Let $L^p(\mathbf{ T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{ T}=\{ z: |z|=1\}\subseteq \mathbb{ C}$. In this paper, we address the problem of finding the best constant in the inequality 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 quasiconformal harmonic mapping. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.
Categories: math.CV