Generalization and development of the Malliavin-Rubel theorem on small entire functions of exponential type with given zeros

Previously, we developed the technique of balayage of measures or charges and ($\delta$-)subharmonic functions of finite order onto an closed system of rays $S$ with a vertex at zero on the complex plane $\mathbb{C}$. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order $1$ and their differences. First, it is a classical balayage of the genus $q=0$ on a system of four closed rays: positive and negative, and real and imaginary semi-axis ${\mathbb{R}}^{+}$, $-{\mathbb{R}}^{+}$, $i{\mathbb{R}}^{+}$, $-i\mathbb{R}$. Second, it is two-sided balayage of genus $q=1$ from the open right and left half-planes ${\mathbb{C}}_{\mathrm{r}\mathrm{h}}$ and ${\mathbb{C}}_{\mathrm{l}\mathrm{h}}$ onto the imaginary axis $i\mathbb{R}$. The classical Malliavin-Rubel theorem gives necessary and sufficient conditions of the existence of an entire function of exponential type (we write e.f.e.t.) $f\equiv ̸0$, vanishing on the given positive sequence ${\sf Z}={{\sf z}k}{k\in \mathbb N}\subset \mathbb R^+$andsatisfyingtheconstraint$|f|\leq |g|$on$i\mathbb R$,where$g$ is an e.f.e.t., vanishing on positive sequence $\mathsf{W}=\{{\mathsf{w}}_k{\}}_{k\in \mathbb{N}}\subset {\mathbb{R}}^{+}$. A combination of these special balayage processes of genus $q=0$ and $q=1$ allows us to extend the Malliavin-Rubel theorem to arbitrary complex sequences ${\sf Z}={{\sf z}k}{k\in \mathbb N}\subset \mathbb C$ separated by a pair of vertical angles from the imaginary axis $i\mathbb{R}$, with much more general restrictions $\ln \mathrm{\mid }f\mathrm{\mid }\le M$ on the imaginary axis $i\mathbb{R}$, where $M$ is an subharmonic function of finite type under the order $1$.