Behavior of subharmonic functions of slow growth outside exclusive sets - Zabolotskyy, M.; Zabolotskyy, T. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC517.53 Let v be a slowly growing function unbounded on [0, +∞), u be subharmonic (in plane) function of zero order, μ be its Riesz measure, n(t,u)=μ({x|x|t}), N(t,u)=_1^tn(τ,u)/τ dτ, and n(r,u)=O(v(r)), r→+∞. A set E C is called a C_0^β-set, 0 < β 1, if E can be covered by a system of disks K(a_n,r_n)={z|z-a_n| < r_n} such that _|a_n| r r_n^β = o(r^β), r→+∞. Then, for every nondecreasing function φ unbounded on [0, +∞), there exists a C_0^β-set E such that equation*u(z)=N(r,u)+o(φ(r)v(r)),r=|z|→+∞,z otin E.equation* It is shown that, in this asymptotic formula, the remainder term o(φ(r)v(r)) cannot be changed by O(v(r)).