Growth of harmonic conjugates in the unit disc

Abstract:

Assuming some mild regularity conditions on a positive nondecreasing function $\psi (x) = O({x^a})$ (for some $a > 0,x \to \infty$), we show that \[ {M_p}(r,u) = O\left ( {\psi \left ( {\frac {1} {{1 - r}}} \right )} \right )(r \to 1,0 < p < 1)\] implies ${M_p}(r,v) = O{({\tilde \psi ^p}(1/(1 - r)))^{1/p}}$, where $u(z) + iv(z)$ is holomorphic in the open unit disc and \[ {\tilde \psi ^p}(x) = \int _{1/2}^x {\frac {{{\psi ^p}(t)}}{t}dt,\quad x \geqslant \frac {1}{2}.} \]