I remember seeing somewhere that whenever $f$ is a holomorphic function with radius of convergence at $z$, $0<R<\infty$ the following holds
$$\limsup\_{n\to \infty}(\max\_{|w-z|\leq \rho R} |S\_n(f,z)(w)|^{1/n})=\rho,$$
where $\rho\geq 1$ and $S\_n(f,z)(w)$ is the $n$-th partial sum of the Taylor series of $f$ with center $z$ at $w$.


Does anyone know where I can find the proof of this?



Reference for result on partial sums of Taylor series
This has an elementary proof, using the formula for the radius of convergence and Cauchy's estimates. It suffices to treat the case $R=1$. We then know that the coefficients satisfy $|a\_n|\lesssim (1+\epsilon)^n$. Thus, if $|z|\le \rho$, then
$$
|S\_N(z)|\le \sum\_{n=0}^N |a\_n| \rho^n \lesssim ((1+\epsilon)\rho)^{N+1}
$$
(because $\rho\ge 1$), and this shows that the LHS is $\le \rho$.


Conversely, we have that
$$
|g^{(k)}(0)| \le \frac{k!}{\rho^k} \max\_{|z|=\rho} |g(z)|
$$
for any holomorphic $g$. So if the LHS of the asserted formula was $\le \rho-\epsilon$, then we could apply this to $g=S\_n$, with $k=n$, and conclude that $|a\_n|\lesssim (1-\epsilon/\rho)^n$, but this contradicts our assumption that $R=1$.


