Weighted Banach spaces of holomorphic functions in the upper half plane

We consider weighted banach spaces of holomorphic functions on the upper half plane that are determined by $\mathrm{\parallel }f\mathrm{\parallel }={\sup }_{y>0,-\mathrm{\infty }<x<\mathrm{\infty }}p(y)\mathrm{\mid }f(x+iy)\mathrm{\mid }<\mathrm{\infty }$ 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.