Journal of the
Korean Mathematical Society JKMS

For any $K>2/\pi$ we determine the optimal constant $L(K)$ for which the following holds. If $u$, $\tilde{u}$ are conjugate harmonic functions on the unit disc with $\tilde{u}(0)=0$, then $$ \int_{-\pi}^{\pi}|\tilde{u}(e^{i\phi})|\frac{\mbox{d}\phi}{2\pi} \leq K\int_{-\pi}^\pi |u(e^{i\phi})|\log^+|u(e^{i\phi})|\frac{\mbox{d}\phi}{2\pi}+L(K).$$ We also establish a related estimate for orthogonal harmonic functions given on Euclidean domains as well as an extension concerning orthogonal martingales under differential subordination.

Keywords: harmonic function, martingale, $L\log L$ inequality, differential subordination, best constants

MSC numbers: Primary 60G44; Secondary 31B05