Research Article General Fritz Carlson s Type Inequality for Sugeno Integrals

Transcription

1 Hindawi Publishing Corporation Journal of Inequalities and pplications Volume, rticle ID 7643, 9 pages doi:.55//7643 Research rticle General Fritz Carlson s Type Inequality for Sugeno Integrals Xiaojing Wang and Chuanzhi Bai Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 33, China Correspondence should be addressed to Chuanzhi Bai, czbai8@sohu.com Received 8 ugust ; Revised 3 November ; ccepted January cademic Editor: László Losonczi Copyright q X. Wang and C. Bai. This is an open access article distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Fritz Carlson s type inequality for fuzzy integrals is studied in a rather general form. The main results of this paper generalize some previous results.. Introduction and Preliminaries Recently, the study of fuzzy integral inequalities has gained much attention. The most popular method is using the Sugeno integral. The study of inequalities for Sugeno integral was initiated by Román-Flores et al., 3 and then followed by the others 4. Now, we introduce some basic notation and properties. For details, we refer the reader to,. Suppose that Σ is a σ-algebra of subsets of X,andletμ : Σ, be a nonnegative, extended real-valued set function. We say that μ is a fuzzy measure if it satisfies μ, E, F Σ and E F imply μ E μ F monotonicity ; 3 {E n } Σ, E E imply lim n μ E n μ n E n continuity from below, 4 {E n } Σ, E E, μ E <, imply lim n μ E n μ n E n continuity from above. If f is a nonnegative real-valued function defined on X, we will denote by L α f {x X : f x α} {f α} the α-level of f for α>, and L f {x B : f x > } supp f is the support of f. Notethatifα β, then{f β} {f α}. Let X, Σ,μ be a fuzzy measure space; by F μ X we denote the set of all nonnegative μ-measurable functions with respect to Σ.

2 Journal of Inequalities and pplications Definition. see. Let X, Σ,μ be a fuzzy measure space, with f F μ X, and Σ, then the Sugeno integral or fuzzy integral of f on with respect to the fuzzy measure μ is defined by fdμ α [ α μ ( { f α }],. where and denote the operations sup and inf on,, respectively. It is well known that the Sugeno integral is a type of nonlinear integral; that is, for general cases, (af bg dμ a fdμ b gdμ. does not hold. The following properties of the fuzzy integral are well known and can be found in. Proposition.. Let X, Σ,μ be a fuzzy measure space, with, B Σ and f, g F μ X ; then χ fdμ μ, kdμ k μ, fork a nonnegative constant, 3 if f g on χ χ χ then fdμ χ gdμ, 4 if B then fdμ fdμ, 5 μ {f α} α fdμ α, 6 χ μ {f α} α fdμ α, 7 χ fdμ < α there exists γ<αsuch that μ {f γ} <α, 8 fdμ > α there exists γ>αsuch that μ {f γ} >α. Remark.3. Let F be the distribution function associated with f on, thatis,f α μ {f α}. By 5 and 6 of Proposition. F α α fdμ α. Thus, from a numerical point of view, the Sugeno integral can be calculated by solving the equation F α α. Fritz Carlson s integral inequality states 3, 4 that.3 f x dx π ( f x dx /4 ( x f x dx /4..4 Recently, Caballero and Sadarangani 8 have shown that in general, the Carlson s integral inequality is not valid in the fuzzy context. nd they presented a fuzzy version of Fritz Carlson s integral inequality as follows.

3 Journal of Inequalities and pplications 3 Theorem.4. Let f :,, be a nondecreasing function and μ the Lebesgue measure on f x dμ x ( x f x dμ x /4 ( f x dμ x /4..5 In this paper, our purpose is to give a generalization of the above Fritz Carlson s inequality for fuzzy integrals. Moreover, we will give many interesting corollaries of our main results.. Main Results This section provides a generalization of Fritz Carlson s type inequality for Sugeno integrals. Before stating our main results, we need the following lemmas. χ Lemma. see. Let X, Σ,μ be a fuzzy measure space, f F X, μ Σ, fdμ,and s.then f s dμ fdμ s.. χ If the fuzzy measure μ in Lemma. is the Lebesgue measure, then fdμ is satisfied readily. Thus, by Lemma., we have the following. Corollary. see 8. Let f :,, be a μ-measurable function with μ the Lebesgue measure and s.then f s x dμ x ( f x dμ x s.. Definition.3. Two functions f, g : X R are said to be comonotone if for all x, y X, ( f x f ( y ( g x g ( y..3 n important property of comonotone functions is that for any real numbers p, q, either {f p} {g q} or {g q} {f p}. Note that two monotone functions in the same sense are comonotone. Theorem.4. Let X, χ Σ,μ be,andχ a fuzzy measure space, f, g F X μ and f and g comonotone functions, Σ with fdμ gdμ.then f gdμ fdμ gdμ..4

