Harmonic and holomorphic Schwarz reflection across the real axis

For the harmonic statement, let $u$ satisfy the hypotheses, and define continuous boundary data $\mathrm{\Phi }$ on the unit circle by taking the upper semicircle values of $u$ and extending them oddly across the real axis. By [L1], $\mathrm{\Phi }$ has a harmonic Poisson extension $H$ to the full unit disc. On the upper half-disc, $H$ and $u$ are continuous harmonic functions with the same boundary values on the upper semicircle and on the diameter $(-1,1)$, so [L1] makes them equal there. By the odd construction of $\mathrm{\Phi }$, the same harmonic function satisfies $H(z)=-u(\bar{z})$ on the lower half-disc. Thus $H$ is exactly the reflected function $U$, so $U$ is harmonic.

For the holomorphic statement, write $f=u+iv$ on ${\mathbb{D}}^{+}$. Since $f$ is real-valued on $(-1,1)$, one has $v=0$ there; applying step 1.1 to $v$ gives a harmonic function $V$ on the full disc that equals $v$ above the axis and $-v(\bar{z})$ below it. Because the disc is star-shaped, [L2] gives a harmonic conjugate $W$ of $V$, so $V+iW$ is holomorphic. Multiplying by $i$ shows that $G:=-W+iV$ is holomorphic and has imaginary part $V$.

On ${\mathbb{D}}^{+}$, the holomorphic functions $G$ and $f$ have the same imaginary part $v$, so their difference is real-valued and holomorphic; [L2] makes $G-f$ a real constant there. Subtracting that constant from $G$, we may assume $G=f$ on ${\mathbb{D}}^{+}$.

For $\mathrm{Im}z<0$, the functions $G(z)$ and $\overline{f(\stackrel{‾}{z})}$ have the same imaginary part $-v(\bar{z})$. Their difference is therefore real-valued and holomorphic on the lower half-disc, hence constant by [L2]; continuity across the diameter, where both functions equal the same real boundary values, forces that constant to be $0$. So $G(z)=\overline{f(\stackrel{‾}{z})}$ below the axis, and the reflected function $F$ is holomorphic on the full disc.