Derivative criterion for Hölder continuity up to a boundary
= Derivative criterion for Hölder continuity up to a boundary
If a holomorphic function on a half-rectangle satisfies $|f'(x+iy)|\leq Cy^{-1+\alpha}$ with $0<\alpha\leq1$, then it extends to an $\alpha$-Hölder continuous function on the closed half-rectangle. Integrate vertically to height $r=|z-w|$, horizontally at that height, and vertically back; each contribution is $O(r^\alpha)$.