Relations for imaginary parts of zeros of entire functions (Q1768064)

For a finite order the entire function \[ f(\lambda) = \sum_{k=0}^\infty \frac{a_k\lambda^k}{(k!)^\gamma}, \quad\quad\lambda\in\mathbb C,\quad a_0=1,\quad\gamma>0, \] with complex coefficients \(a_k\), assume that \(\sum_k | a_k| ^2 <\infty\) and define \[ \psi_f := \biggl[| \text{Im}\,a_1| ^2+\sum_{k=2}^\infty | a_k| ^2\biggr]^{1/2}. \] The main result of the paper is given by the following inequalities: \[ \sum_{k=1}^j \biggl| \text{Im}\,\frac{1}{z_k(f)}\biggr| \leq \psi_f + \sum_{k=1}^j (k+1)^{-\gamma},\qquad j=1,2,\ldots, \] where \(\{z_k(f)\}_{k=1}^m\) (\(m\leq\infty\)) is the set of all zeros of \(f\) taken with their multiplicities, where it is agreed that the zeros are numbered according to \[ \biggl| \text{Im}\,\frac{1}{z_k(f)}\biggr| \geq \biggl| \text{Im}\,\frac{1}{z_{k+1}(f)}\biggr| , \qquad k=1,\ldots,m-1 \] with the further agreement that \(| z_k(f)| ^{-1}=0\) for \(k>m\) in case the zeros are finitely many. Further inequalities of this type are proved under different assumptions, and corresponding inequalities for the real parts as well as for the moduli of the zeros are also given.