Grinshpan, A. Z. Univalent functions with logarithmic restrictions. (English) Zbl 0755.30027 Ann. Pol. Math. 55, No. 1-3, 117-138 (1991). The author generalizes results about analytic functions (1) \(f(z)=c_ 1 z+c_ 2z^ 2+\dots\) of the unit disk that are bounded, have finite range area, or are Bieberbach-Eilenberg functions. Therefore he uses the notion of functions whose range has a finite logarithmic area, and an exponentiation technique based on the Lebedev-Milin inequalities. Moreover for functions \(f(z)\) given by (1) for which \(\alpha_{n,\ell}\) are the Grunsky coefficients \[ \log{{f(z)-f(\zeta)} \over {z- \zeta}}=\sum_{n,\ell=0}^ \infty \alpha_{n,\ell}z^ n\zeta^ \ell, \] and the Grunsky operator \(G_ f=\{\sqrt{n\ell}\alpha_{n,\ell}\}_{n,\ell=1}^ \infty: \ell^ 2\to\ell^ 2\) has norm \((\| x\|=(\sum_{n=1}^ \infty| x_ n|^ 2)^{1/2})\), \(\| G_ f\|\leq k\), coefficient results are obtained by a special exponentiation technique. Reviewer: W.Koepf (Berlin) Cited in 2 Documents MSC: 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C62 Quasiconformal mappings in the complex plane Keywords:Lebedev-Milin exponential inequality; analytic functions; finite range area; Bieberbach-Eilenberg functions; finite logarithmic area; Lebedev- Milin inequalities; Grunsky coefficients; Grunsky operator; coefficient results PDFBibTeX XMLCite \textit{A. Z. Grinshpan}, Ann. Pol. Math. 55, No. 1--3, 117--138 (1991; Zbl 0755.30027) Full Text: DOI