Strengthening of Gaisin’s lemma on the minimum modulus of even canonical products

Abstract

We consider entire functions that are even canonical products of zero genus, all roots of which are located on the real axis.We study the question of lower bound the minimum modulus of such functions on the circle in terms of some negative power of the maximum modulus on the same circle, when the radius of the circle runs through segments with a constant ratio of ends. In 2002 A. M. Gaisin, correcting the erroneous reasoning of M. A. Evgrafov from the book «Asymptotic estimates and entire functions», proved that for each function of the class under consideration there exists a sequence of circles, whose radii tend to infinity, the ratio of the subsequent radius
to the previous one is less than 4, and these circles are such that on each of them the minimum modulus of the function exceeds the −20-th power of the maximum of its modulus. This result is strengthened by us in three directions. First, the exponent −20 has been replaced by −2.
Secondly, we proved that the radii of the circles on which the minimum modulus of the function exceeds the −2-th power maximum of its modulus occur on every interval whose end ratio is 3. Thirdly, we found out that the discussed inequality is true for the functions of the class under study «on average». The latter means that if we take the logarithm of the product of the minimum modulus of a function on a circle and the square of its maximum modulus, divide by the cube of the radius and integrate over all radii belonging to an arbitrary segment with an end ratio of 3, it will be a positive value.