{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:44:36Z","timestamp":1740149076693,"version":"3.37.3"},"reference-count":21,"publisher":"Wiley","license":[{"start":{"date-parts":[[2019,1,3]],"date-time":"2019-01-03T00:00:00Z","timestamp":1546473600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2017R1D1A1B03027869"],"award-info":[{"award-number":["2017R1D1A1B03027869"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Fuzzy Systems"],"published-print":{"date-parts":[[2019,1,3]]},"abstract":"<jats:p>The classical H\u00f6lder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes H\u00f6lder type inequalities and reverse H\u00f6lder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>S<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy=\"false\">\u222b<\/mml:mo><mml:mrow><mml:mn mathvariant=\"normal\">0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mrow><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mi>d<\/mml:mi><mml:mi>\u03bc<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>S<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy=\"false\">\u222b<\/mml:mo><mml:mrow><mml:mn mathvariant=\"normal\">0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mrow><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mi>d<\/mml:mi><mml:mi>\u03bc<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u2264<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mi>q<\/mml:mi><mml:mo>\/<\/mml:mo><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>-<\/mml:mo><mml:msup><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo>\u2228<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>\/<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>-<\/mml:mo><mml:msup><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>S<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy=\"false\">\u222b<\/mml:mo><mml:mrow><mml:mn mathvariant=\"normal\">0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mrow><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>\u03bc<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:mo>,<\/mml:mo><\/mml:math> where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>&lt;<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>&lt;<\/mml:mo><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><mml:mo>,<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>q<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>\u03bc<\/mml:mi><\/mml:mrow><\/mml:math> is the Lebesgue measure on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><mml:mo>,<\/mml:mo><\/mml:math> holds if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><\/mml:math> 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.<\/jats:p>","DOI":"10.1155\/2019\/5080723","type":"journal-article","created":{"date-parts":[[2019,1,3]],"date-time":"2019-01-03T18:31:02Z","timestamp":1546540262000},"page":"1-10","source":"Crossref","is-referenced-by-count":1,"title":["H\u00f6lder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations"],"prefix":"10.1155","volume":"2019","author":[{"given":"Dug Hun","family":"Hong","sequence":"first","affiliation":[{"name":"Department of Mathematics, Myongji University, Yongin, Kyunggido 449-728, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5313-6807","authenticated-orcid":true,"given":"Jae Duck","family":"Kim","sequence":"additional","affiliation":[{"name":"BangMok College of Basic Studies, Myongji University, Yongin, Kyunggido 449-728, Republic of Korea"}]}],"member":"311","reference":[{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(80)90101-8"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5303-5"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2009.08.006"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.01.045"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2008.06.027"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2007.02.006"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.07.066"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2007.04.072"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2007.02.143"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2007.05.032"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2008.04.002"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2007.11.044"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijar.2008.01.004"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2010.05.071"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2011.07.046"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488514500111"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488516500033"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2014.12.056"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2014.11.016"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2010.04.017"},{"year":"1998","key":"2"}],"container-title":["Advances in Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2019\/5080723.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2019\/5080723.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2019\/5080723.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,1,3]],"date-time":"2019-01-03T18:31:04Z","timestamp":1546540264000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/afs\/2019\/5080723\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,3]]},"references-count":21,"alternative-id":["5080723","5080723"],"URL":"https:\/\/doi.org\/10.1155\/2019\/5080723","relation":{},"ISSN":["1687-7101","1687-711X"],"issn-type":[{"type":"print","value":"1687-7101"},{"type":"electronic","value":"1687-711X"}],"subject":[],"published":{"date-parts":[[2019,1,3]]}}}