Density in cos πρ theorems for δ-subharmonic functions
Summary Suppose that u(z) and v(z) are subharmonic functions in the plane, of order at most ρ, where 0 < ρ < 1/2, and that u(z) is nonconstant. Let φ(z) = u(z) − v(z), and suppose that the deficiency δ of φ(z) satisfies 0 ≤ 1 − δ < cos πρ. It is shown that, given σ such that both ρ < σ < 1/2 and 0 ≤ 1 − δ < cos πσ, we have A(r,φ)/B(r,φ) > (cos πσ − (1 − δ))/(1 − (1 − δ)cos πσ), for all r in a set of lower logarithmic density at least 1 − ρ/σ.