Petrovic and Jensen inequaltiy

Question

I have been reading a book on some classic inequalties and i have stumbled upon this: Let $f:[0, +\infty]\rightarrow \mathbf R$ be a convex function and $x_1,x_2,...,x_n$ a sequence of positive numbers. It can be proved that :$$\sum_{i=1}^nf(x_i) \le (n- 1)f(0) + f(\sum_{i=1}^nx_i)$$ This is apparently called Petrovic inequality. I have also read that it can give a two-sided predicition of $\frac1n \sum_{i=1}^nf(x_i)$ when paired with Jensen inequality: $$f(\frac1n\sum_{i=1}^nx_i) \le\frac1n\sum_{i=1}^nf(x_i) \le\frac1n[(n- 1)f(0) + f(\sum_{i=1}^nx_i)]$$ My question is: Did someone actually use this and where?

Answer 1

It's Just Karamata for $(0,x_1,...x_n)$: $$f(0)+f(x_1)+...+f(x_n)\leq nf(0)+f(x_1+...+x_n)$$ Because we can assume that $x_1\geq x_2\geq...\geq x_n>0$ and $$(x_1+x_2,...+x_n,0,...,0)\succ(x_1,x_2,...,x_n,0)$$