Russian Mathematical Surveys, 2023, Volume 78, Issue 6, Pages 1164–1166 DOI: https://doi.org/10.4213/rm10152e (Mi rm10152)

DOI: https://doi.org/10.4213/rm10152e

Funding agency	Grant number
Russian Science Foundation	21-11-00131
This research was carried out at Lomonosov Moscow State University and supported by the Russian Science Foundation under grant no. 21-11-00131.

Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 6(474), Pages 185–186
DOI: https://doi.org/10.4213/rm10152

Bibliographic databases:

Let $\mathscr B$ be the set of holomorphic self-maps of the unit disc $\mathbb D=\{z\in \mathbb C\colon |z|<1\}$. Fixed points of functions $f$ in $\mathscr B$ are of importance for their investigation. An interior point $a\in\mathbb D$ is a fixed point of $f\in \mathscr B$ if $f(a)=a$. A boundary point $a\in \mathbb T$, $\mathbb T=\{z\in \mathbb C\colon |z|=1\}$ is called a fixed point of $f\in \mathscr B$ if $\angle \lim_{z\to a}f(z)=a$. In the latter case a finite or an infinite angular limit $f'(a)=\angle \lim_{z\to a}(f(z)-a)/(z-a)$ exists, which is called the angular derivative of $f$ at $a$; when this quantity is finite, it is positive. The set of all fixed points of a function $f\in \mathscr B$ is non-empty. Furthermore, if $f$ is not linear fractional, then there exists a fixed point with the attraction property, which is, moreover, unique. It is usually called the Denjoy–Wolff point of $f$ (see [1], Chap. VI, § 43, for details). The Denjoy–Wolff point can lie in the interior of $\mathbb D$ (and then $|f'(q)|\leqslant 1$) or on the boundary $\mathbb T$ (and then $0<f'(q)\leqslant 1$). The other fixed points (if any) lie on $\mathbb T$ and are repelling.

Let $\mathscr B[q]$, $q\in \overline{\mathbb D}$, denote the set of functions $f$ in $\mathscr B$ such that $q$ is their Denjoy–Wolff point. If $f\in \mathscr B[q]$, $q\in {\mathbb D}$, and $f'(q)\neq 0$, then $f$ is univalent in a neighbourhood of $q$. It turns out that the size of this neighbourhood can be chosen in dependence on the value of $f'(q)$ alone. For example, if $q=0$ and $|f'(0)|=\beta$, $\beta\in(0,1)$, then $f$ is univalent in the disc $\{z\in \mathbb D\colon |z|<\beta/(1+\sqrt{1-\beta^2}\,)\}$ (see [2]).

Let $\mathscr B\{1\}$ denote the set of functions $f$ in $\mathscr B$ with the fixed boundary point $a=1$: $\mathscr B\{1\}= \{f\in \mathscr B\colon \angle\lim_{z\to 1}f(z)=1\}$. Valiron [1] showed that if $f\in \mathscr B\{1\}$ and $f'(1)<+\infty$, then $f$ is univalent in a sector with vertex $1$ of opening arbitrarily close to $\pi$, whose radius depends on the opening and the function itself. Becker and Pommerenke [3] managed to estimate the size of this sector: in dependence of the function itself, they found a domain of univalence for it which contains all sectors indicated by Valiron.

Given some $\alpha>1$, in the class $\mathscr B\{1\}$ we can distinguish the subclass $\mathscr B_{\alpha}\{1\}$ of functions subject to a constraint on the value of the angular derivative at the fixed point: $\mathscr B_{\alpha}\{1\}=\bigl\{f\in \mathscr B\{1\}\colon f'(1)\leqslant \alpha \bigr\}$. It was shown in [4] that for no $\alpha>1$ does there exist a non-empty univalence domain for the whole of the class $\mathscr B_{\alpha}\{1\}$. Thus, by contrast to an interior fixed point, the class $\mathscr B_{\alpha}\{1\}$ is too wide from the standpoint of the search for domains of univalence, and we must look at more narrow classes to achieve progress in this problem. Since, apart from the repelling fixed point $a=1$, each function in $\mathscr B_{\alpha}\{1\}$ must also have an attracting (Denjoy–Wolff) fixed point, the class $\mathscr B_{\alpha}\{1\}$ can naturally be represented as the union of disjoint subclasses distinguished by the position of the Denjoy–Wolff point (which can lie in the interior or on the boundary of $\mathbb D$): $\mathscr B_{\alpha}\{1\}= \bigcup_{q\in\overline{\mathbb D}}\mathscr B_{\alpha}[q,1]$, where $\mathscr B_{\alpha}[q,1]=\mathscr B_{\alpha}\{1\}\cap \mathscr B[q]$.

That for some values of $\alpha$ the above subclasses of $\mathscr B_{\alpha}\{1\}$ possess domains of univalence was shown in [5]. Now, to have simpler statements we set $q=0$ (in the case of an interior Denjoy–Wolff point) or $q=-1$ (in the case of a boundary Denjoy–Wolff point).

For $\alpha\in(1,4]$ the sharp domain of univalence for the class $\mathscr B_{\alpha}[0,1]$ was found in [6]: $\{z\in \mathbb D\colon |1-2z+|z|^2|\,({1-|z|^2})^{-1}< (\alpha-1)^{-1/2}\}$. In [7] an upper estimate was obtained for a domain of univalence for $\mathscr B_{\alpha}[-1,1]$. It turns out that for $\alpha\in (1,4]$ this estimate is sharp.

This gives a complete answer to the question of a sharp domain of univalence for the subclasses $\mathscr B_{\alpha}\{1\}$, $\alpha\in(1,4]$, when the position of the Denjoy–Wolff point is fixed.

The proof of Theorem 1 is based on a refinement of an inequality due to Becker and Pommerenke [3], which implies that the quantities

On the other hand we can show that a kind of a reverse inequality holds.

Supposing that a function $f\in\mathscr B_\alpha[-1,1]$ is not univalent in the domain $\mathscr U(\alpha)$, taking Lemmas 1 and 2 into account we arrive at a contradiction.

Bibliography
1.	G. Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954, 236 pp.
2.	E. Landau, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., 1926, 467–474
3.	J. Becker and Ch. Pommerenke, Comput. Methods Funct. Theory, 17:3 (2017), 487–497
4.	O. S. Kudryavtseva and A. P. Solodov, Sb. Math., 210:7 (2019), 1019–1042
5.	V. V. Goryainov, Sb. Math., 208:3 (2017), 360–376
6.	A. P. Solodov, Izv. Math., 85:5 (2021), 1008–1035
7.	V. V. Goryainov, O. S. Kudryavtseva, and A. P. Solodov, Dokl. Math., 108:1 (2023), 326–330

Citation in format AMSBIB

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\yr 2023
\vol 78
\issue 6(474)
\pages 185--186
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