Subsequences of zeros for classes of entire functions of exponential type distinguished by growth restrictions

Abstract:

Let $M$ be a function subharmonic on the complex plane $\mathbb {C}$, harmonic outside of the real axis, and such that \[ \limsup _{z\to \infty }\frac {M(z)}{|z|}<+\infty ,\quad \int _{-\infty }^{+\infty } \frac {\max \{0, M(x)\}}{x^2} d x<+\infty , \quad M(0)=0, \] and $M(z)=M(\overline z)$ for all $z\in \mathbb {C}$. A description is given for all sequences of points in $\mathbb C$ that are included in the zero set of some nonzero entire function $f$ with $|f(z)|\leq \exp M(z)$ for all $z\in \mathbb {C}$.