# Functions satisfying $f(2x)+f(3x)\leq k f(x)$

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 $$x>0$$.

• If it's continuous at $x=0$ then taking $x\to 0$ gives $2\leq k$. Try to consider functions that have a divergence as $x\to 0$, for example $e^{-x}/x^n$. – Winther Apr 25 at 16:55