{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:25:26Z","timestamp":1760145926606,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T00:00:00Z","timestamp":1725840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Joint Research Project between China and Serbia","award":["2024-6-7","G2023132005L"],"award-info":[{"award-number":["2024-6-7","G2023132005L"]}]},{"name":"Ministry of Science and Technology of China","award":["2024-6-7","G2023132005L"],"award-info":[{"award-number":["2024-6-7","G2023132005L"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The original Choi\u2013Davis\u2013Jensen\u2019s inequality, known for its extensive applications in various scientific and engineering fields, has inspired researchers to pursue its generalizations. In this study, we extend the Choi\u2013Davis\u2013Jensen\u2019s inequality by introducing a nonlinear map instead of a normalized linear map and generalize the concept of operator convex functions to include any continuous function defined within a compact region. Notably, operators can be matrices with structural symmetry, enhancing the scope and applicability of our results. The Stone\u2013Weierstrass theorem and the Kantorovich function play crucial roles in the formulation and proof of these generalized Choi\u2013Davis\u2013Jensen\u2019s inequalities. Furthermore, we demonstrate an application of this generalized inequality in the context of statistical physics.<\/jats:p>","DOI":"10.3390\/sym16091176","type":"journal-article","created":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T03:04:18Z","timestamp":1725851058000},"page":"1176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Generalized Choi\u2013Davis\u2013Jensen\u2019s Operator Inequalities and Their Applications"],"prefix":"10.3390","volume":"16","author":[{"given":"Shih Yu","family":"Chang","sequence":"first","affiliation":[{"name":"Department of Applied Data Science, San Jose State University, San Jose, CA 95192, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6192-0546","authenticated-orcid":false,"given":"Yimin","family":"Wei","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,9]]},"reference":[{"key":"ref_1","first-page":"345","article-title":"Lower bounds on a generalization of Cesaro operator on time scales","volume":"28","author":"Ahmed","year":"2021","journal-title":"Dyn. Contin. Discret. Impuls. Syst. Ser. Math. Anal"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1186\/s13660-023-02963-9","article-title":"Structure of a generalized class of weights satisfy weighted reverse H\u00f6lder\u2019s inequality","volume":"2023","author":"Saker","year":"2023","journal-title":"J. Inequalities Appl."},{"key":"ref_3","first-page":"565","article-title":"A schwarz inequality for positive linear maps on C*-algebras","volume":"18","author":"Choi","year":"1974","journal-title":"Ill. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1090\/S0002-9939-1957-0084120-4","article-title":"A Schwarz inequality for convex operator functions","volume":"8","author":"Davis","year":"1957","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/0022-247X(89)90325-9","article-title":"Functional differential equations and Jensen\u2019s inequality","volume":"138","author":"Becker","year":"1989","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"052311","DOI":"10.1103\/PhysRevA.79.052311","article-title":"Properties of classical and quantum Jensen-Shannon divergence","volume":"79","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1660","DOI":"10.1109\/TAC.2011.2121410","article-title":"Convergence and equivalence results for the Jensen\u2019s inequality\u2014Application to time-delay and sampled-data systems","volume":"56","author":"Briat","year":"2011","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"052310","DOI":"10.1103\/PhysRevA.72.052310","article-title":"Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states","volume":"72","author":"Majtey","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_9","first-page":"6803","article-title":"Loss function based second-order Jensen inequality and its application to particle variational inference","volume":"34","author":"Futami","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"114533","DOI":"10.1016\/j.cam.2022.114533","article-title":"General tail bounds for random tensors summation: Majorization approach","volume":"416","author":"Chang","year":"2022","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","unstructured":"Mi\u0107i\u0107, J., Moradi, H.R., and Furuichi, S. (2017). Choi-Davis-Jensen\u2019s inequality without convexity. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1405","DOI":"10.1007\/s40995-021-01129-w","article-title":"On Reversing Operator Choi\u2013Davis\u2013Jensen Inequality","volume":"45","author":"Asgari","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1090\/S0002-9939-1959-0113131-7","article-title":"The Stone-Weierstrass theorem","volume":"10","year":"1959","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_14","unstructured":"Pecaric, J., Furuta, T., Hot, J.M., and Seo, Y. (2005). Mond-Pecaric Method in Operator Inequalities, Element Zagreb."},{"key":"ref_15","unstructured":"Fujii, M., and Hot, J.M. (2012). Recent developments of Mond-Pecaric method in operator inequalities. Monographs in Inequalities, Element Zagreb."},{"key":"ref_16","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. 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