Explain to me the concept of global bounds in Jensen's Inequality

Question

I'm trying to understand some journals on Jensen's Inequality, and I noticed that the term "global bounds" is used often. I can't distinguish this from the "usual" bounds I encountered in my analysis courses. I also literally can't find an article explaining global bounds on the internet, so I figured this may be a jargon among mathematicians who work with inequalities. What is it? How is it different from a "local bound" if such a thing exists?

Answer 1

Yes, Jensens inequality states for a convex function $f:[a,b] \to \mathbb{R}$, and a finite sequence $p_i \in [0,1] $ with $\sum p_i = 1$ and a finite sequence $x_i\in[a,b]$ it holds $$ 0 \le \sum_i {p_i} f(x_i) - f( \sum_i p_i x_i) $$ And now you could be tempted to ask how big the right hand side of this inequality might get - or in other words : "is there an upper bound for the Jensen's inequality?"

E.g a well known result in case $f$ is additionally differentable there is an upper bound: $$ \sum_i {p_i} f(x_i) - f( \sum_i p_i x_i) \le \frac{1}{4}(b-a)(f'(b)-f'(a))$$

So we have an upper bound which does not involve the $x_i$ or $p_i$, which is oftenly considered as a 'global bound'. If it involves $x_i$ or $p_i$, then people talk about a 'local bound'.

However the terms 'global bound' and 'local bound' are not strictly defined under all authors publishing papers on this topic.