Edit: cross-posted to the MathOverflow with some with some modifications in order to answer to questions posed in the comments. Now it has an accepted answer by Pietro Majer and a very interesting answer by Alexandre Eremenko.
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 $z\in \Bbb{St}(M)$.
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. For an historical survey on the various proofs and extension of this theorem see the first paragraph of Hardy's paper [1], while Knopp [2] offers a detailed analysis in §8, theorem 4 p.74, theorem 5 p. 74, §52, theorem 1 pp. 391-392.
- The problem was motivated by a research on Fatou's theorem: in particular, during the reading of [3] I found that any $f(z)$ such that $\lim_{x\to 1}f(x)=s$ converges to $s$ along every path $\gamma\subset\Bbb{St}(M)$ with endpoints $0$ and $1$ and is bounded on $\Bbb {St}(M)$ (I had to find a proof by myself, since Privalov consider this fact as obvious and thus does not give an explicit proof).
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.
[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.
[3] Ivan Ivanovich Privalov, "Sur une généralisation du théorème de Fatou" (Russian, French abstract) Recueil Mathématique Moscou (Matematicheskiĭ Sbornik) 31, 232-235 (1923), JFM 49.0225.02.