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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2018 • Question 8

Question

Math 6210 • Real Analysis Spring 2018 • Question 8

Let \(f:D\to D\) be holomorphic. Show that \[ |f(z)-f(0)|\le |z(1-\overline {f(0)}f(z))| \] for all \(z\in D\). In words, the coefficient multiplying \(f(z)\) in the parentheses is the complex conjugate of \(f\) at zero.

Raw LaTeX

Let \(f:D\to D\) be holomorphic. Show that
\[
|f(z)-f(0)|\le |z(1-\overline{f(0)}f(z))|
\]
for all \(z\in D\). In words, the coefficient multiplying \(f(z)\) in the
parentheses is the complex conjugate of \(f\) at zero.

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem