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Math Help - Find all functions satisfying the functional equation

  1. #1
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    Find all functions satisfying the functional equation

    Find all functions f defined on the set of positive real numbers and having real values, that for any real x, y there is satisfied the following equasion:

    f(\sqrt{\frac{x^2+xy+y^2}{3}}) = \frac{f(x) + f(y)}{2}
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  2. #2
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    It is quite interesting question, isn't it? Did someone find the answer?
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  3. #3
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    Can't manage to solve this interesting problem either.. Any ideas?
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  4. #4
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    May be this helps

    <br />
z=\sqrt{\frac{x^2+xy+y^2}{3}}<br />

    <br />
f(z)=(f(x)+f(y))/2<br />

    <br />
f'_z \; z'_x \; = \; f'_z \; \frac{2x+y}{2 \sqrt{3(x^2+xy+y^2})}}  \; = \; f'_x/2 <br />

    <br />
f'_z \; z'_y \;  = \; f'_z \; \frac{2y+x}{2 \sqrt{3(x^2+xy+y^2})}}  \; = \; f'_y/2 <br />

    <br />
\displaystyle { \frac{2x+y}{2y+x}=\frac{f'_x}{f'_y}.<br />
}<br />

    One solution

    <br />
\displaystyle { f'_x =f'_y=0<br />
}<br />

    <br />
\displaystyle { f(x) =const.<br />
}<br />
    Last edited by zzzoak; December 8th 2010 at 01:06 PM.
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  5. #5
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    Hmm.. That's the full solution?
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  6. #6
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    If

    <br />
{<br />
\displaystyle { f'_x =g(x)<br />
}<br />
 }<br />

    <br />
{<br />
\displaystyle { f'_y =g(y)<br />
}<br />
 }<br />

    <br />
(2x+y)g(y)=(2y+x)g(x).<br />

    We get a system

    <br />
2xg(y)=xg(x)<br />

    <br />
yg(y)=2yg(x)<br />

    which has a solution
    <br />
g(x)=g(y)=0.<br />
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