We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$ where $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that for all $x \in ] 0,1] , | F ( x ) | \leqslant \| f \| \left( \mathrm { e } ^ { - 1 } - \ln ( x ) \right) ^ { 1 / 2 }$. One may use Question 19.