Subharmonic test functions and the distribution of zero sets of holomorphic functions

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 a part of 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.