 Solution

Schwarz reflection shows that the Taylor coefficients along the real axis are real. Since $v(x,0)=0$ and $v(x,y)>0$ for $y>0$, one has $v_y(x,0)\ge 0$. The Cauchy–Riemann equations give $h^{\prime }(x)=u_x(x,0)=v_y(x,0)\ge 0$.