- Let \(k\geq0\) and let \(f\) be entire with \[ \max_{|z|=R}|f(z)|\leq AR^k+B \] for positive constants \(A,B\) and all \(R>1000\). Show that \(f\) is a polynomial of degree at most \(k\).
- Define a normal family of holomorphic functions on an open, possibly unbounded, domain \(U\).
- If \(\mathcal F\) is normal on the open unit disk \(D\), prove that \(\mathcal F'=\{f':f\in\mathcal F\}\) is normal on \(D\).
For \(k<0\) the estimate in step <1>1 kills every coefficient with \(m\geq1\), so \(f\) is constant, but \(f\) need not have degree at most \(k\): \(f=1\) with \(A=B=1\) satisfies the bound for every negative \(k\).
Solution
1. The growth bound makes every Taylor coefficient of ordergreater than \(k\) vanish.
Write \(f(z)=\sum_{m\geq0}a_mz^m\), the Taylor expansion valid on the entire plane. Cauchy’s coefficient formula and the length of the circle give, for \(R>1000\), \[ |a_m|=\left|\frac{1}{2\pi i} \int_{|\zeta|=R}\frac{f(\zeta)}{\zeta^{m+1}}\,d\zeta\right| \leq AR^{k-m}+BR^{-m} \] [@SS03]. For an integer \(m>k\geq0\), both terms tend to zero as \(R\to\infty\). Hence \(a_m=0\). All terms above \(\lfloor k\rfloor\) vanish, so \(f\) is a polynomial of degree at most \(k\).
2. Normality requires a locally uniformly convergent subsequencewith finite holomorphic limit from every sequence in the family.
Precisely, a family \(\mathcal F\subseteq\mathcal O(U)\) is normal if every sequence \((f_n)\) in it has a subsequence \((f_{n_j})\) converging uniformly on every compact subset of \(U\) to a holomorphic function \(f:U\to\mathbb C\). The limit need not belong to \(\mathcal F\), and \(U\) may be unbounded [@SS03].
3. The family of derivatives is normal.
Given a sequence \((g_n)\) in \(\mathcal F'\), choose \(f_n\in\mathcal F\) with \(g_n=f_n'\). Normality supplies a subsequence \(f_{n_j}\to f\) locally uniformly on \(D\), where \(f\) is holomorphic. Let \(K\subset D\) be a nonempty compact set. Choose \(\delta>0\) such that the compact set \[ L=\{z+w:z\in K,\ |w|\leq\delta\} \] is contained in \(D\). For instance, take \(\delta=(1-\max_{z\in K}|z|)/2\). For \(z\in K\), the differentiated Cauchy formula gives \[ g_{n_j}(z)-f'(z)=\frac{1}{2\pi i} \int_{|\zeta-z|=\delta} \frac{f_{n_j}(\zeta)-f(\zeta)}{(\zeta-z)^2}\,d\zeta. \] Consequently \[ \sup_{z\in K}|g_{n_j}(z)-f'(z)| \leq\delta^{-1}\sup_{\zeta\in L}|f_{n_j}(\zeta)-f(\zeta)| \longrightarrow0. \] Thus the same subsequence converges locally uniformly to the holomorphic function \(f'\), exactly as required by step 2.