Joint estimates for zeros and Taylor coefficients of entire function

In the paper, for an entire function $f(z)=\sum\limits_{n=0}^{\infty} f_n z^n$, 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 $|f_n|\leqslant\varphi(n)$ with some function $\varphi(x),$ then the absolute values of the zeroes grows faster than $1/\sqrt[n]{\varphi(n)}.$ 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 $f(z)$ 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.