**Context**. The question arises from my former [question on the remainder of a power series](https://mathoverflow.net/questions/396814/on-the-remainder-of-a-power-series-evaluated-on-the-boundary-of-its-convergence). Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in his paper [2] i.e.
$$
\begin{cases}
g(z)\text{ has a limit as $z\to e^{i\theta}$}\\
|g'(z)|< K\big|z -e^{i\theta}\big|^{\alpha-1} 
\end{cases}
$$
where $\Bbb D=\{z\in\Bbb C: |z|<1\}$, $z\in \Bbb D \cap\big\{z\in\Bbb C: |z-e^{i\theta}|<\rho\big\}$, $0<\alpha\le 1$, $\rho>0$ and $K>0$ is more than a simplifying choice (I formerly called it "artificial"), and I found the following interesting lemma in the paper [1] (possibly the first one published by Stanisław Łojasiewicz, reproduced verbatim in the French original language).



> 
> **Lemme II**. Soit $g(z)$ une fonction holomorphe dans un domaine $G$ possédant une limite finie en un point frontiere $\zeta\_0$ de $G$ et $\rho(z)$ la distance du point $z$ à la frontière de $G$. Alors
> $$\DeclareMathOperator{\D}{d\!}
> (z-\zeta\_0)\cdot g^\prime(z) \to 0\text{ pour }z\to \zeta\_0,\text{ de façon que $\frac{|\zeta\_0-z|}{\rho(z)}$ reste borné}.
> $$
> $\quad$*Démonstration*. Posons $A=\lim\_{z\to\zeta\_0} g(z)$, désignons par $C\_z$ la circonférence $ |\zeta\_0-z|= \dfrac{\rho(z)}{2}$ et soit $\eta(z) =\max\_{C\_z}|g(\zeta)-A|$. Pour chaque $z\in G$, on a
> $$
> g^{\prime}(z)=\frac{1}{2 \pi i} \int\limits\_{C\_z} \frac{g(\zeta)}{(\zeta-z)^2} \D\zeta=\frac{1}{2 \pi i} \int\limits\_{C\_z}\frac{g(\zeta)-A}{(\zeta-z)^2} \D \zeta \text {, }
> $$
> d'ou il vient
> $$
> \left|g^{\prime}(z)\right| \leqslant \frac{1}{2 \pi} \pi \rho(z) \frac{\eta(z)}{\left(\frac{1}{2} \rho(z)\right)^2}=2 \frac{\eta(z)}{\rho(z)}
> $$
> et par suite $\left|\left(z-\zeta\_0\right) \cdot g^{\prime}(z)\right| \leqslant 2 \frac{\left|z-\zeta\_0\right|}{\rho(z)} \eta(z)$, ce qui entraîne la thèse car $\lim\_{z\to\zeta\_0} \eta(z) = 0$.
> 
> 
> 


This lemma seems to suggest that *always* $|g^\prime(z)| = \omicron|(z-\zeta\_0)^{-1}|$ as $ z\to\zeta\_0$. Unfortunately this seems not to be the case: by using [a counterexample suggested by Alexandre Eremenko](https://mathoverflow.net/questions/394773/in-search-for-a-counterexample-related-to-the-abel-stolz-theorem#comment1010434_394789) i.e.
$$
g(z)=\frac{1-z}{1+z}\sin\left(\frac{1+z}{1-z}\right)\label{1}\tag{1}
$$
we have that $g(x)\to 0$ as $x$ goes from $0$ to $1$ along the real axis but since
$$
g^\prime(z) = 
-\dfrac{2\left[\left(z-1\right)\sin\left(\frac{z+1}{1-z}\right)+\left(z+1\right)\cos\left(\frac{z+1}{1-z}\right)\right]}{\left(z-1\right)\left(z+1\right)^2}
$$
we have
$$
|(x-1)g^\prime(x)|=O\left|\cos\left(\frac{x+1}{1-x}\right)\right|\nrightarrow 0\quad\text{ as $x\to 1$}
$$


**Question**: what is the exact reason why Lemma II above is not applicable to \eqref{1}?  

The only thing I see in the proof that makes me suspicious is the statement that always $\eta(z)\to 0$ as $z\to\zeta\_0$. This seems not true for the above function as every circle $C\_z$ intersects more and more paths along with $g(z)$ diverges as $z\to\zeta\_0$, thus the maximum of $|g-A|$ is unbounded.


**Bonus question**: what are the additional requirements on $g(z)$ in order for the conclusion of the lemma to hold true?  

The function
$$
g(z)=e^{-\left(\dfrac{1+z}{1-z}\right)^3}
$$
(again given by Alexandre Eremenko in [this answer](https://mathoverflow.net/a/394789/113756)) seems to verify the conclusion of the above lemma and the main difference on the limiting behavior between it and \eqref{1} is that the latter does not have a Stolz approach region with greater than $0$ angular measure. Is this the missing requirement?


**References**


[1] Stanisław Łojasiewicz, "Une demonstration du théorème de Fatou" (French), Annales de la Société Polonaise de Mathématique/Rocznik Polskiego Towarzystwa Matematycznego, 22, 241-244 (1950), [MR0038429](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0038429), [Zbl 0035.33901](https://zbmath.org/?q=an%3A0035.33901).


[2] 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).



On a lemma of Łojasiewicz in complex analysis of one variable
The assumption of the Lemma is that $f$ has a finite limit as $z\to \zeta\_0$. This assumption does not hold in any of the two examples that you mention. In these examples, $f$ has a limit only when $z\to 1$ AND $z$ is real.


The Lemma is also true for non-tangential limits, with essentially the same proof but again none of the examples you cite has a non-tangential limit.


