Let $z \in D$. Show that the function $t \in [ 0,1 ] \mapsto L ( t z )$ is differentiable and give a simple expression for its derivative. Deduce that $t \mapsto ( 1 - t z ) e ^ { L ( t z ) }$ is constant on $[ 0,1 ]$ and conclude that
$$\exp ( L ( z ) ) = \frac { 1 } { 1 - z }$$