{"title":"关于汉德尔曼的一些凸性问题","authors":"Brian Simanek","doi":"10.4153/s0008439524000316","DOIUrl":null,"url":null,"abstract":"<p>We resolve some questions posed by Handelman in 1996 concerning log convex <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$L^1$</span></span></img></span></span> functions. In particular, we give a negative answer to a question he posed concerning the integrability of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$h^2(x)/h(2x)$</span></span></img></span></span> when <span>h</span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$L^1$</span></span></img></span></span> and log convex and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$h(n)^{1/n}\\rightarrow 1$</span></span></img></span></span>.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"2012 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On some convexity questions of Handelman\",\"authors\":\"Brian Simanek\",\"doi\":\"10.4153/s0008439524000316\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We resolve some questions posed by Handelman in 1996 concerning log convex <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$L^1$</span></span></img></span></span> functions. In particular, we give a negative answer to a question he posed concerning the integrability of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$h^2(x)/h(2x)$</span></span></img></span></span> when <span>h</span> is <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$L^1$</span></span></img></span></span> and log convex and <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529095608048-0637:S0008439524000316:S0008439524000316_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$h(n)^{1/n}\\\\rightarrow 1$</span></span></img></span></span>.</p>\",\"PeriodicalId\":501184,\"journal\":{\"name\":\"Canadian Mathematical Bulletin\",\"volume\":\"2012 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Canadian Mathematical Bulletin\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4153/s0008439524000316\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Mathematical Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008439524000316","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We resolve some questions posed by Handelman in 1996 concerning log convex $L^1$ functions. In particular, we give a negative answer to a question he posed concerning the integrability of $h^2(x)/h(2x)$ when h is $L^1$ and log convex and $h(n)^{1/n}\rightarrow 1$.
$L^1$ functions. In particular, we give a negative answer to a question he posed concerning the integrability of 
$h^2(x)/h(2x)$ when h is 
$L^1$ and log convex and 
$h(n)^{1/n}\rightarrow 1$.
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