Let $\alpha>0$, I need to prove that there exists $t_0>0$ such that $$\int_0^1 e^{-tu}(1-u)^{\alpha}du\leq t^{-1}, \forall t>t_0.$$ I received help and found that by Watson's Lemma you could obtain that: $$\int_0^1 e^{-tu}(1-u)^{\alpha}du\sim t^{-1} \text{ as } t\rightarrow+\infty.$$ Nevertheless, that does not concretely solve the inequality.
2026-03-26 08:07:29.1774512449
Prove that $\int_0^1 e^{-tu}(1-u)^{\alpha}du\leq t^{-1}$ for $\alpha,t>0$
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$$ \int_0^1 e^{-tu}(1-u)^{\alpha} \, du \le \int_0^1 e^{-tu}\, du = \frac{1-e^{-t}}{t} < \frac 1t $$ holds for all $t > 0$.