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Univalent functions with logarithmic restrictions. (English) Zbl 0755.30027

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)

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
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