{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:24:42Z","timestamp":1778495082412,"version":"3.51.4"},"reference-count":30,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T00:00:00Z","timestamp":1684454400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation","award":["61772140"],"award-info":[{"award-number":["61772140"]}]},{"name":"National Natural Science Foundation","award":["202103010004"],"award-info":[{"award-number":["202103010004"]}]},{"name":"National Natural Science Foundation","award":["2020KTSCX088"],"award-info":[{"award-number":["2020KTSCX088"]}]},{"name":"Science and Technology Projects in Guangzhou","award":["61772140"],"award-info":[{"award-number":["61772140"]}]},{"name":"Science and Technology Projects in Guangzhou","award":["202103010004"],"award-info":[{"award-number":["202103010004"]}]},{"name":"Science and Technology Projects in Guangzhou","award":["2020KTSCX088"],"award-info":[{"award-number":["2020KTSCX088"]}]},{"name":"Characteristic Innovation Project of Guangdong Provincial Colleges and Universities","award":["61772140"],"award-info":[{"award-number":["61772140"]}]},{"name":"Characteristic Innovation Project of Guangdong Provincial Colleges and Universities","award":["202103010004"],"award-info":[{"award-number":["202103010004"]}]},{"name":"Characteristic Innovation Project of Guangdong Provincial Colleges and Universities","award":["2020KTSCX088"],"award-info":[{"award-number":["2020KTSCX088"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Using weight functions and parameters, as well as applying real analytic techniques, we derive a new Hardy\u2013Hilbert-type integral inequality with the homogeneous kernel 1(x+y)\u03bb+n involving one multiple upper limit function and one derivative function of higher order. Certain equivalent statements of the optimal constant factor related to some parameters are considered. A few particular inequalities and the case of reverses are also provided.<\/jats:p>","DOI":"10.3390\/axioms12050499","type":"journal-article","created":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T10:08:55Z","timestamp":1684490935000},"page":"499","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A New Hardy\u2013Hilbert-Type Integral Inequality Involving One Multiple Upper Limit Function and One Derivative Function of Higher Order"],"prefix":"10.3390","volume":"12","author":[{"given":"Bicheng","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael Th.","family":"Rassias","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece"},{"name":"Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,19]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. (1934). Inequalities, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Yang, B.C. (2009). The Norm of Operator and Hilbert-Type Inequalities, Science Press.","DOI":"10.2174\/97816080505501090101"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yang, B.C. (2009). Hilbert-Type Integral Inequalities, Bentham Science Publishers Ltd.","DOI":"10.2174\/97816080505501090101"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.jmaa.2005.07.071","article-title":"On the norm of an integral operator and applications","volume":"321","author":"Yang","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","first-page":"63","article-title":"Hardy-Hilbert\u2019s inequalities with two parameters","volume":"36","author":"Xu","year":"2007","journal-title":"Adv. Math."},{"key":"ref_6","first-page":"391","article-title":"A new Hilbert-type inequality with the homogeneous kernel of degree-2","volume":"12","author":"Xie","year":"2013","journal-title":"Adv. Appl. Math."},{"key":"ref_7","first-page":"11","article-title":"A new Hilbert-type inequality with the homogeneous kernel of degree-2 and with the integral","volume":"3","author":"Zeng","year":"2014","journal-title":"Bull. Math. Sci. Appl."},{"key":"ref_8","first-page":"70","article-title":"A Hilbert-type integral inequality with the homogeneous kernel of zero degree","volume":"30","author":"Xin","year":"2010","journal-title":"Math. Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1186\/1029-242X-2013-452","article-title":"The connection between Hilbert and Hardy inequalities","volume":"2013","author":"Azar","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"263","DOI":"10.7153\/mia-20-20","article-title":"Sharp bounds for m-linear Hilbert-type operators on the weighted Morrey spaces","volume":"20","author":"Batbold","year":"2017","journal-title":"Math. Inequalities Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1215\/17358787-3495561","article-title":"Multiple Hilbert-type inequalities involving some differential operators","volume":"10","author":"Adiyasuren","year":"2016","journal-title":"Banach J. