= Schwarz integral on the unit disk
{c}
{title2=$H(z)=\int_{\mathbb T}\frac{e^{it}+z}{e^{it}-z}u(t)\,\frac{dt}{2\pi}$}
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 $u\geq0$ 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.
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