Inequalities for imaginary parts of zeros of entire functions (Q1572959)

Let \(f(z):=\sum_{k=0}^{\infty}a_kz^k\), where \(a_k \in \mathbb C\) and \(a_0=1\), be an entire function. Let \[ \psi:=\left (|\text{Im} (a_1)|^2+\sum_{k=2}^{\infty}|a_k|^2\right)^{1/2}. \] Let \(\{z_k\}_{k=1}^{m}\), \(1\leq m \leq \infty\), denote the zeros of \(f\), enumerated such that \[ |\text{Im}(1/z_k)|\geq |Im(1/z_{k+1})|. \] In the paper under review, the author proves (Theorem 1) that for \(N=1,2,3,\dots\), the zeros of \(f\) satisfy the following inequalities \[ \sum_{k=1}^{N}\left |\text{Im}\left(\frac{1}{z_k}\right)\right |\leq N+ \psi. \] The paper concludes with several corollaries.