Locally uniform convergence of holomorphic functions

A locally uniform convergence limit of holomorphic functions is holomorphic, and its derivatives converge locally uniformly as well. The limit is continuous. On each small triangle contained in the domain, uniform convergence allows passage to the limit in its contour integral; the integrals vanish by the Cauchy integral theorem. The Morera theorem therefore makes the limit holomorphic. Applying the Cauchy integral formula on circles in a slightly larger compact subset proves locally uniform convergence of every fixed-order derivative.

The entire functions $g_n(z)=z^n/n$ converge uniformly to zero on the closed unit disc, because their supremum norm is $1/n$. Their derivatives $g_n^{\prime }(z)=z^{n-1}$ instead have supremum norm one there, and equal one at $z=1$. On each smaller disc of radius $0<r<1$, the derivative supremum is $r^{n-1}\to 0$. Thus the Weierstrass convergence theorem controls derivatives on compact subsets of an open domain, not automatically on the boundary of a closed set on which function convergence is known.

For $0<r<R$, suppose $f_n,f$ are holomorphic on a neighborhood of the closed radius-$R$ disc. Subtract their Cauchy integral formula expressions on that circle. Its length is $2\pi R$, and the distance from an interior point with modulus at most $r$ to the circle is at least $R-r$. The estimation lemma gives the displayed bound. Uniform convergence on that circle therefore implies uniform convergence of each fixed-order derivative on the smaller disc. The positive separation from the contour is essential for a uniform bound.