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Keywords:
polynomials; integral inequalities; complex domain
polynomials; integral inequalities; complex domain
Summary:
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.
References:
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