Lower bound for minimum of modulus of entire function of genus zero with positive roots

A. Yu. Popov ; V. B. Sherstyukov

Ufa mathematical journal, Tome 14 (2022) no. 4, pp. 76-95

Résumé

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 $a=\tfrac{9}{4}$ and for $a\in(1,\tfrac{9}{8}]$. We find an asymptotics for $d(a)$ as $a\rightarrow1$. The obtained result differs principally from the classical $\cos(\pi\rho)$-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 $\rho\in[0,1]$ is estimated by a power of the maximum of its modulus.

Détail

Keywords: entire function, minimum of modulus, maximum of modulus.