Stolarsky Type Inequality for Sugeno Integrals on Fuzzy Convex Functions

Transcription

1 International Journal o Mathematical nalysis Vol., 27, no., 2-28 HIKRI Ltd, Stolarsky Type Inequality or Sugeno Integrals on Fuzzy Convex Functions Dug Hun Hong Department o Mathematics, Myongji University Yongin Kyunggido , South Korea Copyright c 26 Dug Hun Hong. This article is distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. bstract Recently, Flores-Franulič et al. [ note on uzzy integral inequality o Stolarsky type, pplied Mathematics and Computation ] proved the Stolarsky s inequality or the Sugeno integral on the special class o continuous and strictly monotone unctions. This result can be generalized to a general class o uzzy convex unctions in this paper. We also give a uzzy integral inequality based on addition. Some illustrated examples are given. Mathematics Subject Classiication: 26E5 Keywords: Fuzzy measure; Sugeno integral; Stolarsky type inequality Introduction and preliminaries number o studies have examined the Sugeno integral since its introduction in 974 [6], it has been exhaustively investigated by many authors. Ralescu and dams [2] generalized a range o uzzy measures and gave several equivalent deinitions o uzzy integrals. Wang and Klir [7] provided an overview o uzzy measure theory. Caballero and Sadarangani [2-4] proved a Hermite-Hadamard type inequality, a Cauchy type inequality, and Fritz Carlson s inequality or uzzy integrals. Román-Flores et al. [3-5] presented several new types o inequalities or

2 22 Dug Hun Hong Sugeno integeals, including a Prekopa-Leindler type inequality, a Jensen type inequality, and some convolution type inequalities. Flores-Franulič et al. [5, 6] presented Chebyshevs inequality, Stolarskys inequality or uzzy integrals. Ouyang and Fang [] generalized their main results to prove some optimal upper bounds or the Sugeno integral o monotone unction in [4]. Ouyang et al. [9] generalized a Chebyshev type inequality or uzzy integral o monotone unctions based on an arbitrary uzzy measure. Hong [7] improved on previous work presenting a Hardy-type inequality or Sugeno integrals. Hong [8] proposed a Liapunov type inequality or Sugeno integrals and ind an optimal constant or which Liapunov type inequality or Sugeno integrals holds or non-increasing concave unctions. Hong [9] proposed a Berwald type inequality and a Favard type inequality or Sugeno integrals. The purpose o this paper is to generalize the main results in [6], that is, we prove a Stolarsky type inequality or Sugeno integrals on uzzy convex unctions. We also give a uzzy integral inequality based on addition. Some examples are provided to illustrate the validity o the proposed inequalities. Deinition. Let Σ be a σ-algebra o subsets o R and let µ : Σ [, ] be a non-negative, extended real-valued set unction. We say that µ is a uzzy measure i and only i a µ =. b E, F Σ and E F imply µe µf monotonicity. c E p } Σ and E E 2 imply lim p µe p = µ p= E p continuity orm below. d E p } Σ, E E 2, and µe < imply lim p µe p = µ p= E p continuity orm above. I is a non-negative real-valued unction deined on R, then we denote by F α = x X x α} = α} the α-level o, or α >, and F = x X x > } = supp is the support o. We note that α β β} α} I µ is a uzzy measure on R, then we deine the ollowing: F µ = : [, is µ-measurable}. Deinition 2. Let µ be a uzzy measure on R, Σ. I F µ R and Σ, then the Sugeno integralor the uzzy integral o on, with respect to the uzzy measure µ, is deined as S dµ = sup [α µ F α ]. α [,

3 Stolarsky type inequality or Sugeno integrals 23 In particular, i = X then S dµ = S R dµ = sup [α µf α ] α [,. The ollowing properties o the Sugeno integral are well known and can be ound in [7]: Proposition [7]. I µ is a uzzy measure on R and, g F µ R, then i S dµ µ; ii S Kdµ = K µ or any constant K [, ; iii S dµ S gdµ, i g on ; iv µ α} α S dµ α; v µ α} α S dµ α; vi S dµ < α there exists γ < α such that µ γ} < α; vii S dµ > α there exists γ > α such that µ γ} > α. 2 Stolarsky type inequality or uzzy convex unctions Flores-Franulič et al. [6] presented the ollowing Stolarsky-type inequality or uzzy integrals under the conditions that is a continuous and strictly monotone unction. Theorem Stolarsky type inequality: monotone case [6]. Let a, b >. I : [, ] [, ] is a continuous and strictly monotone decreasing or increasing unction and µ is the Lebesgue measure on R, then the inequality S dµ S x a dµ S x b dµ holds. This result can be generalized to a general class o uzzy convex unctions in this paper. We need the ollowing lemma. Lemma. Let a, b >. I : [, ] R is a uzzy convex unction, and µ is the Lebesgue measure on R, then the inequalities } } } µ α µ x a α µ x b α and } } } µ α µ x a α + µ x b α 2

