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

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?

• A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase. Jan 25, 2019 at 16:03

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'.