{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T07:50:08Z","timestamp":1781077808326,"version":"3.54.1"},"reference-count":29,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T00:00:00Z","timestamp":1664236800000},"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>Let T\u03f5, 0\u2264\u03f5\u22641\/2, be the noise operator acting on functions on the boolean cube {0,1}n. Let f be a distribution on {0,1}n and let q&gt;1. We prove tight Mrs. Gerber-type results for the second R\u00e9nyi entropy of T\u03f5f which take into account the value of the qth R\u00e9nyi entropy of f. For a general function f on {0,1}n we prove tight hypercontractive inequalities for the \u21132 norm of T\u03f5f which take into account the ratio between \u2113q and \u21131 norms of f.<\/jats:p>","DOI":"10.3390\/e24101376","type":"journal-article","created":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T23:12:12Z","timestamp":1664320332000},"page":"1376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber\u2019s-Type Inequalities for the Second R\u00e9nyi Entropy"],"prefix":"10.3390","volume":"24","author":[{"given":"Niv","family":"Levhari","sequence":"first","affiliation":[{"name":"School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 9103401, Israel"},{"name":"School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alex","family":"Samorodnitsky","sequence":"additional","affiliation":[{"name":"School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 9103401, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"159","DOI":"10.2307\/1970980","article-title":"Inequalities in Fourier Analysis","volume":"102","author":"Beckner","year":"1975","journal-title":"Ann. 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