In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f (z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), ∣∣∣ f ′(0)∣∣∣ , by assuming that the f (z) function is also holomorphic at the boundary point z = 0 on the real axis with f (0) =< f (i)