As a direct consequence of Theorem (3.. we have Corollary 3.. Let (X, Σ, µ be a fuzzy measure space and f : X [0, ] be a measurable function. If S be a semiconorm. Let : [0, ] [0, ] be a continuous and strictly increasing function such that (x x, for every x [0, µ(x], then: ( (fdµ. Acknowledgement I am very grateful to the referee(s for the careful reading of the paper and for the useful comments..
B. DARABY,: GENERAL RELATED JENSEN TYPE INEQUALITIES FOR FUZZY INTEGRALS 7 References [] Daraby, B., Generalization of the Stolarsky type inequality for pseudo-integrals, Fuzzy Sets and Systems 94 (0 90-96. [] Daraby, B., Shafiloo, A. and Rahimi, A., Generalization of the Feng Qi type inequality for pesudointegral, Gazi University Journal of Science 8 (4 (05, 695-70. [3] Daraby, B. and Rahimi, A., Jensen type inequality for seminormed fuzzy integrals, Acta Universitatis Apulensis 46 (06-8. [4] Daraby, B. and Ghazanfary Asll, H. and Sadeqi, I., General related inequalities to Carlson-type inequlity for the Sugeno integral, Applied Mathematics and Computation 305 (07 33-39. [5] Kandel, A., Byatt, W.J., Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proceedings of the IEEE 66 (978 69-639. [6] Klement, E.P., Mesiar, R. and Pap, E., Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval, International Journal of Uncertainty Fuzziness Knowledge-Based Systems 8 (000 70-77. [7] Klement, E.P., Mesiar,R. and Pap, E., Triangular norms, in: Trends in Logic, in: Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 000. [8] Mesiar, R., Choquet-like integrals, Journal of Mathematical Analysis and Applications 94 (995 477-488. [9] Murofushi, T. and Sugeno, M., Fuzzy t-conorm integral with respect to fuzzy measures: Generalization of Sugeno integral and Choquet integral, Fuzzy Sets and Systems 4 (99 57-7. [0] Narukawa, Y., Murofushi, T. and Sugeno, M., Regular fuzzy measure and representation of comonotonically additive functionals, Fuzzy Sets and Systems (000 77-86. [] Ralescu, D. and Adams, G., The fuzzy integral, Journal of Mathematical Analysis and Application 75 (980 56-570. [] Román-Flores, H., Flores-Franulič, A. and Chalco-Cano, Y., A Jensen type inequality for fuzzy integrals, Information Sciences 77 (007 39-30. [3] Román-Flores, H., Flores-Franulič, A. and Chalco-Cano, Y., A Hardy type inequality for fuzzy integrals, Applied Mathematics and Computation 04 (008 78-83. [4] Royden, H.L., Real Analysis, Macmillan, New York, 988. [5] Suárez García, F. and Gil Álvarez, P., Two families of fuzzy integrals, Fuzzy Sets and Systems 8 (986 67-8. [6] Sugeno, M., Theory of Fuzzy Integrals and Its Applications, Ph.D. Dissertation, Tokyo Institute of Technology, 974. [7] Wang, Z. and Klir, G.J., Fuzzy Measure Theory, Plenum Press, New York, 99. [8] Weber, S., Measures of fuzzy sets and measures of fuzziness, Fuzzy Sets and Systems 3 (984 47-7. Bayaz Daraby received his B.S. degree from University of Tabriz, M.S. degree from Tarbiat Modares University in Iran and Ph.D. degree from Puna University, Puna, India. He is an associate professor in Department of Mathematics, University of Maragheh, Iran. His area of interest includes Fuzzy Analysis and Fuzzy Topology.