A Note on Gauss Type Inequality for Sugeno Integrals

Transcription

1 pplied Mathematical Sciences, Vol., 26, no. 8, HIKRI Ltd, Note on Gauss Type Inequality for Sugeno Integrals Dug Hun Hong Department of 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 In this note, we consider a similar type of Gauss inequality for fuzzy integrals. More precisely, we show that the inequality 2 (S) f(t)dµ (S) t 2 f(t)dµ holds for all > c > for some c, where f : [, ) [, ) and f(t), t 2 f(t) are non-increasing functions. n eample is considered. Mathematics Subject Classification: 26E5 Keywords: Fuzzy measure; Sugeno integral; Gauss type inequality Introduction and preliminaries Since Sugeno integral was first presented in 974 [8], it has been ehaustively investigated by many authors. Ralescu and dams [3] generalized a range of fuzzy measures and gave several equivalent definitions of fuzzy integrals. Wang and Klir [9] provided an overview of fuzzy measure theory. Román-Flores et al. [4-7] eamined a Prekopa-Leindler type inequality, a Jensen type inequality, a Young type inequality, and some convolution type inequalities. Caballero and Sadarangani [2-4] proved a Hermite-Hadamarda type inequality, a Cauchy type inequality, and Fritz Carlson s inequality for fuzzy integrals. Flores-Franulič et al. [5, 6] presented Chebyshev s inequality and Stolarsky s

2 88 Dug Hun Hong inequality for fuzzy integrals. Ouyang and Fang [] generalized their main results to prove some optimal upper bounds for the Sugeno integral of the monotone function in [6]. Ouyang et al. [2] generalized a Chebyshev type inequality for the fuzzy integral of monotone functions based on an arbitrary fuzzy measure. Hong [7, 8, ] presented several types of inequalities for Sugeno integrals including a Hardy-type inequality, a Liapunov type inequality, and a Berward and Favard type inequalities for Fuzzy integrals. Hong et al. [9] considered Steffensen s Integral inequality for Sugeno integral. In this note, we also consider a similar type of Gauss inequality for fuzzy integrals. n eample is considered. Definition. Let Σ be a σ-algebra of subsets of R and let µ : Σ [, ] be a non-negative, etended real-valued set function. We say that µ is a fuzzy measure if and only if (a) µ( ). (b) E, F Σ and E F imply µ(e) µ(f ) (monotonicity). ( ) (c) {E p } Σ and E E 2 imply lim p µ(e p ) µ p E p (continuity form below). ( (d) {E p } Σ, E E 2, and µ(e ) < imply lim p µ(e p ) ) µ (continuity form above). p E p If f is a non-negative real-valued function defined on R, then we denote by F α { X f() α} {f α} the α-level of f, for α >, and F { X f() > } supp(f) is the support of f. We note that α β {f β} {f α} If µ is a fuzzy measure on R, then we define the following: F µ () {f : [, ) f is µ-measurable}. Definition.2 Let µ be a fuzzy measure on (R, Σ). If f F µ (R) and Σ, then the Sugeno integral(or the fuzzy integral) of f on, with respect to the fuzzy measure µ, is defined as (S) fdµ sup [α µ( F α )]. α [, ) In particular, if X then (S) fdµ (S) R fdµ sup [α µ(f α )] α [, )

3 Gauss type inequality for Sugeno integrals 88 The following properties of the Sugeno integral are well known and can be found in [9]: Proposition.3 [9] If µ is a fuzzy measure on R and f, g F µ (R), then (i) (S) fdµ µ(); (ii) (S) Kdµ K µ() for any constant K [, ); (iii) (S) fdµ (S) gdµ, if f g on ; (iv) µ ( {f α}) α (S) fdµ α; (v) µ ( {f α}) α (S) fdµ α; (vi) (S) fdµ < α there eists γ < α such that µ ( {f γ}) < α; (vii) (S) fdµ > α there eists γ > α such that µ ( {f γ}) > α. Note. Let F (α) µ ( {f α}). Then by Proposition, (iv), (v), F (α) α (S) f()dµ α. Theorem.4 [] Let f : [, ) [, ) be continuous and non-increasing or non-decreasing functions and µ be the Lebesgue measure on R. Let (S) a f()dµ p. If < p < a, then f(p) p and f(a p) p, respectively. 2 Gauss type inequality type inequality The classical Gauss inequality provides the following inequality []: 2 f(t)dµ 4 9 t 2 f(t)dµ holds where g : [, ) [, ) and f(t), t 2 f(t) are non-increasing functions. We first consider a similar type of Gauss inequality for fuzzy integrals. Theorem 2. Let f(t), t 2 f(t) be a non-increasing function on [, ) and that µ is the Lebesgue measure on R. Then there eists c > such that 2 (S) f(t)dµ (S) t 2 f(t)dµ holds for all c, where f(c) /c. Proof. Let (S) f(t)dµ p. Then by Theorem, f( + p) p and 2 (S) f(t)dµ 2 p 2 f( + p) 2 f().

