# Thread: Find all functions satisfying the functional equation

1. ## 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:

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

2. It is quite interesting question, isn't it? Did someone find the answer?

3. Can't manage to solve this interesting problem either.. Any ideas?

4. May be this helps

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

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

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

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

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

One solution

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

$
\displaystyle { f(x) =const.
}
$

5. Hmm.. That's the full solution?

6. If

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

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

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

We get a system

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

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

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