grandes-ecoles 2021 Q16

grandes-ecoles · France · centrale-maths2__official Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

Let $n$ be a non-zero natural number. Show that $$\forall Q \in \mathbb{C}_{n-1}[X], \quad |Q(1)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|Q(x)\sqrt{1-x^2}\right|.$$ One may consider $f : \theta \mapsto Q(\cos\theta)\sin\theta$ and verify that $f \in \mathcal{S}_n$.

Let $n$ be a non-zero natural number. Show that
$$\forall Q \in \mathbb{C}_{n-1}[X], \quad |Q(1)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|Q(x)\sqrt{1-x^2}\right|.$$
One may consider $f : \theta \mapsto Q(\cos\theta)\sin\theta$ and verify that $f \in \mathcal{S}_n$.

Paper Questions

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