Let $\alpha=2.1$. For a fixed $j$, let $(a_i)_{1\leq i \leq j}$ be a sequence of number, where $$a_i = i^\alpha \cdot \prod_{t=1, t\ne i}^j |t^\alpha - i^\alpha|.$$ Prove $(a_i)_{1\leq i \leq j}$ is increasing as $i$ goes from 1 to $j$.
Ps: For $\alpha=2$, I get $a_i = i^2 \cdot \frac{(j+i)!(j-i)!}{2i^2} = (j+i)!(j-i)!/2$ which is indeed increasing as $i$ goes from 1 to $j$.