Recent zbMATH articles in MSC 30C35https://www.zbmath.org/atom/cc/30C352022-05-16T20:40:13.078697ZWerkzeugA proof of Hall's conjecture on length of ray images under starlike mappings of order \(\alpha\)https://www.zbmath.org/1483.300372022-05-16T20:40:13.078697Z"Hästö, Peter"https://www.zbmath.org/authors/?q=ai:hasto.peter-a"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: Assume that \(f\) lies in the class of starlike functions of order \(\alpha\in[0,1)\), that is, which are regular and univalent for \(|z|< 1\) and such that
\[
\mathrm{Re}\left(\frac{zf'(z)}{f(z)}\right)> \alpha\quad\text{for } |z|<1.
\]
In this paper we show that for each \(\alpha\in[0,1)\), the following sharp inequality holds:
\[
|f(re^{i\theta})|^{-1}\int_0^r|f'(ue^{i\theta})|\,du\leq\frac{\Gamma(\frac{1}{2})\Gamma(2-\alpha)}{\Gamma(\frac{3}{2}-\alpha)} \quad\text{for every } r< 1 \text{ and } \theta.
\]
This settles the conjecture of \textit{R. R. Hall} [Bull. Lond. Math. Soc. 12, 119--126 (1980; Zbl 0442.30007)] positively.Asymptotic upper bound for tangential speed of parabolic semigroups of holomorphic self-maps in the unit dischttps://www.zbmath.org/1483.370572022-05-16T20:40:13.078697Z"Cordella, Davide"https://www.zbmath.org/authors/?q=ai:cordella.davideThe author studies continuous semigroups of holomorphic maps in the unit disc \((\Phi_t)_{t\ge 0}\). For a non-elliptic semigroup, \textit{F. Bracci} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 73, No. 2, 21--43 (2019; Zbl 1436.30007)] introduced and studied three kinds of speeds: the total speed, the orthogonal speed, and the tangential speed. The tangential speed \(v^T(t)\) is related to the slope of convergence of orbits to the Denjoy-Wolff point of the semigroup. In the present paper, the author proves a conjecture in [loc. cit.] claiming that \(\limsup_{t\to\infty}\left(v^T(t)-\frac 12\log t\right)<\infty\) holds for parabolic semigroups.
Reviewer: Barbara Drinovec Drnovsek (Ljubljana)