LashaBukhnikashvili's blog

Math
By LashaBukhnikashvili, 10 years ago, In English

Can someone help me to deduce this?

For any numbers x,y,z:

UPDATE: problem is wrong see coment.

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10 years ago, # |
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http://en.wikipedia.org/wiki/Muirhead%27s_inequality

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10 years ago, # |
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http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means

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    10 years ago, # ^ |
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    I thinking about this but,this is when x,y,z are positive other way no :(

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      10 years ago, # ^ |
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      When numbers can be negative, it's incorrect inequality. For example: x = 0,  y = 2,  z =  - 1

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        10 years ago, # ^ |
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        you'r right,i had a mistake;

        x,y,z is such numbers that in this numbers 2 of them is positive and one any number.

        More formally at first i had such Inequality(where a,b,c>0):

        than mentioned x=b+c-a,y=a+c-b,z=a+b-c numbers,and get:

        may say that a<=b<=c and get that x>0 y>0 z<0

        now I guess that this way is difficult. can somone tell me another easy way to prove first Inequality?

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