4 4 Journal of Inequalities and pplications χ orχ Proof. If fdμ gdμ then the inequality is obvious. Now choose α, β such that fdμ > α >, gdμ > β >. Then by 8 of Proposition.,thereexist>γ α >αand >γ β >βsuch that.5 μ ( { f γ α } >α, μ ( { g γβ } >β..6 s f and g are comonotone functions, then either {f γ α } {g γ β } or {g γ β } {f γ α }. Suppose that {f γ α } {g γ β }. In this case, we have the following: μ ( { fg γ α γ β } μ (( { f γα } ( { g γβ } μ ( { f γα } >α αβ..7 Therefore, by applying 8 of Proposition. again, we find that f gdμ > αβ..8 Since the values of α, β > are arbitrary, we obtain the desired inequality. Similarly, for the case {g γ β } {f γ α } we can get the desired inequality too. From Theorem.4, we get the following. Corollary.5 see 5. Let μ be an arbitrary χ fuzzy measure andχ on,a and f, g :,a R be two a real-valued measurable functions such that fdμ a gdμ.iff and g are increasing (or decreasing functions, then the inequality a f gdμ a a fdμ gdμ.9 holds. χ If the fuzzy measure μ in Corollary.5 is the Lebesgue measure and a, then a andχ fdμ a gdμ are satisfied readily. Thus, by Corollary.5, weobtain Corollary.6 see. Let f, g :, R be two real-valued functions, and let μ be the Lebesgue measure on R. If f, g are both continuous and strictly increasing (decreasing functions, then the inequality f gdμ ( fdμ ( gdμ. holds. The following result presents a fuzzy version of generalized Carlson s inequality.

5 Journal of Inequalities and pplications 5 Theorem.7. Let X, Σ,μ be a fuzzy measure space, f, g, h F μ X, f and g, andf and h are comonotone χ χ,χ χ,χ functions, respectively, Σ with fdμ gdμ, hdμ fgdμ, and fhdμ.then f x dμ x K / p q / p q f p x g x dμ x p f q x h x dμ x q,. χ χ where K g x dμ x p/ p q h x dμ x q/ p q. p f x g x dμ x Proof. By Lemma.,forp, q, we have the following: f x h x dμ x q Multiplying these inequalities, we get that f x g x dμ x p f p x g p x dμ x f p x g p x dμ x, f q x h q x dμ x. q f x h x dμ x f q x h q x dμ x...3 By Theorem.4 f gdμ fdμ gdμ, f hdμ fdμ hdμ..4 Substitutes.4 into.3, weobtain f x dμ x p q f p x g p x dμ x g x dμ x p q h x dμ x f q x h q x dμ x..5 This inequality implies that. holds By Theorem.7, we have the following. Corollary.8. ssume that p, q. Letf, g, h :,, are increasing (or decreasing functions and μ the Lebesgue measure on R. Thenbe f x dμ x K ( f p x g p x dμ x / p q ( f q x h q x dμ x / p q,.6 where K χ g x dμ x p/ p q χ h x dμ x q/ p q.

6 6 Journal of Inequalities and pplications Theorem.9. Let g :,, be a μ-measurable function with μ the Lebesgue measure. If g s (s is a convex function such that, g / g,then { max { g,g } g x dμ x min ( }. g s g s,.7 /s Proof. Firstly, we consider the case of g s <g s. sg s is a convex function, we have by Theorem of Caballero and Sadarangani 7 that { g s g s } x dμ x min g s g s,..8 By Corollary. and.8, weget ( s { g g x dμ x s } min g s g s,,.9 which implies that.7 holds. Similarly, we can obtain.7 by of 7, Theorem for the case of g s >g s. From Theorem.9 and Corollary.8, we have the following. Theorem.. ssume that p, q. Letf, g, h :,, be increasing (or decreasing functions and μ the Lebesgue measure on R. Ifg s ( s orh r ( r is a convex function such that g / g or h / h, then f x dμ x M p/p q K q/p q ( f p x g p x dμ x / p q ( f q x h q x dμ x / p q,. where { max { g,g } M min ( }, g s g s, K /s h x dμ x,. or f x dμ x K p/p q M q/p q ( f p x g p x dμ x / p q ( f q x h q x dμ x / p q,.

