Schwarz lemma

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.Schwarz Lemma. Let D = { z : | z | < 1 } {displaystyle mathbf {D} ={z:|z|<1}} be the open unit disk in the complex plane C {displaystyle mathbb {C} } centered at the origin and let f : D → C {displaystyle f:mathbf {D} ightarrow mathbb {C} } be a holomorphic map such that f ( 0 ) = 0 {displaystyle f(0)=0} and | f ( z ) | ≤ 1 {displaystyle |f(z)|leq 1} on D {displaystyle mathbf {D} } .Let f : H → H be holomorphic. Then, for all z1, z2 ∈ H, In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions. The proof is a straightforward application of the maximum modulus principle on the function which is holomorphic on the whole of D, including at the origin (because f is differentiable at the origin and fixes zero). Now if Dr = {z : |z| ≤ r} denotes the closed disk of radius r centered at the origin, then the maximum modulus principle implies that, for r < 1, given any z in Dr, there exists zr on the boundary of Dr such that As r → 1 {displaystyle r ightarrow 1} we get | g ( z ) | ≤ 1 {displaystyle |g(z)|leq 1} . Moreover, suppose that |f(z)| = |z| for some non-zero z in D, or |f′(0)| = 1. Then, |g(z)| = 1 at some point of D. So by the maximum modulus principle, g(z) is equal to a constant a such that |a| = 1. Therefore, f(z) = az, as desired. A variant of the Schwarz lemma, known as the Schwarz–Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of the unit disc to itself: Let f : D → D be holomorphic. Then, for all z1, z2 ∈ D, and, for all z ∈ D,