Abstract. The main result in this paper is the following: Let $f=u+iv\in H(U)$ with $v(0)=0,$ $p\in[1,\infty )$, $n\in{\msbm N}$ and $\omega $ is an admissible weight with distortion function $\psi.$ Then there are constants $C=C(p,q,n) $ (resp. $C(p,q,n,\omega )$) such that $${\rm(a)} ||u||^q_{\omega,p,q} \leq C\left( |u(0)|^q+\int_0^1M_p^q(f',s)\psi ^q(s)\omega(s)ds\right ) \mbox{ for } q\in[1,\infty );$$ $${\rm(b)} |u(0)|^q+\int_0^1M_p^q(f^{(n)},s)\psi ^{nq}(s)\omega(s)ds \leq C ||u||^q_{\omega,p,q} \mbox{ for } q\in(0,\infty ).$$

AMS Subject Classification (1991): 31A05, 30E99

Keyword(s): Distortion function, admissible weight, integral means, unit disk, conjugate function, harmonic function

Received September 20, 2001, and in revised form April 10, 2002. (Registered under 2887/2009.)