On integrable and bounded automorphic forms

A necessary and sufficient condition that every integrable automorphic form of dimension > − 2 > - 2 be a bounded form is established. Using this condition, it is shown that, for a finitely generated Fuchsian group acting on the unit disc and containing no parabolic elements, every integrable automorphic form of dimension > − 2 > - 2 is bounded. Here the dimension is not required to be integral. In the case of even integral dimension and standard factors of automorphy, this latter result is contained in D. Drasin and C. J. Earle, Proc. Amer. Math. Soc. 19 (1968), 1039-1042, but the present approach is entirely different. Also, using the argument of Drasin and Earle, it is proved that, for finitely generated Fuchsian groups of second kind, every integrable automorphic form of dimension − 2 - 2 is zero.