Zeros of holomorphic functions in the unit disk and ρ-trigonometrically convex functions

Let M be a subharmonic function with Riesz measure mu(M) on the unit disk D in the complex plane C. Let f be a nonzero holomorphic function on D such that f vanishes on Z subset of D, and satisfies vertical bar f vertical bar = exp M on D. Then restrictions on the growth of mu(M) near the boundary of D imply certain restrictions on the distribution of Z. We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using rho-trigonometrically convex functions.