Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce the convergence of the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ for all $z$ in $D$. In what follows, we denote, for $z$ in $D$,
$$P ( z ) : = \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$$