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Thread: Find all functions satisfying the functional equation

  1. #1
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    Find all functions satisfying the functional equation

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

    $\displaystyle f(\sqrt{\frac{x^2+xy+y^2}{3}}) = \frac{f(x) + f(y)}{2}$
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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

    $\displaystyle
    z=\sqrt{\frac{x^2+xy+y^2}{3}}
    $

    $\displaystyle
    f(z)=(f(x)+f(y))/2
    $

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

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

    $\displaystyle
    \displaystyle { \frac{2x+y}{2y+x}=\frac{f'_x}{f'_y}.
    }
    $

    One solution

    $\displaystyle
    \displaystyle { f'_x =f'_y=0
    }
    $

    $\displaystyle
    \displaystyle { f(x) =const.
    }
    $
    Last edited by zzzoak; Dec 8th 2010 at 12:06 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

    $\displaystyle
    {
    \displaystyle { f'_x =g(x)
    }
    }
    $

    $\displaystyle
    {
    \displaystyle { f'_y =g(y)
    }
    }
    $

    $\displaystyle
    (2x+y)g(y)=(2y+x)g(x).
    $

    We get a system

    $\displaystyle
    2xg(y)=xg(x)
    $

    $\displaystyle
    yg(y)=2yg(x)
    $

    which has a solution
    $\displaystyle
    g(x)=g(y)=0.
    $
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