grandes-ecoles 2021 Q2.9

grandes-ecoles · France · x-ens-maths__pc Proof Direct Proof of an Inequality
Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We set for $p > 0$: $$M_p(Q) = \frac{1}{2\pi} \int_0^{2\pi} \left|Q\left(e^{i\theta}\right)\right|^p d\theta$$ Explain why $M_p(Q)$ is strictly positive for all $p > 0$.
Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We set for $p > 0$:
$$M_p(Q) = \frac{1}{2\pi} \int_0^{2\pi} \left|Q\left(e^{i\theta}\right)\right|^p d\theta$$
Explain why $M_p(Q)$ is strictly positive for all $p > 0$.