An equivalent representation for weighted supremum norm on the upper half-plane

In this paper, rstly, we obtain some inequalities which estimates complex polynomials on the circles. Then, we use these estimates and a Moebius transformation to obtain the dual of this estimates for the lines in upper half-plane. Finally, for an increasing weight f on the upper half-plane with certain properties and holomorphic functions f on the upper half-plane we obtain an equivalent representation for weighted supremum norm.