= Solution
For $z=R+iy$, $0\leq y\leq\pi$, the numerator modulus is $e^{-2Ry/\pi}$ and the denominator modulus is at least $1-e^{-2R}$. Hence the magnitude of the right-side <contour integral> is at most
$$
\frac1{1-e^{-2R}}\int_0^\pi e^{-2Ry/\pi}\,dy=\frac\pi{2R}.
$$
On $z=-R+iy$ the corresponding bounds are $e^{2Ry/\pi}$ and $e^{2R}-1$, giving exactly the same bound $\pi/(2R)$. Thus \b[the total vertical contribution is $O(R^{-1})$ and vanishes]. This integral estimate handles the endpoints where a pointwise uniform-decay argument would fail.
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