I am stuck in the first part of problem 2 of the chapter 8 (error estimation) of the book "A probabilistic theory of pattern recognition" by Devroye:
Show that for any $s>0$, and any random variable $X$ with $\mathbf EX=0,\mathbf EX^2=\sigma^2, X\le c$, $$\mathbf E\left\{e^{sX}\right\}\le e^{f(\sigma^2/c^2)}\,,$$ where $$f(u)=\log\left(\frac1{1+u}e^{-csu}+\frac{u}{1+u}e^{cs}\right).$$
The purpose of the problem is to prove Bennett's inequality. I've searched for how Bennett's is usually proved, and it seems like the usual trick is to expand $\mathbb{E}[e^{sX}]$ with the Taylor series, followed by applying an inequality on the terms in $\mathbb{E}[X^k], k \geq 3$. However, this is not what the author has in mind here, and I cannot figure out any way to invoke the term $e^{-csu}$ in any inequality.