Schwarz integral on the unit disk ($H(z)={\int }_{\mathbb{T}}\frac{e^{it}+z}{e^{it}-z}u(t)\frac{dt}{2\pi }$)

For real boundary data $u$, the displayed holomorphic integral has real part equal to the disk Poisson integral of $u$ and imaginary part its normalized harmonic conjugate. If $u\ge 0$ is not identically zero, the real part is strictly positive inside the disk. A holomorphic logarithm then has imaginary part between $-\pi /2$ and $\pi /2$, while its real part records the logarithm of the magnitude. This separates uniform imaginary-part control from large real parts on small boundary sets.