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[수학 정보] 말러의 부등식
△π 2020.07.12 16:49 조회 696
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

원래 민코프스키 부등식을 증명하고 싶었지만...

너무 쌩 노가다가 될 것 같아서,

민코프스키 부등식의 대수적 형태를 곱형태로 변형시킨 '말러의 부등식'을 증명하겠습니다!

양의 정수 x_1,\cdots,x_n과 y_1,\cdots,y_n에 대해 다음과 같은 부등식이 성립합니다.

\prod_{k=1}^n (x_k + y_k)^{1/n} \ge \prod_{k=1}^n x_k^{1/n} + \prod_{k=1}^n y_k^{1/n}

이를 증명해보죠.

 

사실 이건 되게 쉽게 증명됩니다.

그래서 날로 먹으려고 이걸 선택한거죠.

먼저 산술평균-기하평균 사이 부등식에 의해

\prod_{k=1}^n \left({x_k \over x_k + y_k}\right)^{1/n} \le {1 \over n} \sum_{k=1}^n {x_k \over x_k + y_k}

\prod_{k=1}^n \left({y_k \over x_k + y_k}\right)^{1/n} \le {1 \over n} \sum_{k=1}^n {y_k \over x_k + y_k}

이 두 식이 나옵니다.

이제 두 식을 더하면

\prod_{k=1}^n \left({x_k \over x_k + y_k}\right)^{1/n} + \prod_{k=1}^n \left({y_k \over x_k + y_k}\right)^{1/n} \le {1 \over n} n = 1

이 나오고, 여기에서 \prod_{k=1}^n (x_k + y_k)^{1/n}를 곱하면 말러의 부등식이 됩니다.

어때요, 참 쉽죠?

은근히 많은 곳에 사용되는 부등식입니다.

특히 다중루트를 풀 때 많이 사용되죠.

유익하셨다면 좋아요 눌러주세요!

추천 1 목록보기
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  •  
    △π Lv.9 2020.07.12 16:49

    민코프스키 부등식 증명도 누군가 해주세요!

    댓글 작성하기 좋아요1 댓글수0
  •  
    GUN.007 Lv.11 2020.07.13 02:30

    좋아요가 없네요....

    ㅎㅎㅎ

    댓글 작성하기 좋아요1 댓글수0
  •  
    유한의끝도못본남자 Lv.7 2020.07.14 01:40

    코시-슈바르츠부등식 비슷한느낌이나네염

    댓글 작성하기 좋아요0 댓글수1
    •  
      △π Lv.9 2020.07.14 04:39

      네 저도 증명하면서 그런 느낌이 들었습니다.

      제곱을 1/n제곱으로 바꾸고 프로덕트만 들어간 느낌이었어요ㅋㅋ

      좋아요0
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