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111","DOI":"10.7153\/mia-18-07","article-title":"Hilbert\u2013type inequalities involving differential operators, the best constants and applications","volume":"18","author":"Adiyasuren","year":"2015","journal-title":"Math. Inequalities Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"379","DOI":"10.7153\/jmi-2018-12-28","article-title":"A new form of Hilbert integral inequality","volume":"12","author":"Batbold","year":"2018","journal-title":"Math. Inequalities Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.jmaa.2005.11.069","article-title":"Extension of Hilbert\u2019s inequality","volume":"324","author":"Krnic","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1186\/s13660-019-2087-6","article-title":"A new discrete Hilbert-type inequality involving partial sums","volume":"2019","author":"Adiyasuren","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s13660-019-2280-7","article-title":"On a new Hilbert-type integral inequality involving the upper limit functions","volume":"2020","author":"Mo","year":"2020","journal-title":"J. Inequalities Appl."},{"key":"ref_17","first-page":"329","article-title":"A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor","volume":"37","author":"Hong","year":"2016","journal-title":"Ann. Math."},{"key":"ref_18","first-page":"189","article-title":"On the structure character of Hilbert\u2019s type integral inequality with homogeneous kernel and applications","volume":"55","author":"Hong","year":"2017","journal-title":"J. Jilin Univ."},{"key":"ref_19","first-page":"2691816","article-title":"Equivalent property of a Hilbert-type integral inequality related to the beta function in the whole plane","volume":"2018","author":"Xin","year":"2018","journal-title":"J. Funct. Spaces"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liao, J.Q., Wu, S.H., and Yang, B.C. (2020). On a new half-discrete Hilbert-type inequality involving the variable upper limit integral and the partial sum. Mathematics, 8.","DOI":"10.3390\/math8020229"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1186\/s13660-021-02593-z","article-title":"Conditions for the validity of a class of optimal Hilbert-type multiple integral inequalities with non-homogeneous","volume":"2021","author":"He","year":"2021","journal-title":"J. Inequalities Appl."},{"key":"ref_22","first-page":"7414861","article-title":"Equivalent parameter conditions for the validity of half-discrete Hilbert-type multiple integral inequality with generalized homogeneous kernel","volume":"2020","author":"Chen","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1515\/math-2021-0023","article-title":"The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of non-homogeneous kernels and their applications","volume":"19","author":"He","year":"2021","journal-title":"Open Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/s43034-020-00087-5","article-title":"The parameter conditions for the existence of the Hilbert-type multiple integral inequality and its best constant factor","volume":"12","author":"Hong","year":"2020","journal-title":"Ann. Funct. Anal."},{"key":"ref_25","unstructured":"Hong, Y. (2020). Progress in the Study of Hilbert-Type Integral Inequalities from Homogeneous Kernels to Non-Homogeneous Kernels. J. Guangdong Univ. Educ."},{"key":"ref_26","first-page":"779","article-title":"The best matching parameters for semi-discrete Hilbert-type inequality with quasi-homogeneous kernel","volume":"34","author":"Hong","year":"2021","journal-title":"Math. Appl."},{"key":"ref_27","first-page":"252","article-title":"The optimal matching parameter of half-discrete Hilbert-type multiple integral inequalities with non-homogeneous kernels and applications","volume":"36","author":"Hong","year":"2021","journal-title":"Chin. Q. J. Math."},{"key":"ref_28","unstructured":"Wang, Z.X., and Guo, D.R. (1979). Introduction to Special Functions, Science Press."},{"key":"ref_29","unstructured":"Kuang, J.C. (2004). Applied Inequalities, Shangdong Science and Technology Press."},{"key":"ref_30","unstructured":"Kuang, J.C. (2015). Real and Functional Analysis (Continuation), Higher Education Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/499\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:38:36Z","timestamp":1760125116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/499"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,19]]},"references-count":30,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050499"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050499","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,19]]}}}