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

Mohamed Amine Hachani — 2017

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Let P ( z ) be a polynomial of degree n having no zeros in | z | < k , k 1 , and let Q ( z ) : = z n P ( 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 ) | 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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