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.

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