This post concerns the monotonicity of
on with
. The two cases
and
are of special because these are always mentioned in many textbooks as
Clearly, is monotone increasing with respect to
. Hence we are left with the monotonicity of
with respect to
. From the above discussion, the function
is monotone increasing with respect to
when
. When
, as the function
is monotone increasing with respect to
, we deduce that the function
is monotone increasing with respect to
. Hence, we are left with the case
. To study this problem, we examine
with respect to
.
Derivative of . It is easy to get
Hence the sign of is determined by the sign of
on .