Cyclicity and iterated logarithms in the Dirichlet space

Let D ( μ ) D(μ ) denote a superharmonically weighted Dirichlet space on the unit disc D D . We show that outer functions f ∈ D ( μ ) f∈ D(μ ) are cyclic in D ( μ ) D(μ ) , whenever log ⁡ f log f belongs to the Pick-Smirnov class N + ( D ( μ ) ) N^+(D(μ )) . If f f has H ∞ H^∞ -norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions f f are cyclic, whenever log ⁡ ( 1 + log ⁡ ( 1 / f ) ) ∈ N + ( D ( μ ) ) log (1+ log (1/f))∈ N^+(D(μ )) . This condition can be checked by verifying that