# Integral estimate for Hankle´s Contour

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?