 Derivative criterion for Hölder continuity up to a boundary

If a holomorphic function on a half-rectangle satisfies $|f^{\prime }(x+iy)|\le C{y}^{-1+\alpha }$ with $0<\alpha \le 1$, 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 $O(r^{\alpha })$.