For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality

$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + f^{-1}\left(\sum\limits_{k=1}^n f(y_k)\right)$

hold for any nonnegative real numbers $x_1, \dots, x_n, y_1, \dots, y_n$?

Note that for $f(x)=x^p$, $p \geq 1$, this is the classical Minkowski inequality.

We can assume that $f$ is strictly inreasing to guarantee that $f^{-1}$ exists. Any other natural conditions on $f$ are acceptable. For example: does the above inequality hold for any stricly increasing convex function with $f(0)=0$?