On extremal decomposition problems in families of systems of domains of general type (Q1608612)

The author considers extremal decompositions on the sphere in the following context. For the finite number of points in the plane \(c_k\), \(k=1, \dots,k\), there are assigned simply-connected domains structured like end domains for a meromorphic quadratic differential with poles of assigned order \(\geq 3\) at the \(c_k\), as well as a finite number of triangles (trigons) structured like corresponding strip domains. Under approximate analytic conditions at the \(c_k\) by a limiting process there is obtained a corresponding module function. One can also allow ring and strip domains. It is shown that a unique quadratic differential provides an extremal for the module function. The author applies this approach to coefficient problems for the family of functions \(b(z)=z+a_0+ \sum^\infty_{j=1} a_jz^{-j}\) meromorphic and univalent for \(|z|>1\) which omit a simply-connected domain of inner conformal radius \(r\), \(0<r<1\), with respect to the origin. The author considers bounds for the coefficients \(a_1\), a question considered elsewhere by other methods (Zap. Nauchn. Semin. POMI 196, 101-116 (1991; Zbl 0835.30017). The statements at the bottom of p. 3135 require some correction. In the paper of Duren-Schiffer the coefficient normalization is at the origin, not at the point at infinity and thus their results appear to the coefficient \(a_0\), not to \(a_1\), and is an unmediate consequence of Theorem 7.3 in the reviewer's book (Univalent functions and Conformal Mapping (1962; Zbl 0016.04801) as the reviewer pointed out to Duren priori to the publication of that paper. The author does not seem to be aware that the reviewer (Trans. Am. Math. Soc. 95, 387-407 (1960; Zbl 0102.06401), was the first to treat the problem for \(a_1\) and gave a complete solution using the general coefficient theorem.