Wiman and Arima Theorems for Quasiregular Mappings

It follows from the Ahlfors theorem that an entire holomorphic function f of order ρ has no more than 2ρ distinct asymptotic curves where r stands for the largest integer ≤ r. This theorem does not give any information if ρ < 1/2, This case is covered by two theorems: if an entire holomorphic function f has order ρ < 1/2 then lim supr→∞min|z| r |f z | ∞ Wiman 1 and if f is an entire holomorphic function of order ρ > 0 and l is a number satisfying the conditions 0 < l ≤ 2π, l < π/ρ, then there exists a sequence of circular arcs {|z| rk, θk ≤ arg z ≤ θk l}, rk → ∞, 0 ≤ θk < 2π, along which |f z | tends to ∞ uniformly with respect to arg z Arima 2 . Below we prove generalizations of these theorems for quasiregular mappings for n ≥ 2. The next two theorems are generalizations of the theorems of Wiman and of Arima for quasiregular mappings on manifolds.