Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complex
plane $\mathbb C$. The following problem dates back to
F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin,
J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$,
for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, there
exists an entire function $f\neq 0$ of exponential type
that vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis?
The classical Malliavin–Rubel theorem of the early 1960s completely
solves this problem for “positive” $\mathrm Z$ and $\mathrm W$ (which lie only
on the positive semiaxis). Several generalizations of this criterion were
established by the author of the present paper in the late 1980s for “complex”
$\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separated
by angles from the imaginary axis, with some advances in the 2020s.
In this paper, we solve more involved problems in a more general subharmonic
framework for distributions of masses on $\mathbb C$. All the previously
mentioned results can be obtained from the main results of this paper
in a much stronger form (even for the initial formulation for distributions
of points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$
of exponential type).
Some results of the present paper are closely related
to the famous Beurling–Malliavin theorems on the radius of completeness and a multiplier.