4 24 Dug Hun Hong hold. Proo. Suppose that or < α, } α = [α, α 2 ]. Then we have α = α 2 = α, and hence Similarly, suppose that } x a α = [β, β 2 ], then we have x α} = [α, α 2 ]. } x b α = [γ, γ 2 ]. x α} = [β a, β a 2 ] = [γ b, γ b 2 ]. Let x α} = [h, h 2 ]. Then we have [α, α 2 ] = [h, h 2 ], [β, β 2 ] = [h a, h a 2] and [γ, γ 2 ] = [h b, h b 2]. We now consider that µ } } } α µ x a α µ x b α = α 2 α β 2 β γ 2 γ = h 2 h h a 2 h a h b 2 h b = h a h b 2 + h b h a 2 2h and } µ α = α 2 α = β 2 γ 2 β γ = β 2 γ 2 β 2 γ + β 2 γ β γ = β 2 γ 2 γ + γ β 2 β γ 2 γ + β 2 β } } = µ x a α + µ x b α. which completes the proo. By using Lemma, we obtain the ollowing main result. We show that the condition o continuity and strictly monotonicity o can be released. Fuzzy

5 Stolarsky type inequality or Sugeno integrals 25 convexity o is suicient or validity o a Stolarsky type inequality or uzzy integrals. Theorem 2. Let a, b >. I : [, ] R is a uzzy convex unction, and µ is the Lebesgue measure on R, then the inequality S dµ S x a dµ S x b dµ holds. Proo. Let S x a such that minp ε, q ε} >. Then we have S dµ = p and S x a dµ > p ε and S x b dµ = q. nd let ε > x b dµ > q ε holds. By Proposition, there exist α and β such that α > p ε and β > q ε, µ x a α} > p ε and µ x b β} > q ε hold. By Lemma, we have } µ αβ µ µ x a x a } αβ } α > p εq ε. µ µ x b x b } αβ } β Since αβ > p εq ε, by Proposition, Since ε is arbitrary, S S dµ pq = dµ > p εq ε. S dµ S gdµ. In a similar manner, we can prove the ollowing result using Lemma. Theorem 3. Let a, b >. I : [, ] R is a uzzy convex unction, and µ is the Lebesgue measure on R, then the inequality S dµ S x a dµ + S x b dµ

6 26 Dug Hun Hong holds. In the ollowing, we present a example to illustrate the validity o Theorem 2. and 3. Example. Let x = 4x x 2, x [, ] and a = 2, b = /6. Then, a straightorward calculus with the aid o computer work shows that is iis iiis Thereore x a dµ = S.6 = S x b dµ = S dµ = S 2 xdµ = α [, x 6 x 2 dµ = dµ S α [, x x 3 dµ = [α µ4 2 x α] =.68, [α µ4x 6 x 2 α] =.39, α [, x a dµ S [α µ4x 6 3 x 2 3 α] =.6. x b dµ =.97. Example 2. Let x i x [, /4 /2 i x [/4, 3/8 x = 4x i x [3/8, /2 2 x i x [/2, ], and a =, b = 2. Then is not continuous and not monotone but is uzzy convex. straightorward calculus shows that µ x a /2} = /2 and µ x b /2} = /2, and hence by Proposition, we have S x a dµ = /2 and S x b dµ = /2. Now, a straightorward calculus shows that } α/2 µ x 2 α 2 i x [, /4], α = α/2 2 /6 i x /4, /2], α/2 2 /4 + α/4 2 i x /2, ].

7 Stolarsky type inequality or Sugeno integrals 27 By proving the equation α/2 2 /6 = α, we see that µ.4499} =.4499 and hence Thereore.4499 = S S dµ S dµ = x a dµ S x b dµ =.25. Reerences [] P. S. Bullen, Dictionary o Inequalities, ddison Wesley Longman Inc., 998. [2] J. Caballero, K. Sadarangani, Hermite-Hadamard inequality or uzzy integrals, ppl. Math. Comput., 25 29, [3] J. Caballero, K. Sadarangani, Fritz Carlson s inequality or uzzy integrals, Computers and Math. with ppl., 59 2, [4] J. Caballero, K. Sadarangani, Cauchy-Schwarz type inequality or uzzy integrals, Nonlinear nalysis: Theory, Methods and pplications, 73 2, [5]. Flores-Franulič, H. Román-Flores, Chebyshev type inequality or uzzy integrals, ppl. Math. Comput., 9 27, [6]. Flores-Franulič, H. Román-Flores, Y. Chalco-Cano, note on uzzy integral inequality o Stolarsky type, ppl. Math. Comput., 96 28, [7] D. H. Hong, sharp Hardy-type inequality or Sugeno integrals, ppl. Math. Comput., 27 2, [8] D. H. Hong, Liapunov type inequality or Sugeno integrals, Nonlinear nalysis: Theory, Methods and pplications, 27 2,

8 28 Dug Hun Hong [9] D. H. Hong, Berward and Favard type inequalities or Fuzzy integrals, Int. Journal o Uncertainty, Fuzziness and Knowledge-Based Systems, 24 26, [] Y. Ouyang, J. Fang, Sugeno integral or monotone unctions based on Lebesgue measure, Computers and Mathematics with pplications, 56 28, [] Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, Int. J. pprox., 48 28, [2] D. Ralescu, G. dams, The uzzy integral, J. Math. nal. ppl., 75 98, [3] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, Hardy type inequality or uzzy integrals, ppl. Math. Comput., 24 28, [4] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, Jensen type inequality or uzzy integrals, Inorm. Sci., 77 27, [5] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, convolution type inequality or uzzy integrals, ppl. Math. Comput., 95 28, [6] M. Sugeno, Theory o Fuzzy Integrals and Its pplications, Ph.D. Thesis, Tokyo Institute o Technology, 974. [7] Z. Wang, G. Klir, Fuzzy Measure Theory, Springer, New York, Received: December 6, 26; Published: January, 27