A convergent series

Let f be holomorphic on the open unit disk \mathbb{D} and suppose that the integral \displaystyle \iint \limits_{\mathbb{D}} \left| f(z) \right|^2 \, {\rm d}(x, y) converges. If the Taylor expansion of f is of the form \sum \limits_{n=0}^{\infty} a_n z^n then prove that the sum

\displaystyle \mathcal{S}= \sum_{n=0}^{\infty} \frac{|a_n|^2}{n+1}

converges.

Solution

We evaluate the integral using the standard orthogonality results for e^{in \theta}. Thus:

\begin{aligned} \iint \limits_{\mathbb{D}}|f(z)|^2 \, {\rm d}z&=\int_0^1 r\int_0^{2\pi}\left|\sum_{n=0}^{\infty}a_nr^ne^{in\theta}\right|^2\,{\rm d}\theta\,{\rm d}r\\ &=\int_0^1r\int_0^{2\pi}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}a_n\overline{a_m}r^{n+m}e^{i(n-m)\theta}\,{\rm d}\theta\,{\rm d}r\\ &=2\pi\int_0^1r\sum_{n=0}^{\infty}|a_n|^2r^{2n}\,{\rm d}r\\ &=\pi\sum_{n=0}^{\infty}\frac{|a_n|^2}{n+1} \end{aligned}

and thus the series converges.

Note: The set of functions satisfying this is a Hilbert space of functions, but it is not the same as the Hardy space \mathbb{H}^2.

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