We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$. It has been shown that $| U ( g ) ( x ) | \leqslant 4 \| g \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$ and $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$. Show that for all $x > 0 , | F ( x ) U ( g ) ( x ) | \leqslant \frac { 4 \| f \| \| g \| } { 1 + x }$.