Logarithmic inequality
Let . Prove that
Solution
Let and
. Thus,
Thus,
The result follows.
Contour integral
Evaluate the integral
Solution
The function is meromorphic on
. Its only pole is
of order
. Hence,
Therefore,
Nested binomial sum
Prove that
Solution
We may begin with the beta function identity for non negative integer values of .
Hence, for non-negative integers
As a result we may compute the nested summation as,
An arcosine integral
Evaluate the integral