Let D be the open unit disk, b∶ D → D be holomorphic and z ∈ D ̅. We say z is a fixed point of b if and only if lim┬(r→1-)⁡〖b(rz)=z〗. The Caratheodory-Julia-Wolff theorem asserts that lim┬(r→1-)⁡b'(rz) exists at any fixed point z. The Denjoy-Wolff theorem asserts that if b is not the identity map, then b has exactly one fixed point z ∈ D ̅ such that |b'(z)| ≤ 1. In 1982, Cowen and Pommerenke [2] proved important theorems concerning fixed points of holomorphic functions using deep complex analytic methods. In this thesis, we will present alternative proofs of their first theorem using H(b) space method, which was introduced by de Branges in his proof of the Bieberbach conjecture back in 1985. In chapter 1 of the thesis, we will recall the basic definitions, terminologies and facts related to the inequalities of Cowen-Pommerenke’s first theorem. In chapter 2, we will give the proofs of the inequalities using H(b) spaces. In chapter 3, we will discuss the equality cases of the inequalities.