Entire functions of exponential type not vanishing in the half-plane , where
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2017)
- Volume: 71, Issue: 1
- ISSN: 0365-1029
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topMohamed Amine Hachani. "Entire functions of exponential type not vanishing in the half-plane $\Im z > k$, where $k > 0$." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 71.1 (2017): null. <http://eudml.org/doc/289813>.
@article{MohamedAmineHachani2017,
abstract = {Let $P (z)$ be a polynomial of degree $n$ having no zeros in $|z| < k$, $k \le 1$, 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 \[\max \_\{|z| = 1\} |P^\{\prime \}(z)| \le \frac\{n\}\{1 + k^n\} \max \_\{|z| = 1\} |P(z)|.\]
The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.},
author = {Mohamed Amine Hachani},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Inequalities; entire functions of exponential type; polynomial; trigonometric polynomial},
language = {eng},
number = {1},
pages = {null},
title = {Entire functions of exponential type not vanishing in the half-plane $\Im z > k$, where $k > 0$},
url = {http://eudml.org/doc/289813},
volume = {71},
year = {2017},
}
TY - JOUR
AU - Mohamed Amine Hachani
TI - Entire functions of exponential type not vanishing in the half-plane $\Im z > k$, where $k > 0$
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2017
VL - 71
IS - 1
SP - null
AB - Let $P (z)$ be a polynomial of degree $n$ having no zeros in $|z| < k$, $k \le 1$, 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 \[\max _{|z| = 1} |P^{\prime }(z)| \le \frac{n}{1 + k^n} \max _{|z| = 1} |P(z)|.\]
The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.
LA - eng
KW - Inequalities; entire functions of exponential type; polynomial; trigonometric polynomial
UR - http://eudml.org/doc/289813
ER -
References
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- 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).
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- 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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