On the mean growth of the solutions of complex linear differential equations in the disk

Let w be a solution of the differential equation w(z)+q(z)w(z)=0 where q is analytic in the unit disk. We study first the mean square growth of w and give a condition for w to belong to the Hardy space H2 : the proof is based on Carleson measures Then it is shown that implies that w is of bounded characteristic: the proof uses ideas of Hille together with the Hardy Littlewood maximal theorem. This result is finally applied to the case that q is a Г-automorphic form of weight 2 for some Fuchsian group Г this case arises if we consider our differential equation in a multiply connected domain and then use uniformization.