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Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functionsdefined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. Inthis paper, we address the problem of finding the best constant in theinequality of the form: $$\|f\|_{L^p(\mathbf{T})}\le A_{p,b} \|(|P_+ f|^2+b|P_{-} f|^2)^{1/2}\|_{L^p(\mathbf{T})}.$$ Here $p\in[1,2]$, $b>0$, and by $P_{-}f$ and $ P_+ f$ are denoted co-analytic and analytic projection of a function$f\in L^p(\mathbf{T})$. The equality is "attained" for a quasiconformalharmonic mapping. The result extends a sharp version of M. Riesz conjugatefunction theorem of Pichorides and Verbitsky and some well-known estimates forholomorphic functions.

On M. Riesz conjugate function theorem for harmonic functions