Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f$ be a function in $\mathscr { C } _ { b } ^ { 1 }$, such that for all $x \in \mathbb { R }$, we have $\left| f ^ { \prime } ( x ) \right| \leqslant 1$. We denote, for $\lambda \in \mathbb { R }$, $$H ( \lambda ) = \int e ^ { \lambda f ( x ) } m ( x ) d x$$ We admit that $H$ is of class $\mathscr { C } ^ { 1 }$ and that we obtain an expression of $H ^ { \prime } ( \lambda )$ by differentiating under the integral sign in the usual manner.
11a. Show that for all $\lambda \in \mathbb { R }$, $$\lambda H ^ { \prime } ( \lambda ) - H ( \lambda ) \ln H ( \lambda ) \leqslant \frac { C \lambda ^ { 2 } } { 4 } H ( \lambda )$$
11b. Deduce that for $\lambda \geqslant 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ You may study the function $\lambda \mapsto \frac { 1 } { \lambda } \ln H ( \lambda )$.