{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T12:05:33Z","timestamp":1774440333524,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,23]],"date-time":"2021-04-23T00:00:00Z","timestamp":1619136000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the R\u00e9nyi divergence, the Jeffreys\u2013Tsallis divergence and the Jensen\u2013Shannon\u2013Tsallis divergence.<\/jats:p>","DOI":"10.3390\/e23050514","type":"journal-article","created":{"date-parts":[[2021,4,23]],"date-time":"2021-04-23T12:08:30Z","timestamp":1619179710000},"page":"514","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Refined Young Inequality and Its Application to Divergences"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9929-0954","authenticated-orcid":false,"given":"Shigeru","family":"Furuichi","sequence":"first","affiliation":[{"name":"Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicu\u015for","family":"Minculete","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Transilvania University of Bra\u015fov, 500091 Brasov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1098\/rspa.1912.0076","article-title":"On classes of summable functions and their Fourier series","volume":"87","author":"Young","year":"1912","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_2","first-page":"1","article-title":"Learning with Fenchel-Young Losses","volume":"21","author":"Blondel","year":"2020","journal-title":"J. Mach. Learn. Res."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Nielsen, F. (2021). On Geodesic Triangles with Right Angles in a Dually Flat Space. Progress in Information Geometry: Theory and Applications, Springer.","DOI":"10.1007\/978-3-030-65459-7_7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"213","DOI":"10.2298\/AADM0802213M","article-title":"An equivalent form of Young\u2019s inequality with upper bound","volume":"2","author":"Minguzzi","year":"2008","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_5","unstructured":"Nielsen, F. (2020). The \u03b1-divergences associated with a pair of strictly comparable quasi-arithmetic means. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/j.laa.2005.03.005","article-title":"Interpolating the arithmetic\u2013geometric mean inequality and its operator version","volume":"413","author":"Bhatia","year":"2006","journal-title":"Linear Alg. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s40840-017-0483-y","article-title":"Generalized reverse Young and Heinz inequalities","volume":"42","author":"Furuichi","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1090\/S0002-9939-1978-0476971-2","article-title":"A refinement of the arithmetic mean-geometric mean inequality","volume":"71","author":"Cartwright","year":"1978","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_9","first-page":"452","article-title":"On the arithmetic and geometric means and H\u00f6lder inequality","volume":"9","author":"Kober","year":"1958","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.jmaa.2009.08.059","article-title":"Improved Young and Heinz inequalities for matrix","volume":"361","author":"Kittaneh","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","unstructured":"Bobylev, N.A., and Krasnoselsky, M.A. (1981). Extremum Analysis (Degenerate Cases), Institute of Control Sciences. (In Russian)."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"157","DOI":"10.37193\/CMI.2011.02.02","article-title":"A refinement of the Kittaneh\u2013Manasrah inequality","volume":"20","author":"Minculete","year":"2011","journal-title":"Creat. Math. Inform."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"595","DOI":"10.7153\/jmi-05-51","article-title":"Alternative reverse inequalities for Young\u2019s inequality","volume":"5","author":"Furuichi","year":"2011","journal-title":"J. Math. Inequal."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Furuichi, S., and Moradi, H.R. (2020). Advances in Mathematical Inequalities, De Gruyter.","DOI":"10.1515\/9783110643473"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/18.61115","article-title":"Divergence measures based on the Shannon entropy","volume":"37","author":"Lin","year":"1991","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/BF00537520","article-title":"Information radius","volume":"14","author":"Sibson","year":"1969","journal-title":"Z. Wahrscheinlichkeitstheorie Verw Gebiete"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mitroi-Symeonidis, F.C., Anghel, I., and Minculete, N. (2020). Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data. Symmetry, 12.","DOI":"10.3390\/sym12010022"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Niculescu, C.P., and Persson, L.-E. (2018). Convex Functions and Their Applications, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-78337-6"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4868","DOI":"10.1063\/1.1805729","article-title":"Fundamental properties of Tsallis relative entropy","volume":"45","author":"Furuichi","year":"2004","journal-title":"J. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1103\/PhysRevE.58.1442","article-title":"Generalized entropy-based criterion for consistent testing","volume":"58","author":"Tsallis","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_21","unstructured":"Acz\u00e9l, J., and Dar\u00f3czy, Z. (1975). On Measures of Information and Their Characterizations, Academic Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"121907","DOI":"10.1016\/j.physa.2019.121907","article-title":"Inequalities related to some types of entropies and divergences","volume":"532","author":"Furuichi","year":"2019","journal-title":"Physica A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1016\/j.physa.2011.07.052","article-title":"Mathematical inequalities for some divergences","volume":"391","author":"Furuichi","year":"2012","journal-title":"Physica A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"63","DOI":"10.7153\/jmi-07-06","article-title":"Mathematical inequalities for biparametric extended information measures","volume":"7","author":"Mitroi","year":"2013","journal-title":"J. Math. Ineq."},{"key":"ref_25","first-page":"1079","article-title":"Estimates for Tsallis relative operator entropy","volume":"20","author":"Moradi","year":"2017","journal-title":"Math. Ineq. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1186\/s13660-018-1625-y","article-title":"Zipf-Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities","volume":"2018","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3797","DOI":"10.1109\/TIT.2014.2320500","article-title":"R\u00e9nyi Divergence and Kullback -Leibler Divergence","volume":"60","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"023302","DOI":"10.1063\/1.2165744","article-title":"Information theoretical properties of Tsallis entropies","volume":"47","author":"Furuichi","year":"2006","journal-title":"J. Math. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/5\/514\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:51:57Z","timestamp":1760161917000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/5\/514"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,23]]},"references-count":28,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["e23050514"],"URL":"https:\/\/doi.org\/10.3390\/e23050514","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,23]]}}}