For all and for any constants
such that
prove that
Proof. We want to consider the function
If we can show this function is decreasing on the positive real axis then we establish the inequality since this would mean that if then
(So, the trick here is to think of this as a function of the exponent. The and
are some positive fixed constants.) To take the derivative of
we use logarithmic differentiation,
Multiplying both sides by we then obtain
Now we can conclude that for all
since the first term in the product
Since (any real power of a positive number is still positive) and
. For the second term we have
since and
are positive, but both logarithms are negative. We know these logarithms are negative since
implies
Hence, for all
. This means
is a decreasing function. Therefore, if
then we have