Some Zygmund type \(L^q\) inequalities for polynomials. (Q1419757)

Let \(P\) be a polynomial of degree \(n\) which does not vanish in the disk \(| z| <k\), \(k\geq 1\). In [Proc. Indian Acad. Sci. Math. 108, 63--68 (1998; Zbl 0931.30003)], \textit{A. Aziz} and \textit{W. M. Shah} have shown that for \(1\leq s<n\), \[ | | P^{(s)}| | _q\leq \frac{n(n-1)\cdots(n-s+1)}{| | k^s+z| | _q}| | P| | _q,\quad q>0. \] In this paper, the authors establish a refinement of the above inequality and obtain a number of interesting corollaries. They also consider an analogous problem for the class of polynomials of the form \(p(z)=a_nz^n+\sum_{\nu=m}^n a_{n-\nu}z^{n-\nu}\), where the zeros of \(p(z)\) lie inside the disk \(| z| \leq k\), \(k\leq 1\).