• Lp inequalities for polynomials with restricted zeros

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/108/01/0063-0068

• # Keywords

Derivative of a polynomial; Zygmund’s inequality; Lp norm of a polynomial

• # Abstract

LetP(z) be a polynomial of degreen which does not vanish in the disk |z|&lt;k. It has been proved that for eachp&gt;0 andk≥1,$$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered}$$ where$$B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p}$$ andP(s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.

• # Author Affiliations

1. Postgraduate Department of Mathematics and Statistics, University of Kashmir, Hazratbal, Srinagar - 190 006, India

• # Proceedings – Mathematical Sciences

Volume 133, 2023
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019