Rate of growth of polynomials with restricted zeros

Mir, Abdullah and Dawood, Q. M.

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creative_2016_25_2_197_203

DOI: https://doi.org/10.37193/CMI.2016.02.11

Issue no: Vol 25/2016 no. 2 Tags: maximum modulus principle, Growth, inequalities, Polynomials

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In this paper we consider for a fixed $\mu,$ the class of polynomials $P(z)=a_0+\sum\limits_{\nu=\mu}^{n}a_\nu z_\nu,\, 1\leq \mu \leq n,$ of degree at most $n$ not vanishing in the disk $|z|<k, k>0.$ For any $\rho>\sigma\geq 1$ and $0<r\leq R\leq k,$ we investigate the dependence of $\parallel P(\rho z)-P(\sigma z)\parallel_R$ on $\parallel P \parallel_r$ and derive various refinements and generalizations of some well known results.