Let $f$ be a piecewise continuous function, strictly positive, integrable on $\mathbb { R } _ { + }$. Prove that, for all $x > 0$,
$$\exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \leqslant \frac { \mathrm { e } } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t$$
You may note that $\ln ( f ( t ) ) = \ln ( t f ( t ) ) - \ln ( t )$.