show this function have this $2f(\sqrt{xy})=f(x)+f(y),\forall x,y>0$
88683
Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$
26207
If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
156778
Finding functions satisfying $f(f(x))+f(x)+x=0$
48925
BMO1 2009/10 Q5 functional equation: $f(x)f(y) = f(x + y) + xy$
42976
How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$
142873
What does the function f: x ↦ y mean?
32599
Solution to the functional equation $f(x^y)=f(x)^{f(y)}$
31732
If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
147892
Proving if $x<y$ then $\sqrt{x} < \sqrt{y}$
145573
How to show that $f$ is a zero function?
103251
Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions...)
69707
If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
5765
Show that $[2x]+[2y] \geq [x]+[y]+[x+y]$
98255
Find $f(x)$ such that $f(f(x)) = x^2 - 2$
81671
Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$
29647
Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?
137043
To show that function is constant