Entire functions of exponential type not vanishing in the half-plane ℑz>k, where k>0
Mohamed Amine Hachani
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 71 (2017), / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let P(z) be a polynomial of degree n having no zeros in |z|<k, k≤1, and let Q(z):=znP(1/z¯)¯. It was shown by Govil that if max|z|=1|P'(z)| and max|z|=1|Q'(z)| are attained at the same point of the unit circle |z|=1, then max|z|=1|P'(z)|≤n1+knmax|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.

Publié le : 2017-01-01
EUDML-ID : urn:eudml:doc:289813 
@article{bwmeta1.element.ojs-doi-10_17951_a_2017_71_1_31,
     author = {Mohamed Amine Hachani},
     title = {Entire functions of exponential type not vanishing in the half-plane $\Im z > k$, where $k > 0$
            },
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {71},
     year = {2017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2017_71_1_31}
}

Mohamed 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, Tome 71 (2017) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2017_71_1_31/

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.