Prove the inequality
Proof. Define a function . Then, since
for
we have
Since and
is increasing on
(since its derivative is positive) we have
For the inequality on the left (which is much more subtle), we want to show
So, we consider the function
We know
and
Now, if we can show that is decreasing on the whole interval
then we will be done (since this would mean
on the whole interval since if it were less than
somewhere then it would have to increase to get back to
on the right end of the interval).
To show is decreasing on the whole interval we will show that its derivative is negative. To that end, define a function
(This is the numerator in the expression we got for . Since we know the denominator of that expression is always positive, we are going to show this
is always negative to conclude
is always negative.) Now,
and
for . Thus,
is negative, and so
is decreasing. Since
we then have
is negative on the whole interval. Therefore,
on the interval. Hence,
is decreasing. Hence, we indeed have
for all
Very nice proof. Many thanks