Let Δ denote the open unit disc in ℂ and let ℂ ^ = ℂ ∪ ∞ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/inequ35_1.tif"/> be the Riemann sphere. Among the many consequences of the Schwarz lemma, the most fundamental is the fact that every biholomorphic function f on Δ takes the form f(z) = e gc (z) where θ ∈ [0, 2π) and c ∈ Δ axe fixed and g c ( z ) = c + z 1 + z c ¯ . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/unequ35_1.tif"/>