LAUSR

Let M be a subharmonic function with Riesz measure $$\mu _M$$μM on the unit disk $${\mathbb {D}}$$D in the complex plane $${\mathbb {C}}$$C. Let f be a nonzero holomorphic function on $${\mathbb {D}}$$D such that f vanishes on $${\textsf {Z}}\subset {\mathbb {D}}$$Z⊂D, and satisfies $$|f| \le \exp M$$|f|≤expM on $${\mathbb {D}}$$D. Then restrictions on the growth of $$\mu _M$$μM near the boundary of D imply certain restrictions on the distribution of $$\mathsf Z$$Z. We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $$\rho $$ρ-trigonometrically convex functions.