Termwise differentiation of locally uniformly absolutely convergent series of analytic functions

Let \(a _ { n } ( z )\) be an analytic sequence in a domain D such that \(\sum _ { n = 0 } ^ { \infty } | a _ { n } ( z ) |\) converges uniformly on bounded and closed sub-regions of D. Show that \(\sum _ { n = 0 } ^ { \infty } \left| a _ { n } ^ { \prime } ( z ) \right|\) converges uniformly on bounded and closed sub-regions of \(D .\)