Let u(z) be a real_valued harmonic function in the unit disk, we say u(z)∈h p(1<p<+∞), if the p _mean M p(r,u)=12π∫ 2π 0|u(re i θ )| p d θ 1/p is bounded. M.Riesz showed that if u(z)∈h p(1<p<+∞), then there exists a constant A p, depending only on p such that M p(r,v)≤A pM p(r,u), where v(z) is the conjugate harmonic function of u(z).When v(0)=0 and 1<p≤2, W.K.Hayman showed that A p can be given by pp-1 1/p . First, this paper shows that the constant pp-1 1/p can be changed by a smaller constant pp-1 2/p -1 1/2 . Next, if 1<p≤2, then there exists a constant θ 0∈2-p2pπ,π2p such that M p p(r,v)≤ Im p sin pπ2+ tg pπ2pM p p(r,u) for any analytic function f(z)=u(z)+ i v(z) in the unit disk, whenever -θ 0< arg f(z)<π2.
