I'm reading this and there are some details that are missing. I'm asking for those.
First let $f:S^1\to S^1$ a diffeomorphism. By Dini criterion we can write $f(e^{i\theta})=\sum_n \hat{f}(n)e^{in\theta}$. The above post suggest to define $F:D^2\to D^2$ by $F(re^{i\theta})=\sum_n \hat{f}(n)r^ne^{in\theta}$ for $|r|<1.$ By Lagrange inversion theorem we have that $F$ is the diffeomorphism that we are looking for.
The problems that I have is with convergence of those series and the fact that $F$ gives values in $D^2 $. The above post suggest to use Cauchy-Schwarz but I don't know how. I also know Parseval's theorem but again, I don't know how to use it. Can someone fill in the details?
EDIT: My question is now, why $F$ lands in $D^2$?
To establish the series $F(z)=\sum_{n=0}^\infty\hat{f}(n)z^n$, $z=re^{i\theta}$, actually converges in $\mathbb{C}$, we use Cauchy-Schwarz $$ \lvert F(z)\rvert^2=\left\lvert\sum_n \hat{f}(n)z^n\right\rvert^2\leq\left(\sum_n\lvert\hat{f}(n)\rvert^2\right)\cdot\left(\sum_n \lvert z^n\rvert^2\right)= 1\cdot\frac{1}{1-r^2} $$ where the last equality is by Parseval (and summing a geometric series). So $F(z)\in\mathbb{C}$ for all $z\in D$.
Moreover, $F(z)$ is a nonconstant power series. As the power series converges for every $z\in S^1$ to the smooth $f\colon S^1\to S^1$, we have $F\in C(D)\cap C^\omega(\mathbb{D})$. Since $F$ is nonconstant, by the maximum modulus principle, we must have $\lvert F(z)\rvert<\sup_{\lvert\zeta\rvert=1}\lvert F(\zeta)\rvert=1$ for every $z\in\mathbb{D}$.
Finally, $F$ is a diffeomorphism on $\mathbb{D}$: the winding number of $f(e^{it})$, $0\leq t\leq 2\pi$ around any point of $\mathbb{D}$ is clearly $1$, so $F$ is bijective $\mathbb{D}\to\mathbb{D}$ by argument principle. Hence $F\colon\mathbb{D}\to\mathbb{D}$ is univalent.