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Growth of harmonic conjugates in the unit disc. (English) Zbl 0614.31001

Let \(M_p(r,u)\) be the usual \(p\)-mean of \(u(z)\) on \(| z| =r\). ”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 \(M_p(r,u)=O(\psi (1/(1-r))\), \(r\to 1\), \(0<p<1\), 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 \(\tilde{\psi}^p(x)=\int^{x}_{1/2}(\psi^ p(f)/f)\,dt\), \(x\geq\tfrac12.\)”

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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