Let \(\mathbb{D}:=\{z:|z|<1\}\) denote the open unit disk. Suppose that \(f(z): \mathbb{D} \rightarrow \mathbb{D}\) is holomorphic, and that there exists \(a \in \mathbb{D} \backslash\{0\}\) such that \(f(a)=f(-a)=0\).
Prove that \(|f(0)| \leq|a|^{2}\).
What can you conclude when \(|f(0)|=|a|^{2} ?\)
Solution
For \(c\in\DD\), write \[ \phi_c(z)=\frac{z-c}{1-\overline c z}. \]
1. If \(F:\DD\to\DD\) is holomorphic and \(F(c)=0\), then\[ \left|\frac{F(z)}{\phi_c(z)}\right|\le1 \qquad(z\in\DD), \]where the quotient is extended holomorphically at \(z=c\).
The disk automorphism \(\phi_c\) sends \(c\) to \(0\). Thus \[ h(w)=F(\phi_c^{-1}(w)) \] is a holomorphic self-map of \(\DD\) with \(h(0)=0\). Schwarz’s lemma gives \[ |h(w)|\le|w|. \] Putting \(w=\phi_c(z)\) yields \[ |F(z)|\le|\phi_c(z)|. \] Since \(F(c)=\phi_c(c)=0\) and \(\phi_c\) has a simple zero at \(c\), the quotient extends holomorphically there, and the displayed bound persists by continuity.
2. The function\[ g(z)=\frac{f(z)}{\phi_a(z)\phi_{-a}(z)} \]extends holomorphically to \(\DD\) and satisfies \(|g(z)|\le1\).
Apply 1 to \(f\) at the zero \(a\). Then \[ f_1(z)=\frac{f(z)}{\phi_a(z)} \] is holomorphic with \(|f_1|\le1\). Because \(a\ne0\), \(\phi_a(-a)\ne0\), so \(f_1(-a)=0\). Apply 1 again, now to \(f_1\) at \(-a\). The resulting quotient is exactly \(g\).
3. \(|f(0)|\le|a|^2\).
By 2, \[ 1\ge|g(0)| =\frac{|f(0)|}{|\phi_a(0)|\,|\phi_{-a}(0)|}. \] Since \[ |\phi_a(0)|=|a|=|\phi_{-a}(0)|, \] we obtain the claimed inequality.
4. If \(|f(0)|=|a|^2\), then there exists \(\lambda\in\CC\) with \(|\lambda|=1\) such that\[ \boxed{f(z)=\lambda\phi_a(z)\phi_{-a}(z)}. \]
Equality in 3 is equivalent to \[ |g(0)|=1. \] By 2, \(g\) is holomorphic on \(\DD\) and \(|g|\le1\). Thus \(g\) attains its maximum modulus at an interior point, so the maximum modulus principle forces \[ g\equiv\lambda \] for some \(|\lambda|=1\). Substituting the definition of \(g\) gives the formula.
5. Conversely, every function in 4 satisfies the hypotheses and the equality case.
Each \(\phi_{\pm a}\) is an automorphism of \(\DD\), so for \(z\in\DD\), \[ |\lambda\phi_a(z)\phi_{-a}(z)|<1. \] The product vanishes at \(a\) and \(-a\), and at \(0\) its modulus is \[ |\phi_a(0)\phi_{-a}(0)|=|a|^2. \]