Abstract
The Erdős–Lax Theorem states that, if P(z) is a polynomial of degree n having no zeros in \(|z|<1\), then
The problem of generalizing the Erdős–Lax Theorem to the class of polynomials having no zeros in \(|z|<K, K\leq 1\) is still not settled. Motivated by the above open problem, in this paper we prove some inequalities for a class of polynomials having no zeros in \(|z|<K, K\leq 1\), and all these zeros lie either on a ray L emanating from the origin or zeros are symmetrically placed along this line L. Several important consequences are also discussed. Our results are the best possible and examples for the equality cases have been presented.
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The author is grateful to the anonymous referee for valuable suggestions, especially on Hurwitz polynomials.
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The work in the paper was supported by a research grant from the Council for Scientific and Industrial Research, Government of India.
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Kumar, P. On the growth of polynomials. Acta Math. Hungar. 168, 443–453 (2022). https://doi.org/10.1007/s10474-022-01281-8
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DOI: https://doi.org/10.1007/s10474-022-01281-8