Suppose $f(x)$ is a non-decreasing sub-additive convex function
In order words,
$f(x+y)\leq f(x)+f(y)$ for all $x,y$
and $f(x)\leq f(y)$ if $x\leq y$
Let $x_1,x_2\ldots x_i$ are $i$ positive integers such that their sum is $n$.
What will be the minimum and maximum value of $f(x_1)+f(x_2)+f(x_3)+\ldots+f(x_i)$ regardless of values of $x_1,x_2\ldots x_i$?
An ideal solution which I am looking for would of the form devoid of $x_j$ and just in terms of $n,f$ and $i$