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Math Help - prove an inequality

  1. #1
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    prove an inequality

    hi,

    Could somebody please give me an idea of how to prove the following inequality:

    a/b + b/c + c/d + d/a >= b/a + c/b + d/c + a/d

    a,b,c,d being real numbers and 0 < a < b < c < d

    thanx
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  2. #2
    Super Member Bacterius's Avatar
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    Consider doing it by parts : prove that :

    \frac{a}{b} \geq \frac{a}{d}, since a < b < d

    Prove the three other pairs, then add everything together (two or more inequalities of the same sign can be added up). You should prove your inequality just fine
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  3. #3
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    Quote Originally Posted by oeivinr View Post
    hi,

    Could somebody please give me an idea of how to prove the following inequality:

    a/b + b/c + c/d + d/a >= b/a + c/b + d/c + a/d

    a,b,c,d being real numbers and 0 < a < b < c < d

    thanx

    Everything's positive here, so multiply both sides by their common denominator and then you must prove that

    a^2cd+b^2ad+c^2ab+d^2bc\ge b^2cd+c^2ad+d^2ab+a^2bc\Longleftrightarrow a^2c(d-b)+b^2d(a-c)+c^2a(b-d)+d^2b(c-a)\ge 0\Longleftrightarrow (c-a)bd(d-b)+(d-b)ac(a-c)\ge 0

    \Longleftrightarrow (c-a)(d-b)(bd-ac)\ge 0\Longleftrightarrow bd\ge ac , and since this last inequality is (almost trivially) true you can go back and prove the original one...

    Tonio
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