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 $K>0$.
The questions
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 (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.