Estimations de produits de polynômes. (Estimates of products of polynomials) (Q1066298)

Let \(P(z)=\sum_{j\geq 0}a_ jz^ j\) be a polynomial. Let \(P|^ k=\sum^{k}_{j=0}a_ jz^ j\). A typical result of this paper is the following: If \[ \| P\|_ 2=(\int | P(e^{i\theta})|^ 2(d\theta /2\pi))^{1/2}=1 \] and \(\| P|^ k\|_ 2\geq \delta >0\), then \[ \int^{2\pi}_{0}Log| P(e^{i\theta})| d\theta /2\pi \geq 2 Log(\delta /e\cdot 3^{k+1}). \] Some results pertain to the product of two polynomials.