The theorem on the least majorant and its applications.I. Entire and meromorphic functions

The general concept of sweeping out is used to generalize the theorem of Koosis on the least superharmonic majorant in $\mathbb C$ to least majorants with respect to a convex cone of functions defined in a domain in $\mathbb R^k$ or $\mathbb C^n$. This generalization is applied to the description of nontrivial ideals and analytic sets of nonuniqueness of codimension 1 in algebras of entire functions, and to the representation of meromorphic functions of given growth.