Integral mean estimates for polynomials whose zeros are within a circle (Q5915479)

The authors prove two theorems concerning inequalities of \(L_p\) norms on the unit circle of a polynomial with restricted zeros and its derivative. We state here one of the theorems (Theorem 1.3). Let \(p(z):=a_nz^n +\sum_{\nu=\mu }^n a_{n-\nu }z^{n-\nu }\), \(1\leq \mu \leq n\), be a polynomial of degree \(n\) and suppose that all the zeros of \(p(z)\) lie in the closed disk \(|z|\leq K\), \(K\leq 1\). Then for each \(r>0\), \(p>1\), \(q>1\), with \(1/p +1/q =1\), \[ n\left \{\int_0^{2\pi} \bigl|p (e^{i\theta}) \bigr|^r d\theta \right\}^{1 \over r} \leq\left \{\int^{2\pi}_0 |1+K^{\mu}e^{i\theta} |^{ pr} d\theta \right\}^{1 \over pr} \left\{\int_0^{2\pi} \bigl|p' (e^{i\theta}) \bigr|^{qr} d \theta \right\}^{1\over qr} \] holds. This is a generalization of a result due to \textit{A. Aziz} and \textit{N. Ahemad} [Glas. Mat., III. Ser. 31, No. 2, 229-237 (1996; Zbl 0874.30001)].