Locally uniform convergence of analytic functions via $L^1$ convergence on circles

Let \(f _ { n } , f\) be analytic functions on the unit disk D. Show that the following are equivalent.

1. \(f _ { n } ( z )\) converges to f(z) uniformly on compact subsets in D.

2. \(\int _ { | z | = r } | f _ { n } ( z ) - f ( z ) | | d z |\) converges to 0 if \(0 < r < 1\)