**Disclaimer**: I posted this question seven days ago [here on the Math.SE](https://math.stackexchange.com/questions/4157296/in-search-for-a-counterexample-related-to-the-abel-stolz-theorem), with slightly different (however in an inessential way) comments. The question has been upvoted but no answer has been given, so I decided to repost it here to see if the members of this community can offer an answer.


**Edit**: I added a bit more information respect to the OP, answering (hopefully) the question posed by Pietro Majer in [his comment](https://mathoverflow.net/questions/394773/in-search-for-a-counterexample-related-to-the-abel-stolz-theorem#comment1010200_394773). Precisely I have added what I mean for uniform convergence in this context (in the main question text) and why I have considered the limit of $f$ along general curves $\gamma$ (in the "Notes" section).


**Definition 1**. A *curve in the complex plane* is the image $\gamma([a,b])$ of a segment $[a,b]\subseteq\Bbb R$ through a continuous non constant function $\gamma:[a,b]\to\Bbb C$: the points $\gamma(a)$ and $\gamma(b)$ are called *endpoints* of the curve.
With abuse of notation, the curve is identified with its defining continuous function $\gamma$ (which is really simply *a parametrization* of the curve).  

**Definition 2**. A complex sequence $(a\_n)\_{n\in\Bbb N}$ is said *Abel-summable* if, for $x\in [0,1]$, $\lim\_{x\to 1^-}\sum\_{n=0}^\infty a\_n x^n$ is finite: if $s\in\Bbb C$ is the value of the limit, this is usually written as $\sum\_{n=0}^\infty a\_n= s\;(\mathrm{A})$.  

**Definition 3**. A Stolz region $\Bbb {St}(M)$ in the unit disk $\Bbb D\triangleq \{z\in\Bbb C: |z|^2\le1\}$ is the set
$$
\Bbb {St}(M)\triangleq \big\{z\in \Bbb D : |1-z|\leq M(1-|z|)\big\} \quad M>1.
$$
Be it noted that if $M=1$ then $\Bbb {St}(M)=[0,1]$, while if $M<1$ then $\Bbb {St}(M)=\emptyset$, therefore the condition $M>1$ is simply a non-triviality condition.  

**Theorem (Abel-Stolz)**. Let $(a\_n)\_{n\in\Bbb N}\subset\Bbb C$ be a complex sequence such that $\sum\_{n=0}^\infty a\_n=s\in\Bbb C$. Then the power series $f(z)=\sum\_{n=0}^\infty a\_n z^n$ converges to $s$ along every curve $\gamma:[0,1]\to\Bbb C$ with $\gamma(0)=0$ and $\gamma(1)=1$ and $\gamma\subset\Bbb{St}(M)$: moreover the convergence is uniform for every point $z\in \Bbb{St}(M)$.   

In this context, uniform convergence means that, *for any $\varepsilon > 0$ the integer $m\in\Bbb N$ for which the relation*
$$
\left |f(z) -\sum\_{n=0}^m a\_n z^n\right|\triangleq|f(z)-f\_m(z)|<\varepsilon
$$
*holds true is independent from the chosen point $z\in \Bbb{St}(M)$*.


**My question**:



> 
> do there exist a divergent but Abel summable sequence $(a\_n)\_{n\in\Bbb N}\subset\Bbb C$ for which $f(z)$ does not converge uniformly on the "deleted" Stolz region $\Bbb{St}(M)\setminus\{ 1\}$?
> 
> 
> 


**Some notes and why I am interested**


* A (sketchy) proof of Abel-Stolz theorem can be found in the [related wikipedia entry](https://en.wikipedia.org/wiki/Abel%27s_theorem): for an historical survey on the various proofs and extensions of this theorem, I consulted the first paragraph of Hardy's paper [1], while for the summability theory based proof I read Knopp [2] since he offers a detailed analysis in §8, theorem 4 and theorem 5 p. 74, §52, theorem 1 pp. 391-392.
* The problem was motivated by a research on power series on the unit disk: precisely, I am trying to use a result of Giovanni Ricci [4] with different hypothesis. He is able to estimate the remainder (or the limit of the remainder, as Ricci considers also series for which $f(1)$ does not exists but $\lim\_{r\to 1^-} f(r)=s$ does) of a power series which cannot be analytically continued across the point $z=1$. The series I am studying have this property, but the results of Ricci are not directly applicable as he uses several more hypotheses which are not easily checked in my case.  

Then I tried to modify Ricci's method, itself also a modification of a proof method introduced by W.H. Young (see the references in [4], notably Titchmarsch's "Theory of functions") to prove [Fatou's theorem](https://en.wikipedia.org/wiki/Fatou%27s_theorem) for regular boundary points: and since it uses Cauchy's integral formula, I thougt that a judicious use of an integration path touching the boundary could be useful in determining the sought for modification, thus I started considering paths $\gamma$ with one endpoint in $z=1$. However, as pointed out again by Pietro Majer, it is not necessary to consider any curve since *even the standard theorem holds irrespectively of any regularity hypothesis on $\gamma$*: you can have $\gamma\big([0,1]\big)=\Bbb{St}(M)$.
* Dealing with the task described in the previous point, I noticed the version of Fatou's theorem given by Privalov [3]: in my opinion, it is an interesting result since it lists between the *necessary and sufficient conditions* for the theorem to hold the fact that $f$ should be bounded in a non tangential conical region whose vertex is the relevant boundary point. During the reading of [3], I found a simple proof of the necessary part (I had to find a proof by myself, since Privalov considers this fact as obvious and thus does not give an explicit proof). Then I started to wonder if a stronger statement actually holds: and this thought has originated this question.


**References**


[1] Godfrey Harold Hardy, "Some theorems connected with Abel’s theorem on the continuity of power series". (English) Proceedings of the London Mathematical Society (2) 4, 247-265 (1906), [JFM 37.0429.01](https://www.zbmath.org/?q=an%3A37.0429.01).


[2] Konrad Knopp, *Theory and Application of Infinite Series*, Translated from the 2nd ed. and revised in accordance with the fourth by R. C. H. Young, London-Glasgow: Blackie & Son, 1951, XII+563, [Zbl 0042.29203](https://www.zbmath.org/?q=an%3A0042.29203).


[3] Ivan Ivanovich Privalov, "[Sur une généralisation du théorème de Fatou](http://mi.mathnet.ru/eng/msb6849)" (Russian, French abstract) Recueil Mathématique Moscou (Matematicheskiĭ Sbornik) 31, 232-235 (1923), [JFM 49.0225.02](https://www.zbmath.org/?q=an%3A49.0225.02).


[4] 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)



In search for a counterexample related to the Abel-Stolz theorem
In fact a trivial counterexample to the question, as now clarified, is just $a\_n:=(-1)^n$ with $s:=\frac12$. In your notation, $a\_n\to\frac12\;\bf (A)$, because the series $\sum\_{n=0}^\infty (-z)^n$ converges on the open disk to $f(z):=\frac1{1+z}$, which is holomorphic at $z=1$. But the series does not converges uniformly on any Stolz angle, not even on $[1-\epsilon,1)$, since the remainder $f(z)-f\_m(z)$ is $\frac{(-z)^{m+1}}{1+z}$, whose uniform norm on $[1-\epsilon,1)$ is at least $1/2$ .


