I am referring to the Example 2.4 (page 16) in this book chapter https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf

Suppose $Z \sim N(0,1)$ and random variable $X=Z^2$. For $\lambda < \frac{1}{2}$, we have $\mathbb{E}[e^{\lambda(X-1)}] = \frac{e^{-\lambda}}{\sqrt{1-2\lambda}}$.

In Example 2.4 it is stated that based on some calculus we can verify that

$\frac{e^{-\lambda}}{\sqrt{1-2\lambda}} \leq e^{2\lambda^2} $

I cant come up with a calculs to get the above inequality. How can I do it?


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