- Show that if \(f\) is holomorphic on \(\DD\) then \(f\) has a power series expansion that converges uniformly on every compact \(K\subset \DD\).
Solution
Write the Taylor expansion at \(0\) as \[ f(z)=\sum_{n=0}^\infty a_nz^n, \qquad |z|<1. \] Let \(K\subset\mathbb D\) be compact and choose numbers \[ r:=\max_{z\in K}|z|<R<1. \] If \(M_R=\max_{|\zeta|=R}|f(\zeta)|\), Cauchy’s estimates give \[ |a_n|\le\frac{M_R}{R^n}. \] Hence for \(z\in K\), \[ |a_nz^n|\le M_R(r/R)^n. \] The right-hand side is a summable geometric majorant, so the Weierstrass \(M\)-test gives uniform convergence on \(K\).