{"id":1708,"date":"2023-06-10T16:59:08","date_gmt":"2023-06-10T08:59:08","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1708"},"modified":"2023-06-10T17:44:26","modified_gmt":"2023-06-10T09:44:26","slug":"jensenbudengshi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1708","title":{"rendered":"Jensen\u4e0d\u7b49\u5f0f"},"content":{"rendered":"

\u56de\u5fc6\uff0c\u5b9a\u4e49\u5728\u533a\u95f4$I=(a,b)$\u4e0a\u7684\u51fd\u6570$\\varphi$\u79f0\u4e3a\u51f8\u51fd\u6570<\/span>, \u5982\u679c\u5bf9\u4efb\u610f\u7684$a < x < b$, $a < y < b$, \u4ee5\u53ca\u4efb\u610f\u7684$0\\leq\\lambda\\leq1$, \u6210\u7acb\u5982\u4e0b\u4e0d\u7b49\u5f0f\n\\begin{equation}\n \\varphi\\left( (1-\\lambda)x+\\lambda y \\right)\\leq (1-\\lambda)\\varphi(x)+\\lambda\\varphi(y).\n \\label{eq:convex}\n\\end{equation}\n\u4ece\u56fe\u5f62\u4e0a\uff0c \u5047\u8bbe$a < s < t < u < b$, \u4ee4\n$t=(1-\\lambda)s+\\lambda u$, \u5219$\\lambda= \\frac{t-s}{u-s}$, $1-\\lambda= \\frac{u-t}{u-s}$, \u4ece\u800c\\eqref{eq:convex}\u5f97\u5230\n\\[\n \\varphi(t)\\leq (1-\\lambda)\\varphi(s)+\\lambda \\varphi(u)\\iff (1-\\lambda)(\\varphi(t)-\\varphi(s))\\leq \\lambda \\left( \\varphi(u)-\\varphi(t) \\right),\n\\]\n\u6545\n\\[\n \\frac{\\varphi(t)-\\varphi(s)}{t-s}\\leq \\frac{\\varphi(u)-\\varphi(t)}{u-t}.\n\\]\n
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\u5b9a\u7406 1<\/span> (Jensen\u4e0d\u7b49\u5f0f<\/span>).<\/span> \u5047\u8bbe$\\mu$\u662f\u533a\u57df$\\Omega$\u4e0a\u7684Lebesuge\u6d4b\u5ea6\uff0c$f\\in L^1(\\Omega;\\mu)$\u662f$\\Omega$\u4e0a\u7684\u5b9e\u53ef\u79ef\u51fd\u6570\u3002\u5982\u679c$a < f(x) < b$, \u5bf9\u4efb\u610f\u7684$x\\in\\Omega$\u90fd\u6210\u7acb\uff1b\u90a3\u4e48\u5bf9\u4efb\u610f\u5728\u533a\u95f4$I=(a,b)$\u4e0a\u7684\u51f8\u51fd\u6570$\\varphi$, \u90fd\u6709\n \\[\n \\varphi\\left( \\frac{1}{ \\lvert \\Omega \\rvert}\\int_\\Omega fd\\mu \\right)\\leq \\frac{1}{ \\lvert \\Omega \\rvert}\\int_\\Omega (\\varphi\\circ f)d\\mu.\n \\]\n<\/div>\n
\u8bc1\u660e<\/span>.<\/span> \u4ee4$t= \\frac{1}{ \\lvert \\Omega \\rvert}\\int_\\Omega fd\\mu$, \u5219$a< t< b$. \u5b9a\u4e49\n \\[\n \\beta:=\\sup_{a< s< t} \\frac{\\varphi(t)-\\varphi(s)}{t-s},\n \\]\n \u5219\u5bf9\u4efb\u610f\u7684$u$, \u6ee1\u8db3$t< u< b$, \u6211\u4eec\u6709\n \\[\n \\beta\\leq \\frac{\\varphi(u)-\\varphi(t)}{u-t}\\implies\n \\varphi(u)\\geq \\varphi(t)+\\beta(u-t).\n \\]\n \u53e6\u4e00\u65b9\u9762\uff0c\u6309\u7167$\\beta$\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u77e5\u9053, \u5bf9\u4efb\u610f\u7684$a< s< t$, \u6709\n \\[\n \\frac{\\varphi(t)-\\varphi(s)}{t-s}\\leq \\beta\\implies\n \\varphi(s)\\geq \\varphi(t)+\\beta(s-t).\n \\]\n \u6545\u6211\u4eec\u5f97\u5230\u5bf9\u4efb\u610f\u7684$a< s< b$, \u6210\u7acb\n \\[\n \\varphi(s)\\geq \\varphi(t)+\\beta(s-t).\n \\]\n\n \u7279\u522b\u5730\uff0c\u4ee4$s=f(x)$, \u6211\u4eec\u5f97\u5230\n \\[\n \\varphi(f(x))\\geq\\varphi\\left( \\frac{1}{ \\lvert \\Omega \\rvert}\\int_\\Omega fd\\mu \\right)+\\beta\\left( f(x)- \\frac{1}{ \\lvert \\Omega \\rvert}\\int_\\Omega fd\\mu \\right).\n \\]\n \u4e24\u8fb9\u540c\u65f6\u5728$\\Omega$\u4e0a\u79ef\u5206\u5f97\u5230\n \\[\n \\int_{\\Omega}\\varphi(f(x))d\\mu\\geq \\lvert \\Omega \\rvert \\cdot \\varphi\\left( \\frac{1}{ \\lvert \\Omega \\rvert}\\int_\\Omega fd\\mu \\right),\n \\]\n \u53d8\u5f62\u8bb0\u5f97\u5230\u6b32\u8bc1\u4e4b\u4e0d\u7b49\u5f0f\u3002\n<\/div>\n\u8fd9\u6bb5\u6750\u6599\u6765\u6e90\u4e8e[1<\/a>].\n<\/p>\n
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\u53c2\u8003\u6587\u732e<\/div>\n\t
  1. W. Rudin<\/span>, Real and complex analysis<\/a>, Third<\/span>, McGraw-Hill Book Co., New York<\/span>, 1987<\/span>. xiv+416<\/span>. MR924157<\/a><\/li><\/ol><\/div>","protected":false},"excerpt":{"rendered":"

    \u56de\u5fc6\uff0c\u5b9a\u4e49\u5728\u533a\u95f4$I=(a,b)$\u4e0a\u7684\u51fd\u6570$\\varphi$\u79f0\u4e3a\u51f8\u51fd\u6570, \u5982\u679c\u5bf9\u4efb\u610f\u7684$a < x < b$,… \u7ee7\u7eed\u9605\u8bfbJensen\u4e0d\u7b49\u5f0f<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[322,187],"class_list":["post-1708","post","type-post","status-publish","format-standard","hentry","category-math","tag-jensen","tag-budengshi","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1708","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1708"}],"version-history":[{"count":27,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1708\/revisions"}],"predecessor-version":[{"id":1735,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1708\/revisions\/1735"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1708"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1708"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}