To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$ and $\lim_{x\to 0} U(f)(x) = 0$. Deduce from the above that $U$ is an endomorphism of $E$ and that, for all $f \in E$ and all $x > 0$, $$| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$$
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$ and $\lim_{x\to 0} U(f)(x) = 0$. Deduce from the above that $U$ is an endomorphism of $E$ and that, for all $f \in E$ and all $x > 0$,
$$| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$$