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

as stated).

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 .

I suggest

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.