  We consider weighted banach spaces of holomorphic functions on the upper half
plane that are determined by $
\|f\|=\sup_{y>0,-\infty<x<\infty}p(y)|f(x+iy)|<\infty $ for a very large class
of weight functions p. We completely solve the problem whether such banach
spaces are trivial or not by giving necessary and sufficient conditions stated
in terms of some simple properties of the weight function. Further, we
investigate the behaviour at infinity of some functions that belong to some of
the banach spaces under consideration.
