- Let f be a holomorphic function in the punctured disk \(\{ z : 0 < | z | < 2 \}\) satisfying
\[ | f ( z ) | \leq ( \log { \frac { 1 } { | z | } } ) ^ { 1 0 0 } \mathrm { { i n } } \left\{ | z | \leq 1 / 2 \right\} , \]
\[ | f ( z ) | = 1 \ \mathrm { o n } \ | z | = 1 . \]
Show that f has a removable singularity at the origin.
Show that if \(f ( z ) \neq 0\) in \(| z | < 1\) , then f is constant.
(Extra credit) True or false, explain.
\(f = \alpha z ^ { n }\) for \(\alpha \in \mathbb { C } , | \alpha | = 1\) and an integer \(n \geq 0\)
Solution
1. (a) \(f\) has a removable singularity at \(0\).
With \(t=\log(1/\abs z)\), the hypothesis gives \(\abs{zf(z)}\le e^{-t}t^{100}\le101!/t\) for \(\abs z\le1/2\), using \(e^t\ge t^{101}/101!\), so \(zf(z)\to0\) as \(z\to0\). By the removable singularity theorem \(h(z)=zf(z)\) extends holomorphically with \(h(0)=0\), so \(h(z)=zH(z)\) with \(H\) holomorphic near \(0\), and \(H\) extends \(f\) [@SS03].
2. (b) If \(f\), extended by step 1, has no zero in \(\abs z<1\), then \(f\) is constant.
Both \(f\) and \(1/f\) are holomorphic on \(\abs z\le1\) with modulus \(1\) on \(\abs z=1\), so the maximum modulus principle gives \(\abs f\le1\) and \(\abs{1/f}\le1\) there. So \(\abs f\equiv1\) on \(\abs z<1\), where \(f\) attains its maximum modulus at an interior point; hence \(f\) is constant on \(\abs z<1\), and on the connected disk \(\abs z<2\) by the identity theorem.
3. (c) False: \(f(z)=z^{100}\frac{z-1/3}{1-z/3}\) satisfies both hypotheses and is not of the form \(\alpha z^n\).
\(f\) is holomorphic on \(\abs z<2\), since its only pole is \(z=3\). For \(a=1/3\), \(\abs{1-az}^2-\abs{z-a}^2=(1-a^2)(1-\abs z^2)\), so the rational factor has modulus \(1\) on \(\abs z=1\) and at most \(1\) on \(\abs z\le1\). Hence \(\abs f=1\) on \(\abs z=1\), and for \(0<r=\abs z\le1/2\), \(\abs{f(z)}\le r^{100}\le(\log(1/r))^{100}\), because \(\log(1/r)\ge\log2>1/2\ge r\). Finally \(f(1/3)=0\), while \(\alpha z^n\) with \(\abs\alpha=1\) has no zero other than \(0\).
4. Q.E.D.
Steps 1, 2 and 3 answer parts (a), (b) and (c).