4 882 Dug Hun Hong Now, let (S) t 2 f(t)dµ c, then by Theorem, c 2 f(c) c, that is, f(c) /c. Therefore, since t 2 f(t) is a non-increasing function, sup c 2 (S) f(t)dµ c 2 f(c) (S) t 2 f(t)dµ. Theorem 3. Let f : [, ) [, ) be a conve strictly decreasing function and µ be the Lebesgue measure on R. Then the function (S) f(t)dµ is f() strictly increasing and lim f() (S) f(t)dµ. Proof. Let f : [, ) [, ) be a conve strictly decreasing function and µ be the Lebesgue measure on R. Let < < 2 and L() be the line equation connecting two points (, f( )) and ( 2, f( 2 )). Let θ (S) f(t)dµ, θ 2 (S) and θ (S) L(t)dµ, θ2 (S) Then we have by Theorem, and 2 f( + θ ) θ, f( 2 + θ 2 ) θ 2, L( + θ ) θ, L( 2 + θ 2) θ 2. 2 f(t)dµ L(t)dµ. Since f is strictly decreasing and conve, θ < θ and θ 2 > θ 2, and hence θ2 θ < θ 2 θ < θ 2 θ. Now, considering similar trapezoids, ((, ), ( +θ, ), ( +θ, θ ), (, f( ))) and (( 2, ), ( 2 + θ 2, ), ( 2 + θ 2, θ 2), ( 2, f( 2 ))), we have θ2 θ f( 2) f( ). Therefore, we have f( 2 ) (S) 2 f(t)dµ θ 2 f( 2 ) θ 2 θ f( 2 ) θ

5 Gauss type inequality for Sugeno integrals 883 Now, we note that f() (S) > θ 2 θ f( 2 ) θ f( 2 ) f( 2 ) f( ) θ θ f( ) f( ) (S) f(t)dµ f( + ) f() f(t)dµ, < f() f() where (S) f(t)dµ. Since is decreasing and f is non-negative and strictly decreasing and conve, it is easy to check that which complete the proof. f( + ) lim f(), y * 2 Figure : The graph of f() * Eample. Let f(t) /t 2. Then c and it is easy to check that (S) t 2 f(t)dµ. By Theorem 2, we have sup { 2 p ( + p) 2 p }.

6 884 Dug Hun Hong More precisely, from Theorem 3, we also have sup > { 2 p ( + p) 2 p }, and hence, the inequality in Theorem 2 is optimal. References [] P. S. Bullen, Dictionary of Inequalities, ddison Wesley Longman Inc., 998. [2] J. Caballero, K. Sadarangani, Hermite-Hadamarda inequality for fuzzy integrals, ppl. Math. Comput., 25 (29), [3] J. Caballero, K. Sadarangani, Fritz Carlson s inequality for fuzzy integrals, Computers and Math. with ppl., 59 (2), [4] J. Caballero, K. Sadarangani, Cauchy-Schwarz type inequality for fuzzy integrals, Nonlinear nalysis, 73 (2), [5]. Flores-Franulič, H. Román-Flores, Chebyshev type inequality for fuzzy integrals, ppl. Math. Comput., 9 (27), [6]. Flores-Franulič, H. Román-Flores, Y. Chalco-Cano, note on fuzzy integral inequality of Stolarsky type, ppl. Math. Comput., 96 (28), [7] D. H. Hong, sharp Hardy-type inequality for Sugeno integrals, ppl. Math. Comput., 27 (2), [8] D. H. Hong, Liapunov type inequality for Sugeno integrals, Nonlinear nalysis: Theory, Methods and pplications, 74 (2), [9] D. H. Hong, E-L. Moon, J. D. Kim, Steffen s Integral Inequality for the Sugeno integral, Int. J. of Uncertainty Fuzziness and Knowledge-Based Systems, 22 (24),

7 Gauss type inequality for Sugeno integrals 885 [] D. H. Hong, Berward and Favard type inequalities for Fuzzy integrals, Int. J. of Uncertainty Fuzziness and Knowledge-Based Systems, 24 (26), [] Y. Ouyang, J. Fang, Sugeno integral for monotone functions based on Lebesgue measure, Computers and Math. with ppl., 56 (28), [2] Y. Ouyang, J. Fang, L. Wang, Fuzzy Chebyshev type inequality, Int. J. ppro. Reasoning, 48 (28), [3] D. Ralescu, G. dams, The fuzzy integral, J. Math. nal. ppl., 75 (98), [4] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, Hardy type inequality for fuzzy integrals, ppl. Math. Comput., 24 (28), [5] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, Jensen type inequality for fuzzy integrals, Inform. Sci., 77 (27), [6] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, The fuzzy integral for monotone functions, ppl. Math. Comput., 85 (27), [7] H. Román-Flores,. Flores-Franulič, Y. Chalco-Cano, convolution type inequality for fuzzy integrals, ppl. Math. Comput., 95 (28), [8] M. Sugeno, Theory of Fuzzy Integrals and its pplications, Ph.D Thesis, Tokyo Institute of Technology, 974. [9] Z. Wang, G. J. Klir, Fuzzy Measure Theory, Springer Science and Business Media, New York, Received: January 23, 26; Published: March 5, 26