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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
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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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bwmeta1.element.ojs-doi-10_17951_a_2017_71_1_31