# On the remainder of a power series evaluated on the boundary of its convergence disk

Background

This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily regular point of its convergence disk.

Ricci proved an estimate for the remainder of a power series $$f(z)$$ evaluated at a regular point $$z=e^{i\theta}$$ of the boundary of its convergence disk (the radius is assumed equal to unity without restrictions in generality) in the paper [2]: however, in his review Alexander Peyerimhoff [1] pointed out that the following simpler and more beautiful formula for the same quantity $$\left|\sum_{k=0}^na_ke^{i\theta k}-f\big(e^{i\theta}\big)\right|= C\left(\frac{1}{n^2}+\sum_{\nu=0}^\infty\frac{|a_\nu|}{(|\nu-n|+1)^2}\right) \label{1}\tag{1}$$ where $$C$$ depends on the length of the compact arc containing $$e^{i\theta}$$ and contained in the arc of regularity of $$f$$, was developed implicitly by Marcel Riesz in his 1911 proof [4] of the now called Fatou-Riesz theorem. Ricci, perhaps following Peyerimhoff suggestion, proved a formula similar to \eqref{1} in [3] (p. 236, formula 3.1) significantly weakening the hypothesis on $$e^{i\theta}$$, which is no more a regular point but only a point for which the radial limit $$\lim_{r\to 1^-} f\big(re^{i\theta}\big)$$ exists and is finite, despite introducing a growth requirement on the first derivative of $$f$$, namely $$|f'(z)|< K\big|z -e^{i\theta}\big|^{\alpha-1} \quad z\in \Bbb D \cap\big\{z\in\Bbb C: |z-e^{i\theta}|<\rho\big\}\label{2}\tag{2}$$ for three given real numbers $$0<\alpha\le 1$$, $$\rho>0$$ and $$K>0$$ (where $$\Bbb D=\{z\in\Bbb C: |z|<1\}$$).

The question.

Has this problem being further studied? I have not been able to find any other reference on this topic. In particular, I'd be very happy to find out that it is possible to remove the requirement \eqref{2} since it seems somewhat artificial.

An update

I recently asked this very same question to Enrico Bombieri, who was one of Ricci's students. He remarked that he's not aware of any further research on this topic but also points out that, while having been involved on similar matters when he was studying under the supervision of Ricci, later his mathematical interests developed in other directions. He suggest to have a look at the brief Mathematical Reviews surveys on such topics as "Power series" written in the years after the publication of [3]. I currently do not have access to the MR, so I am reporting here his suggestion in order to give a complete information on what I did to further investigate the question and on where possibly an answer may lie unnoticed.

References

[1] Alexander Peyerimhoff, "Zbl 0058.05903 (review of [2])", Zentralblatt Für Mathematik 58, p. 59 (February 1957).

[2] Giovanni Ricci, "Maggiorazione del resto delle serie di potenze sul cerchio di convergenza" (Italian) Annali della Scuola Normale Superiore di Pisa. Scienze Fisiche e Matematiche. III Serie, 8, pp. 121-131 (1954), MR0070707, Zbl 0058.05903.

[3] Giovanni Ricci, "Sul resto delle serie di potenze alla periferia del cerchio di convergenza" (Italian) in Scritti Matematici in Onore di Filippo Sibirani, Bologna: Cesare Zuffi, pp. 233-242 (1957), MR0086864, Zbl 0077.28403.

[4] Marcel Riesz, "Über einen Satz des Herrn Fatou" (German) Journal für die Reine und Angewandte Mathematik 140, 89-99 (1911), JFM 42.0277.01.

• interesting question
– mick
Jul 4, 2021 at 22:05