## 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