Notation and premises. Here it is a list of notations more or less explicitly used in the question:
- If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(2\pi it)$,
- $\Bbb D \triangleq\{z\in \Bbb C : |z|<1\}$ and consequently $\partial\Bbb D \triangleq \{z\in \Bbb C : |z|=1\}$ and $\bar{\Bbb D} \triangleq\{z\in \Bbb C : |z|\le1\}$.
- An approach region to the point $z_o\in\partial D$ for a holomorphic function $f(z)$ in $\Bbb D$ is a subregion of $\bar{\Bbb D}$ containing $z_o$ for which the limit of $f$ along any continuous curve ending at $z_o$ and contained in it is equal to its (assumed finite) radial limit.
- A tangential region is an approach region which has a first order contact with $\partial\Bbb D$ at $z_o$, i.e. for which the boundary around $z_o$ is smooth and the tangent to the boundary coincide with the tangent to $\partial\Bbb D$ at $z_o$.
The following theorem was proved by Andrew M. Rockett and Peter Szüsz ([2], pp. 447-448, theorem 1):
Theorem. Let $f(z)=\sum_{k=0}^\infty a_kz^k$ be a power series converging in $\Bbb D\cup \{z=1\}$. Then the region $$ \left\{z\in\Bbb C :\sum_{l=0}^\infty r^{l/t}\max_{l/t\le j<(l+1)/t} \left|\sum_{k\ge l/t}^ja_k r^k\right|\ll 1\right\} $$ is a tangential region for the function $f(z)$ to the point $z=1$.
Question: has this result be used and/or improved further in subsequent researches on approach regions? I am asking this because its ZBMath Open review does not list any citations in subsequent works (be it reviews or documents), nor it is cited in the bibliography of the standard monograph [1] of Fausto Di Biase: moreover I asked to Prof. Rockett and even him is not aware of any further development.
Notes
- This question was inspired by this one of Gagar and it is related to the researches that inspired my former question: precisely, while searching a characterization for power series converging at $z=1$ involving their Stolz region, I found this paper.
- The paper is interesting in that it shows that any power series converging at a boundary point of its convergence disk has a tangential approach region. This is not true if $f(z)$ has "only" a radial limit.
References
[1] Fausto Di Biase, Fatou type theorems. Maximal functions and approach regions, (English) Progress in Mathematics, 147. Boston, MA: Birkhäuser Verlag, pp. viii+152 (1998), ISBN: 0-8176-3976-4, MR1483892, Zbl 0889.31002.
[2] Andrew M. Rockett and Peter Szüsz, "On tangential regions for power series" (English), Archiv der Mathematik 60, No. 5, 446-450 (1993), MR1213514, Zbl 0778.30004.
