Let \(D(0,r) = \{z \in \mathbb{C} : |z| < r\}\). Assume \(r > 1\) and \(f : \overline{D(0,r)} \setminus D(0,1) \to \mathbb{C}\) is a continuous function, holomorphic on \(D(0,r) \setminus \overline{D(0,1)}\) that satisfies
\[\max_{\partial D(0,1)} |f(z)| = 1 \quad \text{and} \quad \max_{\partial D(0,r)} |f(z)| = R.\]
Prove \(\log |f(z)| \leq \frac{\log R}{\log r} \log |z|\).
Solution
Write \(A=\{z:1<|z|<r\}\). We prove \(|f(z)|\leq |z|^{\log R/\log r}\) on \(\overline A\). At zeros of \(f\), the requested logarithmic form uses \(\log0=-\infty\).
1. The outer boundary maximum satisfies \(R>0\).
Continuity on the compact outer circle gives \(0\leq R<\infty\). Suppose \(R=0\). For each positive integer \(n\), the function \(z^n f(z)\) is continuous on \(\overline A\) and holomorphic on \(A\). Its modulus is at most one on the inner boundary and zero on the outer boundary. The maximum modulus principle gives \(|z|^n|f(z)|\leq1\) throughout \(A\) [@SS03]. For fixed \(|z|>1\), letting \(n\to\infty\) forces \(f(z)=0\). Continuity then makes \(f=0\) on the inner boundary as well, contradicting its maximum modulus one. Thus \(R>0\).
2. Integer powers give the bound without choosing a logarithm branch on the annulus.
Set \(a=\log R/\log r\) and, for each positive integer \(n\), let \(m_n=\lfloor na\rfloor\). The function \[ H_n(z)=z^{-m_n}f(z)^n \] is single-valued and holomorphic on \(A\), and continuous on \(\overline A\), even when \(m_n\) is negative. On \(|z|=1\) its modulus is at most one. On \(|z|=r\) it is at most \[ r^{-m_n}R^n=r^{na-m_n}\leq r, \] since \(0\leq na-m_n<1\). The maximum modulus principle therefore gives \(|H_n(z)|\leq r\) on \(\overline A\). Taking \(n\)th roots yields \[ |f(z)|\leq r^{1/n}|z|^{m_n/n}. \] Because \(m_n/n\to a\) and \(r^{1/n}\to1\), passage to the limit gives \(|f(z)|\leq|z|^a\) at every point of \(\overline A\). At nonzero values of \(f\), taking logarithms gives the required inequality. At zeros it holds with the stated extended-real convention. This includes both boundary circles and all signs of \(\log R\).