Lower bound for minimum of modulus of entire function of genus zero with positive roots in terms of degree of maximal modulus at frequent sequence of points

Авторы: Popov A.Yu, Sherstyukov V.B.
Журнал: Ufa Mathematical Journal
Том: 14
Номер: 4
Год издания: 2022
Первая страница: 76
Последняя страница: 95
DOI: 10.13108/2022-14-4-76
Аннотация: We consider entire function of genus zero, the roots of which are located at a single ray. On the class of all such functions, we obtain close to optimal lower bounds for the minimum of the modulus on a sequence of the circumferences in terms of a negative power of the maximum of the modulus on the same circumferences under a restriction on the quotient a>1 of the radii of neighbouring circumferences. We introduce the notion of the optimal exponent d(a) as an extremal exponent of the maximum of the modulus in this problem. We prove two-sided estimates for the optimal exponent for a "test" value a=9/4 and for a∈(1,9/8]. We find an asymptotics for d(a) as a→1. The obtained result differs principally from the classical cos(πρ)-theorem containing no restrictions for the frequencies of the radii of the circumferences, on which the minimum of the modulus of an entire function of order ρ∈[0,1] is estimated by a power of the maximum of its modulus.
Добавил в систему: Шерстюков Владимир Борисович

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