Uniform sign of concave and homogeneous functions

Question

Let $f:\mathbb{R}^n_+\mapsto\mathbb{R}$ be concave and homogeneous of degree $\lambda\in(-\infty,0)\cup(0,1)$. Does the inequality $\lambda f(x)>0$ necessarily hold for all $x>0$? I suspect that this is the case, because I can't find a counter example. I know that for $\lambda=1$ the answer is NO as can be seen from the example $f(x_1,x_2)=\sqrt{x_1x_2}-x_2$. I was trying to find a reference that deals with this kind of question but to no avail.