Cauchy derivative formula

Cauchy’s derivative formula expresses derivatives as contour integrals with higher-order Cauchy kernels.

If $f$ is analytic and bounded by $K$ on $\mathrm{Re}z>0$, then Cauchy's derivative estimate giveson $\mathrm{Re}z>c$. Integrating along line segments gives a Lipschitz bound with the same constant in that smaller half-plane.