Let $p , q , r : \mathbb { R } \rightarrow \mathbb { R } _ { * } ^ { + }$ be three continuous functions, with strictly positive values and integrable on $\mathbb { R }$. 14a. Show that there exists a function $u : ] 0,1 [ \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 1 }$ bijective such that $$\forall t \in ] 0,1 [ , \quad u ^ { \prime } ( t ) p ( u ( t ) ) = \int p ( x ) d x$$ Similarly, there exists an analogous function $v : ] 0,1 [ \rightarrow \mathbb { R }$ for $q$. 14b. We assume that $$\forall x , y \in \mathbb { R } , \quad p ( x ) q ( y ) \leqslant \left( r \left( \frac { x + y } { 2 } \right) \right) ^ { 2 } . \tag{4}$$ Show that $$\left( \int p ( x ) d x \right) \left( \int q ( x ) d x \right) \leqslant \left( \int r ( x ) d x \right) ^ { 2 } \tag{5}$$ You may use, after having justified its validity, the change of variable defined by $x = \frac { u ( t ) + v ( t ) } { 2 }$ in the right-hand side of inequality (5).
Let $p , q , r : \mathbb { R } \rightarrow \mathbb { R } _ { * } ^ { + }$ be three continuous functions, with strictly positive values and integrable on $\mathbb { R }$.
14a. Show that there exists a function $u : ] 0,1 [ \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 1 }$ bijective such that
$$\forall t \in ] 0,1 [ , \quad u ^ { \prime } ( t ) p ( u ( t ) ) = \int p ( x ) d x$$
Similarly, there exists an analogous function $v : ] 0,1 [ \rightarrow \mathbb { R }$ for $q$.
14b. We assume that
$$\forall x , y \in \mathbb { R } , \quad p ( x ) q ( y ) \leqslant \left( r \left( \frac { x + y } { 2 } \right) \right) ^ { 2 } . \tag{4}$$
Show that
$$\left( \int p ( x ) d x \right) \left( \int q ( x ) d x \right) \leqslant \left( \int r ( x ) d x \right) ^ { 2 } \tag{5}$$
You may use, after having justified its validity, the change of variable defined by $x = \frac { u ( t ) + v ( t ) } { 2 }$ in the right-hand side of inequality (5).