Question 85
Find the smallest real constant $\alpha$ such that for all positive integers $n$ and real numbers $0 = y_{0} \lt y_{1} \lt \cdots \lt y_{n}$, the following inequality holds:
$$
\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}} \text{.}
$$
$$
\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}} \text{.}
$$