A convergent series
A convergent series
Let $f$ be holomorphic on the open unit disk $\mathbb{D}$ and suppose that
$$\iint \limits_{\mathbb{D}} \left| f(z) \right|^2 \, {\rm d}(x, y) < +\infty$$
If the Taylor expansion of $f$ is of the form $\displaystyle \sum_{n=0}^{\infty} a_n z^n$ , prove that the series $\displaystyle \sum_{n=0}^{\infty} \frac{|a_n|^2}{n+1}$ converges
$$\iint \limits_{\mathbb{D}} \left| f(z) \right|^2 \, {\rm d}(x, y) < +\infty$$
If the Taylor expansion of $f$ is of the form $\displaystyle \sum_{n=0}^{\infty} a_n z^n$ , prove that the series $\displaystyle \sum_{n=0}^{\infty} \frac{|a_n|^2}{n+1}$ converges
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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