Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 4 / 17B / b / Solution

On the circular arc, $|z|=R$, the reverse triangle inequality gives $|1+z^n|\ge R^n-1$. The arc length is $2\pi R/n$, so the ML inequality givesThe strict assumption $n>m+1$ is exactly what makes the exponent negative; it also ensures convergence of the positive-real-axis integral at infinity.