Lower bounds on the values of an entire function of exponential type at certain integers, in terms of a least superharmonic majorant

In this,paper and the following one, it is shown that if $A<\pi$ and $\eta>0$ is sufficiently small (depending on $A$), the entire functions $f(z)$ of exponential type $\le A$ satisfying $\sum^{\infty}_{m=-\infty}(\log^+|f(n)|/(1+n^2))\le\eta$ form a normal family (in $\mathbb C$). General properties of least superharmonic majorants are used to obtain this result, and from it the multiplier theorem of Beurling and Malliavin is readily derived.