Let $I\subset\mathbb{R}^{\ge 0}$ be an interval and let $f:I\rightarrow\mathbb{R}^{\ge 0}$ be concave (and smooth enough). I'm wondering, weather the following inequality holds:
$f(a+b) \le f(a) + f(b).$
Is this true? I could not find a proof for it. I came to this question for the map $f(x) = x^{\frac{1}{p}}$, where $1\le p < \infty$. The inequality is true in this case, isn't it? I'm especially interested in this case.
I'd be thankful for any hint.