Let $X$ be a random variable such that $|X|\le{K}$ for some $K>0$. I am trying to prove the following bound on the moment generating function of X:
$$\mathbb{E}[\exp(\lambda{}X)]\le{}\exp(g(\lambda)\mathbb{E}[X^2])$$ where
$$g(\lambda):=\frac{\lambda^2\space{}/\space{}2}{1-|\lambda|K\space{}/\space{}3}$$ provided we have $|\lambda|<3\space{}/\space{}K.$
I've managed to show that if $X\le{}K$ a.e. then $$\mathbb{E}[\exp(\lambda(X-\mathbb{E}[X]))]\le{\exp\bigg(\frac{\lambda^2/2\mathbb{E[X]}}{1-K\lambda/3}\bigg)}$$ for $\lambda\in{[0,3/K]}$. But can't see where to go from here or if this is even the right direction.
p.s. I got to that inequality by considering the function $h(\lambda):=2\big(\frac{e^X-X-1}{2}\big)$