Let $f$ be a piecewise continuous function, strictly positive, integrable on $\mathbb { R } _ { + }$. Deduce that $x \mapsto \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right)$ is integrable on $\mathbb { R } _ { + } ^ { * }$ and that
$$\int _ { 0 } ^ { + \infty } \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \mathrm { d } x \leqslant \mathrm { e } \int _ { 0 } ^ { + \infty } f ( x ) \mathrm { d } x$$