{"id":654,"date":"2013-02-24T13:44:30","date_gmt":"2013-02-24T04:44:30","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=654"},"modified":"2021-09-10T19:06:40","modified_gmt":"2021-09-10T10:06:40","slug":"osr200702","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr200702\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2007\uff1a\u7b2c2\u554f"},"content":{"rendered":"
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\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(x\\) \u304c\u6b63\u306e\u6570\u306e\u3068\u304d \\(| \\log x | \\leqq \\dfrac{|x-1|}{\\sqrt{x}}\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(p , q , r\\) \u304c \\(p+q+r = 1\\) \u3092\u6e80\u305f\u3059\u6b63\u306e\u6570\u306e\u3068\u304d\r\n\\[\r\np^2+q^2+r^2 \\geqq \\dfrac{1}{3}\r\n\\]\r\n\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a , b , c\\) \u304c\u76f8\u7570\u306a\u308b\u6b63\u306e\u6570\u3067, \\(\\sqrt{a} +\\sqrt{b} +\\sqrt{c} = 1\\) \u3092\u6e80\u305f\u3059\u3068\u304d\r\n\\[\r\n\\dfrac{ab}{b-a} \\log \\dfrac{b}{a} +\\dfrac{bc}{c-b} \\log \\dfrac{c}{b} +\\dfrac{ca}{a-c} \\log \\dfrac{a}{c} \\leqq \\dfrac{1}{3}\r\n\\]\r\n\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(x \\geqq 1\\) \u306e\u3068\u304d
      \r\n\\(f(x) = \\sqrt{x} -\\dfrac{1}{\\sqrt{x}} -\\log x\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'(x) & = \\dfrac{1}{2 \\sqrt{x}} +\\dfrac{1}{2 x^{\\frac{3}{2}}} -\\dfrac{1}{x} \\\\\r\n& = \\dfrac{x -2\\sqrt{x} +1}{2 x^{\\frac{3}{2}}} \\\\\r\n& = \\dfrac{\\left( 1 -\\sqrt{x} \\right)^2}{2 x^{\\frac{3}{2}}} \\geqq 0 \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306f \\(x \\geqq 1\\) \u306b\u304a\u3044\u3066\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a\r\n\\[\\begin{align}\r\nf(x) & \\geqq f(1) = 0 \\\\\r\n\\text{\u2234} \\quad \\sqrt{x} -\\dfrac{1}{\\sqrt{x}} & -\\log x \\geqq 0 \\\\\r\n\\text{\u2234} \\quad \\log x & \\leqq \\dfrac{x-1}{\\sqrt{x}} \\ .\r\n\\end{align}\\]\r\n\\(\\log x \\geqq 0\\) , \\(x-1 \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\left| \\log x \\right| \\leqq \\dfrac{| x-1 |}{\\sqrt{x}} \\ .\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(0 \\lt x \\lt 1\\) \u306e\u3068\u304d
      \r\n\\(x= \\dfrac{1}{t}\\) \u3068\u304a\u304f\u3068, \\(t \\gt 1\\) \u306a\u306e\u3067 1*<\/strong> \u306e\u3068\u304d\u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\left| \\log t \\right| \\leqq \\dfrac{| t-1 |}{\\sqrt{t}} \\ .\r\n\\]\r\n\\(t =\\dfrac{1}{x}\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\left| \\log \\dfrac{1}{x} \\right| & \\leqq \\dfrac{\\left| \\frac{1}{x}-1 \\right|}{\\sqrt{\\frac{1}{x}}} \\\\\r\n\\left| -\\log x \\right| & \\leqq \\dfrac{| 1-x |}{\\sqrt{x}} \\\\\r\n\\text{\u2234} \\quad \\left| \\log x \\right| & \\leqq \\dfrac{| x-1 |}{\\sqrt{x}} \\ .\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u6b63\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\left| \\log x \\right| \\leqq \\dfrac{| x-1 |}{\\sqrt{x}} \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n

