Given $a<0$ such that $k:=2^{a}+3^{a}\in (0,1)$ and define $f(x):=x^{a}$ for all $x>0$. Then,
$f(2x)+f(3x)=x^{a}(2^{a}+3^{a})=kf(x)$.
Somebody know another example of a function $f:(0,+\infty)\longrightarrow [0,+\infty)$ (not identically null) such that
$f(2x)+f(3x)\leq k f(x)$,
for some $k\in (0,1)$ and $x>0$.
Thank you in advance for your comments!