Polynomial Markov versus Chernoff Bound for random variables

Question

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?

Answer 1

Let $b$ denote the LHS. Expanding $e^{\lambda X}$ in a power series you can deduce that $$E(e^{\lambda X}) \ge \sum_{k \ge 0} \frac {b \lambda^k \delta^k}{k!}=b e^{\lambda \delta} \,.$$