Vinay Kumar Jain: Inequalities for a polynomial with prescribed zeros, p.451-461

Abstract:

For a polynomial $p(z)$ of degree $n$ with a zero of order $k (\geq 1)$ at $\beta$, it is known that

$$\nonumber \max_{\vert z\vert=1} \vert\frac{p(z)}{(z-\beta)^k... ...rt}\Bigr)^k \max_{1 \leq l \leq n-k+1}\vert p(\gamma'_l)\vert,$$

$\gamma'_1,\gamma'_2,\hdots, \gamma'_{n-k+1}$ being the roots of $z^{n-k+1}+e^{i\gamma(n-k+1)}=0$, with $\gamma = \arg \beta$ ( $\gamma = 0 \textrm{ for } \beta = 0$). By considering a polynomial $p(z)$ of degree $n$ with zeros $\beta_1, \beta_2, \hdots, \beta_k$ we have obtained certain inequalities thereby giving a refinement of the known result.

Key Words: Polynomial, prescribed zeros, inequalities, refinement.

2000 Mathematics Subject Classification: Primary: 30C10;
Secondary: 30A10.

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