Entire functions of exponential type not vanishing in the half-plane \(\Im z > k\), where \(k > 0\)

Mohamed Amine Hachani

Abstract


Let \(P (z)\) be a polynomial of degree \(n\) having no zeros in \(|z| < k\), \(k \leq 1\), and let \(Q (z) := z^n \overline{P (1/{\overline {z}})}\). It was shown by Govil that if \(\max_{|z| = 1} |P^\prime (z)|\) and \(\max_{|z| = 1} |Q^\prime (z)|\) are attained at the same point of the unit circle \(|z| = 1\), then \[\max_{|z| = 1} |P'(z)| \leq \frac{n}{1 + k^n} \max_{|z| = 1} |P(z)|.\]
The main result of the present article is a generalization of Govil's polynomial inequality to a class of entire functions of exponential type.


Keywords


Inequalities; entire functions of exponential type; polynomial; trigonometric polynomial

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References


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DOI: http://dx.doi.org/10.17951/a.2017.71.1.31
Data publikacji: 2017-06-30 17:33:54
Data złożenia artykułu: 2017-06-29 19:04:26

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