$L_{p}$ inequalities for the growth of polynomials with restricted zeros

Rather, Nisar A. ; Gulzar, Suhail ; Bhat, Aijaz A.

Archivum mathematicum, Tome 58 (2022) no. 3, pp. 159-167.

Résumé

Let $P(z)=\sum _{\nu =0}^{n}a_{\nu }z^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $|z|1$, then for $1\le p\infty $ and $R>1$, Boas and Rahman proved \[\left\Vert P(Rz)\right\Vert _{p}\le \big (\left\Vert R^{n}+z\right\Vert_{p}/\left\Vert1+z\right\Vert_{p}\big )\left\Vert P\right\Vert _{p}.\] In this paper, we improve the above inequality for $0\le p \infty $ by involving some of the coefficients of the polynomial $P(z)$. Analogous result for the class of polynomials $P(z)$ having no zero in $|z|>1$ is also given.

Détail

DOI : 10.5817/AM2022-3-159

Classification : 26D10, 30C15, 41A17
Keywords: polynomials; integral inequalities; complex domain