{"id":1007,"date":"2022-03-19T09:13:05","date_gmt":"2022-03-19T09:13:05","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1007"},"modified":"2022-03-19T09:57:45","modified_gmt":"2022-03-19T09:57:45","slug":"karamatabudengshi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1007","title":{"rendered":"Karamata\u4e0d\u7b49\u5f0f"},"content":{"rendered":"

Karamata\u4e0d\u7b49\u5f0f\u662f\u51f8\u4f18\u5316\u4e2d\u91cd\u8981\u7684\u4e0d\u7b49\u5f0f\uff0c\u5b83\u63a8\u5e7f\u4e86\u79bb\u6563\u7684Jensen\u4e0d\u7b49\u5f0f\uff0c\u8fd8\u53ef\u8fdb\u4e00\u6b65\u63a8\u5e7f\u5f97\u5230Shur\u51f8\u51fd\u6570\u7684\u6982\u5ff5.
\n

\u5b9a\u7406 1<\/span>.<\/span> \u5047\u8bbe$I$\u662f\u5b9e\u6570\u8f74\u4e0a\u7684\u4e00\u4e2a\u533a\u95f4\u3002\u82e5$\\left\\{ x_i \\right\\}_{i=1}^n$\u4ee5\u53ca$\\left\\{ y_i \\right\\}_{i=1}^n$\u90fd\u662f$I$\u4e0a\u7684\u5b9e\u6570\u5e8f\u5217\uff0c\u6ee1\u8db3:
\n
  1. $x_1\\geq x_2\\geq \\cdots\\geq x_n$\u4ee5\u53ca$y_1\\geq y_2\\geq\\cdots\\geq y_n$;
    \n <\/li>
  2. $\\sum_{i=1}^kx_i\\geq\\sum_{i=1}^ky_i$, \u5bf9\u4efb\u610f\u7684$k=1,2,\\ldots, n$\u90fd\u6210\u7acb;
    \n <\/li>
  3. $\\sum_{i=1}^nx_i=\\sum_{i=1}^ny_i$.
    \n <\/li><\/ol> \u5219\u79f0$\\left\\{ x_i \\right\\}_{i=1}^n$\u4f18\u4e8e<\/span>(majorize)$\\left\\{ y_{i=1}^n \\right\\}$, \u8bb0\u4f5c$\\left\\{ x_i \\right\\}_{i=1}^n\\succ \\left\\{ y_{i} \\right\\}_{i=1}^n$. <\/p>\n

    \u82e5$f:I\\to \\mathbb{R}$\u662f\u4e00\u4e2a\u51f8\u51fd\u6570\u3002\u5373\uff0c$f(\\lambda x+(1-\\lambda)y)\\leq \\lambda f(x)+(1-\\lambda)f(y)$\uff0c\u5bf9\u4efb\u610f\u7684$x,y\\in I$\u90fd\u6210\u7acb\u3002\u5219\uff0c\u5bf9$\\left\\{ x_i \\right\\}_{i=1}^n\\succ \\left\\{ y_{i} \\right\\}_{i=1}^n$, \u6709
    \n\\[
    \n f(x_1)+\\cdots+f(x_n)\\geq f(y_1)+\\cdots+f(y_n).
    \n\\]
    \n<\/div>
    \n
    \n\u6211\u4eec\u9996\u5148\u7ed9\u51fa\u51f8\u51fd\u6570\u7684\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a
    \n