7 Journal of Inequalities and pplications 7 where K g x dμ x, { } max{h,h } M min h r h r,..3 /r Theorem.. ssume that p, q. Letf, g, h :,, be increasing (or decreasing functions and μ the Lebesgue measure on R. Ifg s s and h r r are two convex functions such that g / g and h / h, then, f x dμ x M p/p q M q/p q ( f p x g p x dμ x / p q ( f q x h q x dμ x / p q,.4 where M and M are as in. and.3, respectively. Straightforward calculus shows that x dμ x 3 5, xdμ x, dμ x..5 If p q, g x x and h x, g x x and h x x, g x x,andh x, respectively, then Corollary.8 reduces to Theorem.4, and the following Corollaries. and.3. Corollary.. Let f :,, be a nondecreasing function and μ the Lebesgue measure on f x dμ x 3 5 ( x 4 f x dμ x /4 ( x f x dμ x /4..6 Corollary.3. Let f :,, be a nondecreasing function and μ the Lebesgue measure on f x dμ x 6 5 ( x 4 f x dμ x /4 ( f x dμ x /4..7 Remark.4. Corollary.8 is a generalization of the main result in 8, Theorem. If p q, g x h x x,thencorollary.8 reduces to the following corollary. Corollary.5. Let f :,, be a nondecreasing function and μ the Lebesgue measure on R. Then f x dμ x 3 5 x f x dμ x..8

8 8 Journal of Inequalities and pplications Consider g x e x on,. This function is nonincreasing g x / x e x <, nonnegative and convex g x /4 x e x / x. Let p q, g x h x e x,ands r. s g /e > /e g and h >h, we have the following M M e e e..9 Thus, by Theorem. we can get the following corollary. Corollary.6. Let f :,, be a nonincreasing function and μ the Lebesgue measure on f x dμ x e e e e x f x dμ x..3 Consider g x x ln x and h x x arc tan x on,. Obviously, g and h are nonnegative, nondecreasing and convex on the interval,. Lets r, then, we have the following: { max { g,g } } M min ( g s g s, ln /s ln, { } max{h,h } M min h r h r, /r 4 π 8 π..3 Thus, by Theorem. set p q we can get the following corollary. Corollary.7. Let f :,, be a nondecreasing function and μ the Lebesgue measure on f x dμ x ln 8 π ln 4 π ( ( x ln x f x dμ x x arctan x f x dμ x /. /.3 Consider g x x x /8 on,. Obviously, this function is nonnegative, nondecreasing g x x / x x /8 /, and nonconvex g x /8 x x /8 3/.Butg x x x /8isconvex.Sets, then we obtain M 7/8 34 ( 7/8 (..33 /8 8 7

9 Journal of Inequalities and pplications 9 Thus, by Theorem. set g x x /8, h x x, s, p,q we can get the following corollary. Corollary.8. Let f :,, be a nondecreasing function and μ the Lebesgue measure on R. Then ( f x dμ x 7 ( x f x dμ x /3. /3 ( /3 x x /8 f x dμ x.34 cknowledgments The authors would like to thank the referees for reading this work carefully, providing valuable suggestions and comments. This work is supported by the National NaturalScience Foundation of China no. 77. References M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Institute of Technology, Flores-Franulič andh.román-flores, Chebyshev type inequality for fuzzy integrals, pplied Mathematics and Computation, vol. 9, no., pp , 7. 3 H. Román-Flores,. Flores-Franulič, and Y. Chalco-Cano, Jensen type inequality for fuzzy integrals, Information Sciences, vol. 77, no. 5, pp. 39 3, 7. 4 R. Mesiar and Y. Ouyang, General Chebyshev type inequalities for Sugeno integrals, Fuzzy Sets and Systems, vol. 6, no., pp , 9. 5 H. Román-Flores,. Flores-Franulič, and Y. Chalco-Cano, Hardy-type inequality for fuzzy integrals, pplied Mathematics and Computation, vol. 4, no., pp , 8. 6 H. gahi, R. Mesiar, and Y. Ouyang, General Minkowski type inequalities for Sugeno integrals, Fuzzy Sets and Systems, vol. 6, no. 5, pp ,. 7 J. Caballero and K. Sadarangani, Hermite-Hadamard inequality for fuzzy integrals, pplied Mathematics and Computation, vol. 5, no. 6, pp , 9. 8 J. Caballero and K. Sadarangani, Fritz Carlson s inequality for fuzzy integrals, Computers and Mathematics with pplications, vol. 59, no. 8, pp ,. 9 H. Román-Flores,. Flores-Franulič, and Y. Chalco-Cano, The fuzzy integral for monotone functions, pplied Mathematics and Computation, vol. 85, no., pp , 7. H. Román-Flores,. Flores-Franulič, and Y. Chalco-Cano, convolution type inequality for fuzzy integrals, pplied Mathematics and Computation, vol. 95, no., pp , 8. J. Caballero and K. Sadarangani, Cauchy-Schwarz type inequality for fuzzy integrals, Nonlinear nalysis. Theory, Methods and pplications. Series, vol. 73, no., pp ,. Z. Wang and G. Klir, Fuzzy Measure Theory, Plenum Press, New York, NY, US, F. Carlson, Une ineqalite, rkiv för Matematik, vol. 5, pp. 5, G. H. Hardy, note on two inequalities, Journal of the London Mathematical Society, vol., pp. 67 7, Y. Ouyang, J. Fang, and L. Wang, Fuzzy Chebyshev type inequality, International Journal of pproximate Reasoning, vol. 48, no. 3, pp , 8.