Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote
$$S_{k} = \sum_{i=1}^{k} U_{i}$$
Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by $\varphi(\lambda) = \ln\left(\mathbb{E}\left[e^{\lambda U_{1}}\right]\right)$. Establish that
$$\forall \lambda \in \mathbb{R}, \quad \varphi(\lambda) \leqslant \frac{\lambda^{2}}{2}.$$