Bounds for Zeros of Entire Functions

Abstract

Let f be an entire function. Denote by z1(f),z2(f),… the zeros of f with their multiplicities. In the paper, estimates for the sums $$\sum_{k=1}^{j}\frac{1}{|z_{k}(f)|}\quad (j=1,2,\ldots )$$ and for the counting function of the zeros of f are established. If f is of finite order ρ(f), we derive bounds for the series $$S_{p}(f):=\sum_{k=1}^{\infty}\frac{1}{|z_{k}(f)|^{p}}\quad (p\geq \rho(f))\quad \mbox{and}\quad \sum_{k=1}^{\infty}\biggl(\mathop{Im}\frac{1}{z_{k}(f)}\biggr)^{2}\quad (\rho(f)<2),$$ as well as relations between the series $$\sum_{k=1}^{\infty}\frac{1}{z_{k}^{m}(f)}\quad (m\geq \rho(f))$$ and the traces of certain matrices. The contents of the paper is closely connected with the following well-known results: the Hadamard theorem on the convergence exponent of the zeros of an entire function, the Jensen inequality for the counting function, the Cauchy theorem on the comparison of the zeros of polynomials, Ostrowski’s inequalities for the real and imaginary parts of the zeros of polynomials and the Cartwright–Levinson theorem. The suggested approach is based on the recent development of the spectral theory of linear operators.