Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 604217, 29 pages doi:10.1155/2010/604217 Research Article Wiman and Arima Theorems for Quasiregular Mappings O. Martio, 1 V. M. Miklyukov, 2 and M. Vuorinen 3 1 Department of Mathematics and Statistics, University of Helsinki, 00014 Helsinki, Finland 2 Department of Mathematics, Volgograd State University, 2 Prodolnaya 30, Volgograd 400062, Russia 3 Department of Mathematics, University of Turku, 20014 Turku, Finland Correspondence should be addressed to M. Vuorinen, vuorinen@utu.fi Received 28 December 2009; Accepted 11 February 2010 Academic Editor: Shusen Ding Copyright q 2010 O. Martio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wiman’s theorem says that an entire holomorphic function of order less than 1/2 has a minimum modulus converging to ∞ along a sequence. Arima’s theorem is a refinement of Wiman’s theorem. Here we generalize both results to quasiregular mappings in the manifold setup. The so called fundamental frequency has an important role in this study. 1. Main Results 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 sup r →∞ 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|  r k ,θ k ≤ arg z ≤ θ k  l},r k → ∞, 0 ≤ θ k < 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. Theorem 1.1. Let M, N be n-dimensional noncompact Riemannian manifolds without boundary. Assume that h : M → 0, ∞ is a special exhaustion function of the manifold M and u is a nonnegative growth function on the manifold N, which is a subsolution of 3.4 with the structure conditions 3.2, 3.3 and the structure constants p  n, ν 1 , ν 2 .