The following inequality is known as Petrovic's inequality:
Prop 1. Let $f : [0,\infty) \rightarrow \mathbb{R}$ be a convex function, and
$(x_i)_{i=1}^n$, be a sequence of positive numbers. Then the
inequality \\
$f(x_1) + ... + f(x_n) \leq f(x_1 + ... + x_n) + (n-1)f(0)$ holds.
Proof.
Set $s_k= x_1 + ... + x_k$, $k=1,2,...,n$, and $s_0=0$. Then $x_k=s_k-s_{k-1}$ and
by convexity $f(s_k)- f(s_{k-1})\geq f(x_k)- f(0)$. Summing these inequalities we get
$f(s_n)- f(0))\geq \sum_{k=1}^n f(x_k)- n f(0)$.
Although this inequality looks very simple there is something intriguing about it and our impression is that
there are the other approaches, generalization, geometric interpretation and application of it.