    \u5f15\u7406 2<\/span>.<\/span> \u5047\u8bbe$f:I\\to \\mathbb{R}$\u662f\u51f8\u51fd\u6570\uff0c\u82e5\u5b9a\u4e49$\\Gamma(x,y):=\\frac{f(y)-f(x)}{y-x}$, \u5219\u5bf9\u4efb\u610f\u7684$x,x_1,x_2\\in I$, $x_1\\leq x_2$, \u6709$\\Gamma(x_1,x)\\leq \\Gamma(x_2,x)$.
    \n<\/div>
    \n
    \u8bc1\u660e<\/span>.<\/span> \u53ea\u9700\u5206\u7c7b\u8ba8\u8bba\u3002\u4f8b\u5982\uff0c\u5f53$x_1\\leq x\\leq x_2$\u65f6\uff0c\u6211\u4eec\u6709$x=\\lambda x_1+(1-\\lambda)x_2$, \u4ece\u800c$\\lambda=\\frac{x-x_2}{x_1-x_2}$, $1-\\lambda=\\frac{x_1-x}{x_1-x_2}$. \u5229\u7528\u51f8\u51fd\u6570\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u5f97\u5230
    \n \\[
    \n \\lambda f(x)+(1-\\lambda)f(x)=f(x)\\leq \\lambda f(x_1)+(1-\\lambda)f(x_2),
    \n \\]
    \n \u5373
    \n \\[
    \n \\lambda \\left( f(x)-f(x_1) \\right)\\leq(1-\\lambda)(f(x_2)-f(x)),
    \n \\]
    \n \u4ee3\u5165$\\lambda$\u4ee5\u53ca$1-\\lambda$, \u6574\u7406\u5373\u5f97\u7ed3\u8bba\u3002
    \n<\/div>
    \n\u73b0\u5728\uff0c\u6211\u4eec\u6765\u8bc1\u660e\u5b9a\u7406\u3002\u4e3a\u6b64\uff0c\u4ee4$c_i=\\Gamma(x_i,y_i)$, \u5e76\u4ee4$A_i:=\\sum_{j=1}^ix_i$, $A_0=0$\u4ee5\u53ca$B_i:=\\sum_{j=1}^iy_i$, $B_0=0$. \u5219\u6839\u636e\u5f15\u7406\uff0c\u5e76\u6ce8\u610f\u5230$x_{i+1}\\leq x_i$, $y_{i+1}\\leq y_i$, \u6211\u4eec\u5f97\u5230
    \n\\[
    \n c_{i+1}=\\Gamma(x_{i+1},y_{i+1})\\leq \\Gamma(x_i,y_{i+1})=\\Gamma(y_{i+1},x_i)\\leq\\Gamma(y_i,x_i)=c_i.
    \n\\]
    \n\u4ece\u800c
    \n\\begin{align*}
    \n \\sum_{i=1}^nf(x_i)-\\sum_{i=1}^nf(y_i)&=\\sum_{i=1}^n \\left( f(x_i)-f(y_i) \\right)=\\sum_{i=1}^nc_i(x_i-y_i)\\\\
    \n\t\t\t\t &=\\sum_{i=1}^nc_i\\left( A_i-A_{i-1}-B_i+B_{i-1} \\right)\\\\
    \n\t\t\t\t &=\\sum_{i=1}^nc_i(A_i-B_i)-\\sum_{i=0}^{n-1}c_{i+1}(A_i-B_i)\\\\
    \n\t\t\t\t &=A_n-B_n+\\sum_{i=1}^{n-1}(c_i-c_{i+1})(A_i-B_i)-c_1(A_0-B_0)\\\\
    \n\t\t\t\t &=\\sum_{i=1}^{n-1}(c_i-c_{i+1})(A_i-B_i)\\geq0.
    \n\\end{align*}
    \n\u4f5c\u4e3a\u5e94\u7528\uff0c\u6211\u4eec\u8003\u5bdf\u5982\u4e0b
    \n
    \u4f8b\u5b50 1<\/span>.<\/span> \u8bc1\u660e\u4e0d\u7b49\u5f0f
    \n \\[
    \n (x+y)^p\\geq x^p+y^p,\\quad p\\geq1, x,y\\geq0.
    \n \\]
    \n<\/div>
    \n
    \u8bc1\u660e<\/span>.<\/span> \u8003\u5bdf$I=[0,+\\infty)$\u4e0a\u5230\u51f8\u51fd\u6570$f(t)=t^p$, $p\\geq1$. \u6ce8\u610f\u5230$(1,0)\\succ \\left( \\frac{x}{x+y},\\frac{y}{x+y} \\right)$, \u7531Karamata\u4e0d\u7b49\u5f0f\u5f97\u5230
    \n \\[
    \n f(1)+f(0)\\geq f\\left( \\frac{x}{x+y} \\right)+f\\left( \\frac{y}{x+y} \\right),
    \n \\]
    \n \u5373
    \n \\[
    \n 1\\geq\\left( \\frac{x}{x+y} \\right)^p+\\left( \\frac{y}{x+y} \\right)^p.
    \n \\]
    \n \u53d8\u5f62\u5373\u5f97
    \n \\[
    \n (x+y)^p\\geq x^p+y^p.
    \n \\]
    \n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"

    Karamata\u4e0d\u7b49\u5f0f\u662f\u51f8\u4f18\u5316\u4e2d\u91cd\u8981\u7684\u4e0d\u7b49\u5f0f\uff0c\u5b83\u63a8\u5e7f\u4e86\u79bb\u6563\u7684Jensen\u4e0d\u7b49\u5f0f\uff0c\u8fd8\u53ef\u8fdb\u4e00\u6b65\u63a8\u5e7f\u5f97\u5230Shur\u51f8\u51fd… \u7ee7\u7eed\u9605\u8bfbKaramata\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":[186,187],"class_list":["post-1007","post","type-post","status-publish","format-standard","hentry","category-math","tag-karamata","tag-budengshi","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1007","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=1007"}],"version-history":[{"count":7,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1007\/revisions"}],"predecessor-version":[{"id":1015,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1007\/revisions\/1015"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}