Abstract. Let m, n ≥ 1 are integers and D be a domain in the complex plane C or in the m-dimensional real space R m . We build positive subharmonic functions on D vanishing on the boundary ∂D of domain D. We use such (test) functions to study the distribution of zero sets of holomorphic functions f on D ⊂ C n with restrictions on the growth of f near the boundary ∂D.

results and their implementation in more or less concrete situations are new not only for subharmonic functions u, and also for holomorphic functions f even in the case when D is C, the unit disk, an annulus etc. Thus this is not a review.

Let m, n ≥ 1 are integers and D be a domain in the complex plane C or in the m-dimensional real space R m . We build positive subharmonic functions on D vanishing on the boundary ∂D of domain D. We use such (test) functions to study the distribution of zero sets of holomorphic functions f on D ⊂ C n with restrictions on the growth of f near the boundary ∂D.