Joint estimates for zeros and Taylor coefficients of entire function

Abstract

In the paper, for an entire function ( ) = ∞ ∑︀ =0 , we provide asymptotic and uniform bounds of commensurability of the growth of zeroes and the decaying of the Taylor coefficients one with respect to the other. As an initial point for these studies, the following Hadamard statement serves: if the coefficients of the series obey the inequality | | ( ) with some function ( ), then the absolute values of the zeroes grows faster than 1/ √︀ ( ). In the present work we improve recently obtained lower bound for the joint growth of the zeroes and the coefficients via the maximal term of the Taylor series of the function ( ) or via the counting function of its zeroes. The employing of Hadamard-rectified coefficients of the series give an opportunity to establish corresponding two-sided estimates. By the methods developing classical ideas we find a numerical dependence of such estimates on the sizes of the gaps of the power series representing the entire function. In particular, we find asymptotic identities relating the zeroes and the coefficients of an entire function. The obtained estimates are sharp and strengthen the known results by other authors.

On the Type of Subharmonic Functions of Finite Order

“…For functions of nonintegral order, the degree of the polynomial in ( 5) is less than . Therefore, the order of the function coincides with the order of the canonical integral (6), and therefore…”

“…We note some of them. G. G. Braichev in [6] considered the problem of joint estimation of the zeros and Taylor coefficients of an entire function. In the series of works [7][8][9][10], he studied problems on the lower indicator of an entire function with zeros of zero lower density lying on a ray, two-sided estimates of the relative growth of functions and their derivatives, and also the smallest type of an entire function with zeros of given averaged densities and exact estimates for the types of entire…”

“…functions with zeros on the rays. We note the joint works by G. G. Braichev and V. B. Sherstyukov [11,12], which dealt with the problem of estimating the indicators of an entire function with negative zeros and exact estimates of the asymptotic characteristics of the growth of entire functions with zeros on given sets, as well as the work by G. G. Braichev and O. V Sherstyukova [13], in which the problem of the smallest type of an entire function with a given subsequence of zeros was considered. A series of works by B. N. Khabibullin are devoted to this issue (see [14][15][16]).…”