Suppose that a rv $X$ is such that $|X| \leq c$, $E[X] = 0$ and $E[X^2] = \sigma^2 < \infty$. Prove that for any $\theta > 0$ $$E\left[e^{\theta X} \right]\leq \exp\left(\sigma^2 \left(\frac{e^{\theta c} - 1 - \theta c}{c^2}\right)\right).$$
I am trying to prove the inequality using a similar approach to Hoeffding's lemma, but I am struggling to obtain the result. Could someone point me in the right direction?
Let $Y = X/c$. Then $Y \le 1$, $\mathbb{E}[Y]=0$ and $\mathbb{E}[Y^2] = \frac{\sigma^2}{c^2}$.
From the result in [1] (page 218 in the link), we have, for $s > 0$, $$\mathbb{E}[\mathrm{e}^{sY}] \le \mathrm{exp}((\mathrm{e}^{s} - s - 1)\mathbb{E}[Y^2])$$ which results in $$\mathbb{E}[\mathrm{e}^{\theta X}] \le \mathrm{exp}(\sigma^2(\mathrm{e}^{\theta c} - \theta c - 1)/c^2)$$ where $\theta = s/c$. We are done.
[1] Boucheron, Lugosi, and Bousquet, "Concentration Inequalities". http://www.econ.upf.edu/~lugosi/mlss_conc.pdf