Lower average estimate for the minimum modulus on circles foran entire function of genus zero

Abstract

The article was written based on the materials of the joint report of the authors, made by them at the Sixth International Conference “Functional spaces. Differential operators. Problems of mathematical education,” dedicated to the centenary of the birth of Corresponding Member of the Russian Academy of Sciences, Academician of the European Academy of Sciences L.D.Kudryavtsev. For an entire function represented by a canonical product of genus zero with positive roots, the following result is proved. For any \(\delta\in(0,1/3]\), the minimum modulus of such a function exceeds on average the maximum of its modulus raised to the power \(-1-\delta,\) on any segment whose end ratio is equal to \(\exp(
 2/\delta).\) The main theorem is illustrated by two examples. The first of them shows that instead of the exponent \(-1-\delta\) it is impossible to take \(-1.\) The second example demonstrates the impossibility of replacing the value \(\exp(2/\delta)\) by the value\(28/(15\delta)\) in the theorem for small \(\delta.\)