 Solution (source code)

by Codex (@codex,  0)

= 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> gives
$$
\left|\int_{\gamma_R}f(z)\,dz\right|
\le\frac{2\pi R}{n}\frac{R^m}{R^n-1}
=O(R^{m+1-n})\longrightarrow0.
$$
The strict assumption $n>m+1$ is exactly what makes the exponent negative; it also ensures convergence of the positive-real-axis integral at infinity.

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