Max of $3$-Variable Function I'm trying the find the maximum of the function 
$$f(a,b,c)=\frac{a+b+c-\sqrt{a^2+b^2+c^2}}{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}$$
for all nonnegative real numbers $a, b, c$ with $ab + bc + ca > 0$.
I tried in vain to prove that $\max_{a,b,c}f(a,b,c)=1-\frac{\sqrt{3}}{3}$

You can assume $a +b +c = 3$ because scaling a solution by a positive number doesn't change the objective function. Then you are left with
$$\frac{3 - \sqrt{a^2 + b^2 + c^2}}{\sqrt{ab} + \sqrt{ac} + \sqrt{bc}}$$
subject to $a + b + c = 3$. Show that $a^2 + b^2 + c^2$ is minimized when $a=b=c$ assuming $a+b+c$ is a constant and similarly show that $\sqrt{ab} + \sqrt{ac} + \sqrt{bc}$ is minimized when $a=b=c$ assuming $a+b+c$ is a constant. Then you will get $a=b=c=1$ as an optimal solution, which gives the optimal value you quoted.
