I didn't try, but maybe by induction?
For example this method shows the inequality, but you cannot not use it to go from
I guess I wasnt clear enough in my post, sorry
If that was convoluted just ignore it, there is no need to prove this, its just out of curiosity
of the following beautiful result: let be such that exists on we know that by the mean
value theorem now it can be proved that:
1) if then
2) if then
the proposer, i guess, was professor Alex Lupas, a Romanian mathematician who sadly passed away few months
ago. my proof of this result is easy to follow and i can post it here, but some of you might want to think about it
a little bit first?
then the beauty of the result is that it gives a better location of the mean value theorem can
only tell us that but under those conditions that i mentioned, we can tell if is in the first half of the
interval or in the second half. probably it'd help if you see it in a couple of examples.