      (2)<\/strong><\/p>\r\n

      \\[\\begin{align}\r\n( 1^2 +1^2 +1^2 )( p^2 +q^2 +r^2 ) & \\geqq ( 1 \\cdot p +1 \\cdot q +1 \\cdot r )^2 \\\\\r\n& = 1^2 = 1 \\quad ( \\ \\text{\u2235} \\ p+q+r = 1 ) \\\\\r\n\\text{\u2234} \\quad p^2 +q^2 +r^2 & \\geqq \\dfrac{1}{3} \\ .\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(\\dfrac{p}{1} = \\dfrac{q}{1} = \\dfrac{r}{1}\\) \u3059\u306a\u308f\u3061 \\(p = q = r = \\dfrac{1}{3}\\) \u306e\u3068\u304d.<\/p>\r\n

      (3)<\/strong><\/p>\r\n

      (1)<\/strong> \u306e\u7d50\u679c\u306b\u5bfe\u3057\u3066, \\(x = \\dfrac{b}{a} \\gt 0\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\left| \\log \\dfrac{b}{a} \\right| & \\leqq \\dfrac{\\left| \\frac{b}{a} -1 \\right|}{\\sqrt{\\frac{b}{a}}} \\\\\r\n\\left| \\log \\dfrac{b}{a} \\right| & \\leqq \\dfrac{|b-a|}{\\sqrt{ab}} \\\\\r\n\\left| \\dfrac{ab}{b-a} \\log \\dfrac{b}{a} \\right| & \\leqq \\sqrt{ab} \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(b-a\\) \u3068 \\(\\log \\dfrac{b}{a}\\) \u306e\u6b63\u8ca0\u306f\u4e00\u81f4\u3059\u308b\u306e\u3067\r\n\\[\r\n\\dfrac{ab}{b-a} \\log \\dfrac{b}{a} \\geqq 0 \\ .\r\n\\]\r\n\u306a\u306e\u3067, [1] \u3088\u308a\r\n\\[\r\n\\dfrac{ab}{b-a} \\log \\dfrac{b}{a} \\leqq \\sqrt{ab} \\quad ... [2] \\ .\r\n\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(a = b\\) \u306e\u3068\u304d.\r\n\\(x = \\dfrac{c}{b} , \\ \\dfrac{a}{c}\\) \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{bc}{c-b} \\log \\dfrac{c}{b} & \\leqq \\sqrt{bc} \\quad ... [3] , \\\\\r\n\\dfrac{ca}{a-c} \\log \\dfrac{a}{c} & \\leqq \\sqrt{ca} \\quad ... [4] \\ .\r\n\\end{align}\\]\r\n[2] \uff5e [4] \u306e\u8fba\u3005\u3092\u52a0\u3048\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{ab}{b-a} \\log \\dfrac{b}{a} +\\dfrac{bc}{c-b} \\log \\dfrac{c}{b} & +\\dfrac{ca}{a-c} \\log \\dfrac{a}{c} \\\\\r\n& \\leqq \\sqrt{ab} +\\sqrt{bc} +\\sqrt{ca} \\quad ... [5] \\ .\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f \\(a = b = c\\) \u306e\u3068\u304d.
      \r\n\u3055\u3089\u306b, \\(\\sqrt{a} +\\sqrt{b} +\\sqrt{c} = 1\\) \u3068 (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\na+b+c \\geqq \\dfrac{1}{3} \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\sqrt{ab} +\\sqrt{bc} +\\sqrt{ca} & = \\dfrac{\\left( \\sqrt{a} +\\sqrt{b} +\\sqrt{c} \\right)^2 -(a+b+c)}{2} \\\\\r\n& = \\dfrac{1 -(a+b+c)}{2} \\\\\r\n& \\leqq \\dfrac{1 -\\frac{1}{3}}{2} =\\dfrac{1}{3} \\quad ... [6] \\ .\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(a = b = c\\) \u306e\u3068\u304d.
      \r\n\u3088\u3063\u3066, [5] [6] \u3088\u308a\r\n\\[\r\n\\dfrac{ab}{b-a} \\log \\dfrac{b}{a} +\\dfrac{bc}{c-b} \\log \\dfrac{c}{b} +\\dfrac{ca}{a-c} \\log \\dfrac{a}{c} \\leqq \\dfrac{1}{3} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(x\\) \u304c\u6b63\u306e\u6570\u306e\u3068\u304d \\(| \\log x | \\leqq \\dfrac{|x-1|}{\\sqrt{x}}\\) \u3092\u793a\u305b. (2)\u3000\\(p , q , r\\) \u304c \\(p+q+r = 1\\ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[89],"tags":[142,109],"class_list":["post-654","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2007","tag-osaka_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/654","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=654"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/654\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}