Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. We assume throughout this question that $f \in \mathscr { C } _ { b } ^ { 2 }$ and that there exists $\delta > 0$ such that $$\forall x \in \mathbb { R } , \quad f ( x ) \geqslant \delta .$$ We denote $g = \left( f ^ { \prime } \right) ^ { 2 } / f$.
7a. Show that $J$ is then of class $\mathscr { C } ^ { 1 }$ on $\mathbb { R }$ and that $$\forall t \in \mathbb { R } , \quad J ^ { \prime } ( t ) \cos t = - \frac { \sin t } { 2 } \int \frac { \left( \partial _ { x } \Phi _ { f } ( t , x ) \right) ^ { 2 } } { \Phi _ { f } ( t , x ) } \mu ( x ) d x$$
7b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Show that $$\Phi _ { f ^ { \prime } } ( t , x ) ^ { 2 } \leqslant \Phi _ { f } ( t , x ) \Phi _ { g } ( t , x )$$
7c. Conclude that $$\int h ( f ( x ) ) \mu ( x ) d x - h \left( \int f ( y ) \mu ( y ) d y \right) \leqslant \frac { 1 } { 4 } \int g ( x ) \mu ( x ) d x$$