I believe you are right, in fact if (for
instance and then the sequence increases in the first terms , but and then the hypothesis seems to hold (for
However, if we have the constant sequence
I conjecture without doing any more calculations that this is the unique problem with the hypothesis, we have to require .
a) Try to prove the inequality if , it seems an easy induction at least for the first one.
b) Try to show using differential calculus that
maps into . This would solve the case fos "small 's"
because it would allow to apply a)
I haven't made the calculations but it seems to me the way of attacking it.