**Background**

This question is related to [this one](https://mathoverflow.net/questions/394773/in-search-for-a-counterexample-related-to-the-abel-stolz-theorem), 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.


**References**

[1] Alexander Peyerimhoff, "[Zbl 0058.05903](https://www.zbmath.org/?q=an%3A0058.05903) (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](http://www.numdam.org/item?id=ASNSP_1954_3_8_3-4_121_0)" (Italian) Annali della Scuola Normale Superiore di Pisa. Scienze Fisiche e Matematiche. III Serie, 8, pp. 121-131 (1954), [MR0070707](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0070707), [Zbl 0058.05903](https://www.zbmath.org/?q=an%3A0058.05903).

[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](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0086864), [Zbl 0077.28403](https://www.zbmath.org/?q=an%3A0077.28403).

[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](https://www.zbmath.org/?q=an%3A42.0277.01).