- This exercise shows how the mean square convergence dominates the uniform convergence of analytic functions. If U is an open subset of C we use the notation
\[ \| f \| _ {L ^ {2} (U)} = \left(\int_ {U} | f (z) | ^ {2} d x d y\right) ^ {1 / 2} \]
for the mean square norm, and
\[ \| f \| _ {L ^ {\infty} (U)} = \sup _ {z \in U} | f (z) | \]
for the sup norm.
- If f is holomorphic in a neighborhood of the disc \(D _ { r } ( z _ { 0 } )\) , show that for any \(0 < s < r\) there exists a constant \(C > 0\) (which depends on s and \(r )\) such that
\[ \| f \| _ {L ^ {\infty} (D _ {s} (z _ {0}))} \leq C \| f \| _ {L ^ {2} (D _ {r} (z _ {0}))}. \]
- Prove that if \(\left\{ f _ { n } \right\}\) is a Cauchy sequence of holomorphic functions in the mean square norm \(\| \cdot \| _ { L ^ { 2 } ( U ) }\) , then the sequence \(\left\{ f _ { n } \right\}\) converges uniformly on every compact subset of \(\dot { U }\) to a holomorphic function.
[Hint: Use the mean-value property.]
Solution
Fix \(0<s<r\) and put \[ \rho=\frac{r-s}{2}>0. \] For every \(z\in D_s(z_0)\), the closed disc \(\overline{D_\rho(z)}\) lies in \(D_r(z_0)\). By the mean-value property, \(f(z)\) is the average of \(f\) over \(D_\rho(z)\), so the Cauchy–Schwarz inequality gives \[ |f(z)|^2\le \frac1{\pi\rho^2}\int_{D_\rho(z)}|f(w)|^2\,dA(w) \le \frac1{\pi\rho^2}\|f\|_{L^2(D_r(z_0))}^2. \] Thus \[ \|f\|_{L^\infty(D_s(z_0))} \le \frac1{\sqrt\pi\rho}\|f\|_{L^2(D_r(z_0))}. \] So one may take \(C=(\sqrt\pi\rho)^{-1}\).
Let \(K\Subset U\). Choose finitely many discs \[ K\subset\bigcup_{j=1}^N D_{s_j}(z_j), \qquad \overline{D_{r_j}(z_j)}\subset U, \qquad 0<s_j<r_j. \] Applying part (a) to \(f_n-f_m\) on each disc gives \[ \|f_n-f_m\|_{L^\infty(K)} \le C_K\|f_n-f_m\|_{L^2(U)} \] for a constant \(C_K\) independent of \(m,n\). Since \((f_n)\) is Cauchy in \(L^2(U)\), it is uniformly Cauchy on \(K\). Hence it converges uniformly on every compact subset of \(U\) to a function \(f\). By the Weierstrass theorem on locally uniform limits of holomorphic functions, \(f\) is holomorphic on \(U\).