Product inequalities for norms of linear factors (Q1023014)

Let \(T\) be the unit circle in the complex plane, and let \(L^p(T)\) be the usual \(L^p\)-space on \(T\) with \[ \|f\|_p=\left(\frac{1}{2\pi} \int^{2 \pi}_0|f(e^{i\theta})|^pd \theta\right)^{\frac 1p} \] for \(0<p< \infty\) and \(f\in L^p(T)\). For \(p=\infty\), \(\|f\|_\infty\) denotes the usual sup norm of \(f\in L^\infty(T)\). Let \(0<p< \infty\), and let \(f(z)= \prod^n_{k=1} (z-r_k)\) be a complex monic polynomial of degree \(n\). In this paper, the author obtains the following sharp inequalities: \[ \prod^n_{k=1}\|z-r_k\|_p\leq\|z-1\|_p^{n-1}\|f(z)\|_p \] with equality if and only if \(f(z)=z^n-e^{i\theta}\) for some \(\theta\in[0,2 \pi)\). This result generalizes the available sharp results for \(p=2\) or \(p=\infty\).