Exercise 2.3: Polynomial Markov Versus Chernoff
Using that \(X \ge 0\), perform a series expansion:
\begin{equation} \E[e^{\lambda X}] = \sum_{n=0}^\infty \frac{\lambda^n}{n!} \E[\abs{X}^n] \ge \parens{\sum_{n=0}^\infty \frac{(\lambda \delta)^n}{n!}} \inf_{k = 0,1,2,\ldots} \frac{1}{\delta^k} \E[\abs{X}^k] \ge e^{\lambda \delta} \inf_{k=0,1,2,\ldots} \frac{1}{\delta^k} \E[\abs{X}^k]. \end{equation}
Therefore,
\begin{equation} \inf_{k = 0, 1, 2, \ldots} \frac{\E[\abs{X}^k]}{\delta^k} \le \inf_{\lambda > 0} \frac{\E[e^{\lambda X}]}{e^{\lambda \delta}}. \end{equation}