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Let $\{f_{j} \}_{j=1}^{\infty}$ be a sequence of holomorphic function from $D(0,1) \rightarrow D(0,1) \backslash \{ 0 \}$ so that $\sum_{j=1}^{\infty} |f_{j}(0)| < \infty$.

Find a sequence of holomorphic functions satisfying the above condition but $\sum_{j=1}^{\infty} f_{j}(z)^{2}$ does not converge uniformly on $D(0,r)$ for any $r > \frac{1}{3}$.

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Is this homework? –  Greg Martin Apr 4 '12 at 3:50

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