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 $f(0)=0$?

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Recall that the Minkowski inequality turns into equality when $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)$ are proportional. Thus, if you perturb the function $x^p$ a bit in a neighborhood of some point $a$ (keeping monotonicity, convexity, whatnot) then you may set $a=x_n$ or $a=x_n+y_n$ refuting your inequality for the perturbed function. – Ilya Bogdanov Mar 3 '14 at 19:43

The question that you are asking was asked in "On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality" by F. Mulholland (1949). In that paper, Mulholland established a sufficient condition on $f$, namely that it should satisfy $f(0)=0$, be increasing on $x \ge 0$ and be g-convex, i.e., $\log f(e^x)$ is convex on the reals.