Suppose that $X\geq0$, and that the moment generating function of $X$ exists in an interval around 0. Given some $\delta>0$ and integer $k=1,2,...$, show that
$$\inf\_{k=0,1,...}\frac{E(|X|^k)}{\delta^k} \leq \inf\_{\lambda>0} \frac{E(e^{\lambda X})}{e^{\lambda \delta}}. $$


Consequently, an optimized bound based on polynomial moments is always at least as good
as the Chernoff upper bound. Could anyone enlighten me how to prove this?



Polynomial Markov versus Chernoff Bound for random variables
Let $b$ denote the LHS. Expanding $e^{\lambda X}$ in a power series you can deduce that
$$E(e^{\lambda X}) \ge \sum\_{k \ge 0} \frac {b \lambda^k \delta^k}{k!}=b e^{\lambda \delta} \,.$$


