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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 6

Question

Math 6210 • Real Analysis Fall 2009 • Question 6

Prove Schwarz’s lemma: If \(f:\Delta \to \Delta \) is a holomorphic function from the open unit disk \(\Delta \) to itself with \(f(0)=0\) show that \(|f(z)|\le |z|\), \(|f'(0)|\le 1\) and if \(|f(z)|=|z|\) for some \(z\ne 0\) or \(|f'(0)|=1\) then \(f(z)=\lambda z\) with \(|\lambda |=1\).

(Hint: Apply the maximum principle to the function \(f(z)/z\).)

Raw LaTeX

Prove Schwarz's lemma: If \(f:\Delta\to\Delta\) is a holomorphic
function from the open unit disk \(\Delta\) to itself with \(f(0)=0\) show that
\(|f(z)|\le |z|\), \(|f'(0)|\le 1\) and if \(|f(z)|=|z|\) for some \(z\ne0\) or
\(|f'(0)|=1\) then \(f(z)=\lambda z\) with \(|\lambda|=1\).

(\textit{Hint:} Apply the maximum principle to the function \(f(z)/z\).)

Topic Tags

Argument Principle and Rouche's Theorem Holomorphic Functions and Cauchy-Riemann Equations Maximum Modulus Principle Schwarz Lemma and Schwarz-Pick Lemma