Integral mean estimates for polynomials with restricted zeros (Q2770408)

Let \(p(z)\) be a polynomial of degree \(n\) and suppose that \(p(z)\) does not vanish in the disk \(|z|<K\). Then for \(K=1\) and \(0<q<\infty\), it is known that NEWLINE\[NEWLINE \left\{\frac{1}{2\pi}\int_0^{2\pi}|p(\text{Re}^{i\theta})|^q d\theta\right\} ^q\leq B_q \left\{\frac{1}{2\pi}\int_0^{2\pi}|p(e^{i\theta})|^q d\theta\right\}, NEWLINE\]NEWLINE where \(B_q\) is a constant which depends on \(R\), \(n\) and \(q\). In the paper under review, the authors present a generalization of this result when \(K\geq 1\). They also prove similar results for polynomials all of whose zeros lie in \(|z|\leq K\), where \(K\geq 1\).