Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations
The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex fun...
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Wiley
2019-01-01
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| Series: | Advances in Fuzzy Systems |
| Online Access: | http://dx.doi.org/10.1155/2019/5080723 |
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| author | Dug Hun Hong Jae Duck Kim |
| author_facet | Dug Hun Hong Jae Duck Kim |
| author_sort | Dug Hun Hong |
| collection | DOAJ |
| description | The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities. |
| format | Article |
| id | doaj-art-8321daa2b2984b5dbfaaf1096533b2cc |
| institution | OA Journals |
| issn | 1687-7101 1687-711X |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
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| series | Advances in Fuzzy Systems |
| spelling | doaj-art-8321daa2b2984b5dbfaaf1096533b2cc2025-08-20T02:09:00ZengWileyAdvances in Fuzzy Systems1687-71011687-711X2019-01-01201910.1155/2019/50807235080723Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication OperationsDug Hun Hong0Jae Duck Kim1Department of Mathematics, Myongji University, Yongin, Kyunggido 449-728, Republic of KoreaBangMok College of Basic Studies, Myongji University, Yongin, Kyunggido 449-728, Republic of KoreaThe classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.http://dx.doi.org/10.1155/2019/5080723 |
| spellingShingle | Dug Hun Hong Jae Duck Kim Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations Advances in Fuzzy Systems |
| title | Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations |
| title_full | Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations |
| title_fullStr | Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations |
| title_full_unstemmed | Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations |
| title_short | Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations |
| title_sort | holder type inequalities for sugeno integrals under usual multiplication operations |
| url | http://dx.doi.org/10.1155/2019/5080723 |
| work_keys_str_mv | AT dughunhong holdertypeinequalitiesforsugenointegralsunderusualmultiplicationoperations AT jaeduckkim holdertypeinequalitiesforsugenointegralsunderusualmultiplicationoperations |