Is there any "direct" proof of the following asymptotic inequality: let $\alpha\le 1$ and consider $$Q_n(x)=\sum_{k=1}^n\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+k-1)}{k!}x^k$$
Then, $$\int_0^1 Q_n^{\frac{1}{\alpha}}(x) \, dx\ge C(\alpha)\log n$$ for some universal constant $C(\alpha)$ that does not depend on $n.$
I believe I can prove it myself using combinatorics but I would like to see as direct approach as possible.