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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: initBoardView

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Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Line: 315
Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Line: 315
Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: initBoardView

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Line: 315
Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: initBoardView

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Line: 315
Function: require_once

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Filename: libraries/Common.php

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: initBoardView

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Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: initBoardView

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Line: 315
Function: require_once

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Filename: libraries/Common.php

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: initBoardView

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Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 585
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 425
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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[수학 정보] 헬더의 부등식
△π 2020.07.14 20:11 조회 1095
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헬더의 부등식!

헬더의 부등식은, 코시-슈바르츠 부등식의 일반화된 형태로,

{a_i >0\;\;b_i >0\;(i=1,2,3,\cdots,n)}이고,{p>0,q>0,\;\frac{1}{p}+\frac{1}{q}=1}일 때,

\sum_{i=1}^{n} a_i b_i\leq\bigg(\sum_{i=1}^{n}{a_i}^p\bigg)^{\frac{1}{p}}\bigg(\sum_{i=1}^{n}{b_i}^q\bigg)^{\frac{1}{q}}가 성립한다는 부등식입니다!

여기서 p=q=2를 집어넣으면 코시-슈바르츠의 부등식이 되죠.

이를 증명해보겠습니다!

0<1/p<1이므로, f(x)=x^1/p는 위로 볼록한 함수입니다. 따라서 젠센부등식에 의해서

{\lambda_i >0,\;\;\sum_{i=1}^{n}\lambda_i =1}이고 xi가 양수일 때,

\sum_{i=1}^{n}\lambda_i {x_i}^{\frac{1}{p}}\leq\bigg(\sum_{i=1}^{n}{\lambda_i}x_i\bigg)^{\frac{1}{p}}이 성립합니다!

 

이제,i=1,2,3,4...,n에 대해 

{{a_i}^p =A_i,\;\;a_i={A_i}^{\frac{1}{p}}\;\;\sum_{i=1}^{n}{a_i}^p =\sum_{i=1}^{n} A_i =A}

{{b_i}^q =B_i,\;\;b_i={B_i}^{\frac{1}{q}}=B_i {B_i}^{-\frac{1}{p}}\;\;\sum_{i=1}^{n}{b_i}^p =\sum_{i=1}^{n} B_i =B}

{\lambda_i =\frac{B_i}{B}}

이라 하면, {\lambda_i >0,\;\sum_{i=1}^{n}{\lambda=1}}입니다. 따라서 이를 통해 헬더의 부등식을 증명할 수 있습니다. 어떻게요? 바로 이렇게요!

\sum_{i=1}^{n} a_i b_i=\sum_{i=1}^{n} {A_i}^{\frac{1}{p}} {B_i}^{1-\frac{1}{p}}=\sum_{i=1}^{n}B_i \bigg(\frac{A_i}{B_i}\bigg)^{\frac{1}{p}}=B\sum_{i=1}^{n}\lambda_i \bigg(\frac{A_i}{B_i}\bigg)^{\frac{1}{p}}\leq B\bigg(\sum_{i=1}^{n}\lambda_i \frac{A_i}{B_i}\bigg)^{\frac{1}{p}}

=B\bigg(\sum_{i=1}^{n} \frac{A_i}{B}\bigg)^{\frac{1}{p}}=B^{1-\frac{1}{p}}\bigg(\sum_{i=1}^{n}A_i\bigg)^{\frac{1}{p}}=A^{\frac{1}{p}}B^{\frac{1}{q}}=\bigg(\sum_{i=1}^{n}{a_i}^p\bigg)^{\frac{1}{p}}\bigg(\sum_{i=1}^{n}{b_i}^q\bigg)^{\frac{1}{q}}

등호는 {a_i}^{p}=\lambda{b_i}^{q}일 때 성립합니다.

 

네, 이로서 헬더의 부등식 증명이 끝났습니다!

이걸 왜 증명했냐고요?

앞으로 올라올 '민코프스키 부등식 증명' 글을 올리기 위해서 꼭 필요한 증명이기 때문입니다!

 

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