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Home » Olympiad Section » Inequalities » Inequalities Unsolved Problems
 
An intersting ineq
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miwalin
Poincare Conjecture
Poincare Conjecture

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#1
An intersting ineq
Let a,b>0 and 0\le v\le 1. Show (a^vb^{1-v}-a^{1-v}b^v)^2\le (1-2r)(a-b)^2, where r=min\{v,1-v\}. maybe

PostPosted: Sun Nov 01, 2009 10:25 am  Back to top 
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miwalin
Poincare Conjecture
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#2
The proposer has got a proof, but it is lengthy. Blush

PostPosted: Sun Nov 08, 2009 2:59 pm  Back to top 
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spanferkel
Yang-Mills Theory
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#3
Re: An intersting ineq
miwalin wrote:
Let a,b > 0 and 0\le v\le 1. Show (a^vb^{1 - v} - a^{1 - v}b^v)^2\le (1 - 2r)(a - b)^2, where r = min\{v,1 - v\}. maybe

Why does it need a lengthy proof? Wlog v\le\frac12. Putting x: =\sqrt{\frac{a}{b}} and k: =1-2v\in[0,1], it becomes
(x^k-x^{-k})^2\le k\left(x-\frac1x\right)^2. Wlog assume x\ge1. Replacing k by k^2, we have thus to prove for k\in[0,1]
x^{k^2}-x^{-k^2}\le k\left(x-\frac1x\right) which is easy with derivatives,Blush fixing x\ge1 and considering it as a function of k. Smile
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PostPosted: Mon Nov 09, 2009 2:37 am  Back to top 
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