Exercise 3.7: Entropy for Bounded Variables
The entropy satisfies the following variational characterisation:
\begin{equation} \Hb(e^{\lambda X}) = \min_{t \in \R}\, \E[\psi(\lambda(X - t)) e^{\lambda X}] \end{equation}
with \(\psi(u) = e^{-u} - 1 + u\). Setting \(t = a\), we obtain
\begin{equation} \Hb(e^{\lambda X}) \le \psi(\lambda(b - a)) \varphi_X(\lambda) \le \frac12 (b - a)^2 \lambda^2 \varphi_X(\lambda), \end{equation}
using that \(\varphi(u) \le \tfrac12 u^2\) for \(u \ge 0\).
Unfortunately, this misses the result by a constant \(\tfrac14\). It does not work to set \(t = \tfrac12(a + b)\), because \(\varphi(u)\) grows exponentially for \(u \le 0\).