We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$. Demonstrate that, for all $p \in \mathbb { N }$, there exist a real $K _ { p }$ and a natural integer $N _ { p }$ such that $$\forall n \geqslant N _ { p } , \quad \left| a _ { n } \right| \leqslant \frac { K _ { p } } { n ^ { p } }$$
We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Demonstrate that, for all $p \in \mathbb { N }$, there exist a real $K _ { p }$ and a natural integer $N _ { p }$ such that
$$\forall n \geqslant N _ { p } , \quad \left| a _ { n } \right| \leqslant \frac { K _ { p } } { n ^ { p } }$$