TITLE: Reference for result on partial sums of Taylor series
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: 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 $R=1$.