On an inequality (Q5906767)

Let \(f(z)= \sum^\infty_{k= -\infty} a_kz^k\) be a Laurent series convergent in the ring \(\{z|r<|z|<R\}\) \((0\leq r<R \leq+\infty)\) and let \(\Gamma\) be a closed Jordan curve in the ring and enclosing \(\{z\mid |z|\leq r\}\). The author proves that \(\forall n\in \mathbb{Z}\), \(\exists z_0\in \Gamma\) such that the inequality \(|f(z) |\leq |\sum_{k \leq n} a_kz^k |+ |\sum_{k \geq n+1} a_kz^k|\) turns into an equality at \(z=z_0\) and that for \(n\in\mathbb{Z}\) and verifying simultaneously \[ \sum_{k\leq n} a_kz^k \not\equiv 0 \quad \text{and} \quad \sum_{k\geq n+1} a_kz^k\not \equiv 0. \] \(\exists z_1\in \Gamma\) such that the above inequality turns into a strong one at \(z=z_1\).