Geodesic
$L_{p}$ inequalities for the growth of polynomials with restricted zeros
Archivum mathematicum, Tome 58 (2022) no. 3, pp. 159-167 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2022-3-159
Classification : 26D10, 30C15, 41A17
Keywords: polynomials; integral inequalities; complex domain 
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Rather, Nisar A.; Gulzar, Suhail; Bhat, Aijaz A. $L_{p}$ inequalities for the growth of polynomials with restricted zeros. Archivum mathematicum, Tome 58 (2022) no. 3, pp. 159-167. doi: 10.5817/AM2022-3-159

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