On Schwarz-Christoffel mappings
Let \(f\) be a Schwarz-Christoffel mapping of the unit disk \(\mathbb D\) onto the interior of an \((n+1)\)-gon. The first author et al. [Ann. Acad. Sci. Fenn., Math. 36, No. 2, 449--460 (2011; Zbl 1239.30003); Proc. Am. Math. Soc. 140, No. 10, 3495--3505 (2012; Zbl 1283.30048)] have shown that the pre-Schwarzian of \(f\) has the form NEWLINE\[NEWLINE\frac{f''}{f'}=\frac{2\frac{B_1}{B_2}}{1-z\frac{B_1}{B_2}}NEWLINE\]NEWLINE for some finite Blaschke products \(B_1\), \(B_2\) without common zeros, with respective degrees \(d_1,\;d_2\) satisfying \(d_1+d_2=n\). It is known that NEWLINE\[NEWLINE\frac{f''}{f'}=-2\sum_{k=1}^{n+1}\frac{\beta_k}{z-z_k},\tag{1}NEWLINE\]NEWLINE where \(z_k\) are prevertices and \(2\pi\beta_k\) are the exterior angles ot \(z_k\). Theorem 2 says that \(d_2\) is equal to the number of concave vertices, while \(d_1+1\) is equal to the number of convex vertices. In Theorem 3 the authors characterize the mappings \(f\) onto the exterior of an \((n+2)\)-gon. Theorem 4 solves the problem of the univalence of solutions of (1). Namely, let \(0\leq t_1<\cdots <t_{n+1}<2\pi\), \(z_k=e^{it_k}\), \(\beta_k\in\mathbb R\) and \(\sum_{k=1}^{n+1} \beta_k=1\), \(\sum_{k=1}^{n+1} |\beta_k|\leq 2\). Then \(f\) defined by NEWLINENEWLINENEWLINE\[NEWLINEf(z)=a\prod_{k=1}^{n+1}(z-z_k)^{-2\beta_k},\quad a\in\mathbb C\setminus\{0\},NEWLINE\]NEWLINE is univalent in \(\mathbb D\).NEWLINENEWLINEIn the final part of the paper the authors study the geometric interplay between the location of the zeros of the Blaschke products and the separation of the prevertices.
scientific article; zbMATH DE number 6361405