On Szegö-type covering problem of K-Q.C. (English) Zbl 0703.30018

Cheng, Baolong

Northeast. Math. J. 5, No. 3, 272-276 (1989).

Let \(w=f(z)\) be a K-q.c. in the unit disk \(D=\{z\); \(| z| <1\}\), normalized by the conditions: \(f(0)=0\), and there exist n small positive numbers \(r_ 1,r_ 2,...,r_ n\), such that \[ \arg f(r_ k \exp (2k\pi i/n))=2k\pi /n,\quad k=1,2,...,n; \]
\[ \min_{1\leq k\leq n}f(r_ k \exp (2k\pi i/n))=\max_{1\leq k\leq n}r_ k^{1/k}. \] The author applies He Chengqi’s piecewise symmetrization theorem [J. Fudan Univ., Nat. Sci. 24, 445-451 (1985; Zbl 0609.30020)] to prove \[ J=(\prod^{n}_{k=1}| W^*_ k|)^{1/n}>4^{(1-2K)/nK}, \] where \(W^*_ k=\inf \{| W|\), arg W\(=2k\pi /n\), \(W\in \partial f(D)\}\). When f is a univalent function of class S, we have \(K=1\), and \(r=\max_{1\leq k\leq n}r_ k\to 0\), then the result degenerates to \(J>4^{1/n}\), the classical Szegö covering theorem for univalent functions.

Reviewer: Jixiu Chen

MSC:

30C62

Quasiconformal mappings in the complex plane

Keywords:

Teichmüller function; class S; Grötzsch function; symmetrization; Szegö covering

Citations:

Zbl 0609.30020