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## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2017 | 71 | 1 |
Tytuł artykułu

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

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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.
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2017
online
2017-06-30
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Bibliografia
• Besicovitch, A. S., Almost Periodic Functions, Cambridge University Press, London, 1932.
• Boas, R. P. Jr., Entire Functions, Academic Press, New York, 1954.
• Boas, R. P. Jr., Inequalities for asymmetric entire functions, Illinois J. Math. 3 (1957), 1-10.
• Bohr, H., Almost Periodic Functions, Chelsea Publishing Company, New York, 1947.
• van der Corput, J. G., Schaake G., Ungleichungen fur Polynome und trigonometrische Polynome, Composito Math. 2 (1935), 321-61.
• Govil, N. K., On a theorem of S. Bernstein, Proc. Nat. Acad. Sci. India 50 (A) (1980), 50-52.
• Levin, B. Ya., On a special class of entire functions and on related extremal properties of entire functions of finite degree, Izvestiya Akad. Nauk SSSR. Ser. Math. 14 (1950), 45-84 (Russian).
• Qazi, M. A., Rahman, Q. I., The Schwarz–Pick theorem and its applications, Ann. Univ. Mariae Curie-Skłodowska Sect. A 65 (2) (2011), 149-167.
• Rahman, Q. I., Schmeisser, G., Analytic Theory of Polynomials, Clarendon Press, Oxford, 2002.
• Riesz, M., Formule d’interpolation pour la derivee d’un polynome trigonometrique, C. R. Acad. Sci. Paris 158 (1914), 1152-1154.
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