{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T16:41:14Z","timestamp":1764175274577,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,19]],"date-time":"2022-08-19T00:00:00Z","timestamp":1660867200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["11688101","2018YFA0704705","Z190004","2021NTST32"],"award-info":[{"award-number":["11688101","2018YFA0704705","Z190004","2021NTST32"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NKRDP","award":["11688101","2018YFA0704705","Z190004","2021NTST32"],"award-info":[{"award-number":["11688101","2018YFA0704705","Z190004","2021NTST32"]}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["11688101","2018YFA0704705","Z190004","2021NTST32"],"award-info":[{"award-number":["11688101","2018YFA0704705","Z190004","2021NTST32"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["11688101","2018YFA0704705","Z190004","2021NTST32"],"award-info":[{"award-number":["11688101","2018YFA0704705","Z190004","2021NTST32"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper studies the properties of the derivatives of differential entropy H(Xt) in Costa\u2019s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m\u22651, (\u22121)m+1(dm\/dtm)H(Xt)\u22650, while McKean conjectured a stronger statement, whereby (\u22121)m+1(dm\/dtm)H(Xt)\u2265(\u22121)m+1(dm\/dtm)H(XGt). Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: C1(m,n):(\u22121)m+1(dm\/dtm)H(Xt)\u22650, where n denotes that Xt is a random vector taking values in Rn, and similarly, C2(m,n):(\u22121)m+1(dm\/dtm)H(Xt)\u2265(\u22121)m+1(dm\/dtm)H(XGt)\u22650. In this paper, we prove some new multivariate cases: C1(3,i),i=2,3,4. Motivated by our results, we further propose a weaker version of McKean\u2019s conjecture C3(m,n):(\u22121)m+1(dm\/dtm)H(Xt)\u2265(\u22121)m+11n(dm\/dtm)H(XGt), which is implied by C2(m,n) and implies C1(m,n). We prove some multivariate cases of this conjecture under the log-concave condition: C3(3,i),i=2,3,4 and C3(4,2). A systematic procedure to prove Cl(m,n) is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.<\/jats:p>","DOI":"10.3390\/e24081155","type":"journal-article","created":{"date-parts":[[2022,8,21]],"date-time":"2022-08-21T22:23:13Z","timestamp":1661120593000},"page":"1155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1537-3689","authenticated-orcid":false,"given":"Laigang","family":"Guo","sequence":"first","affiliation":[{"name":"Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3349-533X","authenticated-orcid":false,"given":"Chun-Ming","family":"Yuan","sequence":"additional","affiliation":[{"name":"KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Shan","family":"Gao","sequence":"additional","affiliation":[{"name":"KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communications","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. 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