I have to proof the following estimate

$\vert \int\limits_{H_k}z^{s-1}(e^{z}-1)^{-1}dz\vert \leq k^{\sigma}$

Where $H_k$ ist the Hankel Contour with radius $\rho_k = (2k+1) \pi$

From another estimate, which I already have proofen I got

$\vert \frac{z^{s-1}}{e^z-1}\vert \leq \rho^{\sigma-2}$ for every radius $> 0$. where $\sigma = Re(s)$

For the searched estimate I tried

$\vert \int\limits_{H_k}z^{s-1}(e^{z}-1)^{-1}dz\vert \leq \int\limits_{H_k}\vert z^{s-1}(e^{z}-1)^{-1}\vert dz \leq \int\limits_{H_k} \rho_k^{\sigma-2}dz$

but that don´t lead me further. Anyone has a clue for me hwo I can go on?


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