Recent zbMATH articles in MSC 30C20https://www.zbmath.org/atom/cc/30C202022-05-16T20:40:13.078697ZWerkzeugConformal welding for critical Liouville quantum gravityhttps://www.zbmath.org/1483.300292022-05-16T20:40:13.078697Z"Holden, Nina"https://www.zbmath.org/authors/?q=ai:holden.nina"Powell, Ellen"https://www.zbmath.org/authors/?q=ai:powell.ellenSummary: Consider two critical Liouville quantum gravity surfaces (i.e., \(\gamma\)-LQG for \(\gamma =2)\), each with the topology of \(\mathbb{H}\) and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent \(\mathrm{SLE}_4\). Combined with the proof of uniqueness for such a welding, recently established by \textit{O. McEnteggart, J. Miller} and \textit{W. Qian} [``Uniqueness of the welding problem for SLE and Liouville quantum gravity'', Preprint, \url{arXiv:1809.02092}], this shows that the welding operation is well-defined. Our result is a critical analogue of \textit{S. Sheffield}'s quantum gravity zipper theorem [Ann. Probab. 44, No. 5, 3474--3545 (2016; Zbl 1388.60144)], which shows that a similar conformal welding for subcritical LQG (i.e., \(\gamma\)-LQG for \(\gamma\in (0,2))\) is well-defined.Time-reversal of multiple-force-point \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) with all force points lying on the same sidehttps://www.zbmath.org/1483.300302022-05-16T20:40:13.078697Z"Zhan, Dapeng"https://www.zbmath.org/authors/?q=ai:zhan.dapeng.1|zhan.dapengSummary: We define intermediate \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) and reversed intermediate \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the time-reversal of multiple-force-point chordal \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) curves in the case that all force points are on the boundary and lie on the same side of the initial point, and \(\kappa\) and \(\underline{\rho}=(\rho_1,\dots,\rho_m)\) satisfy that either \(\kappa\in (0,4]\) and \(\sum_{{j=1}^k}{\rho_j}> -2\) for all \(1\le k\le m\), or \(\kappa\in (4,8)\) and \(\sum_{{j=1}^k}\rho_j\ge \frac{\kappa}{2}-2\) for all \(1\le k\le m\).A 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.