ThmDex – An index of mathematical definitions, results, and conjectures.
Modulus sum upper bound to distance between two finite complex products for complex numbers in the unit disc
Formulation 0
Let $x_1, w_1, \ldots, z_N, w_N \in \mathbb{C}$ each be a D1207: Complex number such that
(i) \begin{equation} \forall \, n \in \{ 1, \ldots, N \} : |z_n|, |w_n| \leq 1 \end{equation}
Then \begin{equation} \left| \prod_{n = 1}^N z_n - \prod_{n = 1}^N w_n \right| \leq \sum_{n = 1}^N |z_n - w_n| \end{equation}
Proofs
Proof 0
Let $x_1, w_1, \ldots, z_N, w_N \in \mathbb{C}$ each be a D1207: Complex number such that
(i) \begin{equation} \forall \, n \in \{ 1, \ldots, N \} : |z_n|, |w_n| \leq 1 \end{equation}