Subharmonicity of p|f| for quasiregular harmonic functions, with applications

We prove that if f is a quasiregular harmonic function, then there exists a number q∈(0,1) such that q|f| is subharmonic, and use this fact to generalize a result of Rubel, Shields, and Taylor, and Tamrazov, on the moduli of continuity of holomorphic functions.