Asymptotic growth of $\prod_{k=1}^n \frac{k^\alpha}{\lambda + k^\alpha}$? Can you please explain why for $\lambda > 0$ and $0 < \alpha < 1$
$$\prod_{k=1}^n \frac{k^\alpha}{\lambda + k^\alpha} \sim \exp\left(-\frac{\lambda}{1-\alpha}n^{1-\alpha}+o(n^{1-\alpha})\right)$$
holds? I'm stuck at the moment.
EDIT: added $o()$ term for correctness.

Not really a answer but a quick very informal outline:
$$\log \prod_{k=1}^n \frac{k^\alpha}{\lambda+k^\alpha } = \sum_{k=1}^n - \log(1 +\lambda \, k^{-\alpha}) \approx - \lambda \sum_{k=1}^n k^{-\alpha} \approx - \lambda \int_0^n x^{-\alpha}dx = -\lambda \frac{n^{1-\alpha}}{1-\alpha}$$
