Samwise1
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We are given $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in \mathcal{O} (ann (z_0, r, R)), \ \ 0<r<R< \infty $$.
Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d \lambda(z) = \sum _{n \neq -1} \frac{R^{2n+2} - r^{2n+2}}{n+1}|a_n|^2 + 2 \log \frac{R}{r}|a_{-1}|^2$$.
We know that the series above is convergent, so $$R = \frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{|a_n|}}$$ and $$r = \limsup_ {n \rightarrow \infty} \sqrt[n]{|a_{-n}|}$$.
In the series $$\sum _{n \neq -1} \frac{R^{2n+2}}{n+1}|a_n|^2, \ \ \sum _{n \neq -1} \frac{r^{2n+2}}{n+1}|a_n|^2$$ we have $$b_n = \frac{|a_n|^2}{n+1}$$ and radii of convergence are $R'=\frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{\frac{|a_n|^2}{n+1}}} \ge \frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{|a_n|^2}} \ge \frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{|a_n|}}^2 = R^2$, so $$R' \ge R^2$$.
Similarly, $$r' = \limsup_ {n \rightarrow \infty} \sqrt[n]{\frac{|a_{-n}|^2}{n+1}} \le \limsup_ {n \rightarrow \infty} \sqrt[n]{|a_{-n}|^2} \le r^2$$.
So the two series are convergent - they have the form $$\sum b_n (z-z_0)^n$$ with $z=R^2$ or $-r^2$.
Does that make sense? Could you tell me how to prove the equality of the integral and the series?
Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d \lambda(z) = \sum _{n \neq -1} \frac{R^{2n+2} - r^{2n+2}}{n+1}|a_n|^2 + 2 \log \frac{R}{r}|a_{-1}|^2$$.
We know that the series above is convergent, so $$R = \frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{|a_n|}}$$ and $$r = \limsup_ {n \rightarrow \infty} \sqrt[n]{|a_{-n}|}$$.
In the series $$\sum _{n \neq -1} \frac{R^{2n+2}}{n+1}|a_n|^2, \ \ \sum _{n \neq -1} \frac{r^{2n+2}}{n+1}|a_n|^2$$ we have $$b_n = \frac{|a_n|^2}{n+1}$$ and radii of convergence are $R'=\frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{\frac{|a_n|^2}{n+1}}} \ge \frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{|a_n|^2}} \ge \frac{1}{\limsup_ {n \rightarrow \infty} \sqrt[n]{|a_n|}}^2 = R^2$, so $$R' \ge R^2$$.
Similarly, $$r' = \limsup_ {n \rightarrow \infty} \sqrt[n]{\frac{|a_{-n}|^2}{n+1}} \le \limsup_ {n \rightarrow \infty} \sqrt[n]{|a_{-n}|^2} \le r^2$$.
So the two series are convergent - they have the form $$\sum b_n (z-z_0)^n$$ with $z=R^2$ or $-r^2$.
Does that make sense? Could you tell me how to prove the equality of the integral and the series?