Two Equivalent Presentations for the Norm of Weighted Spaces of Holomorphic Functions on the Upper Half-plane

Introduction In this paper, we intend to show that without any certain growth condition on the weight function, we always able to present a weighted sup-norm on the upper half plane in terms of weighted sup-norm on the unit disc and supremum of holomorphic functions on the certain lines in the upper half plane. Material and methods We use a certain transform between the unit dick and the upper half-plane, a translation operator between weighted spaces of holomorphic functions toghther with Phragmen-Lindelof theorem in order to obtain our main results. Results and discussion We prove 3 Lemma which enable us to get our main results in Theorem 3. Conclusion The following conclusions were drawn from this research. We find lower bound and upper bound for weighted sup-norm in terms of supremum of the function on the lines in the upper half-plane. We obtain lower and upper bounds for translation operator in terms of  weighted Sup- norm./files/site1/files/51/%D8%A7%D8%B1%D8%AF%D9%84%D8%A7%D9%86%DB%8C.pdf