Univalent functions with logarithmic restrictions

A. Z. Grinshpan

Annales Polonici Mathematici (1991)

  • Volume: 55, Issue: 1, page 117-139
  • ISSN: 0066-2216

Abstract

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It is known that univalence property of regular functions is better understood in terms of some restrictions of logarithmic type. Such restrictions are connected with natural stratifications of the studied classes of univalent functions. The stratification of the basic class S of functions regular and univalent in the unit disk by the Grunsky operator norm as well as the more general one of the class 𝔐 * of pairs of univalent functions without common values by the τ-norm (this concept is introduced here) are given in the paper. Moreover, some properties of univalent functions whose range has finite logarithmic area are considered. To apply the logarithmic restrictions a special exponentiation technique is used.

How to cite

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A. Z. Grinshpan. "Univalent functions with logarithmic restrictions." Annales Polonici Mathematici 55.1 (1991): 117-139. <http://eudml.org/doc/262298>.

@article{A1991,
abstract = {It is known that univalence property of regular functions is better understood in terms of some restrictions of logarithmic type. Such restrictions are connected with natural stratifications of the studied classes of univalent functions. The stratification of the basic class S of functions regular and univalent in the unit disk by the Grunsky operator norm as well as the more general one of the class 𝔐 * of pairs of univalent functions without common values by the τ-norm (this concept is introduced here) are given in the paper. Moreover, some properties of univalent functions whose range has finite logarithmic area are considered. To apply the logarithmic restrictions a special exponentiation technique is used.},
author = {A. Z. Grinshpan},
journal = {Annales Polonici Mathematici},
keywords = {univalent functions; logarithmic restrictions; Grunsky operator; quasiconformal extension; exponentiation technique; functions without common values; homeomorphic assembling; Lebedev-Milin exponential inequality; analytic functions; finite range area; Bieberbach-Eilenberg functions; finite logarithmic area; Lebedev- Milin inequalities; Grunsky coefficients; coefficient results},
language = {eng},
number = {1},
pages = {117-139},
title = {Univalent functions with logarithmic restrictions},
url = {http://eudml.org/doc/262298},
volume = {55},
year = {1991},
}

TY - JOUR
AU - A. Z. Grinshpan
TI - Univalent functions with logarithmic restrictions
JO - Annales Polonici Mathematici
PY - 1991
VL - 55
IS - 1
SP - 117
EP - 139
AB - It is known that univalence property of regular functions is better understood in terms of some restrictions of logarithmic type. Such restrictions are connected with natural stratifications of the studied classes of univalent functions. The stratification of the basic class S of functions regular and univalent in the unit disk by the Grunsky operator norm as well as the more general one of the class 𝔐 * of pairs of univalent functions without common values by the τ-norm (this concept is introduced here) are given in the paper. Moreover, some properties of univalent functions whose range has finite logarithmic area are considered. To apply the logarithmic restrictions a special exponentiation technique is used.
LA - eng
KW - univalent functions; logarithmic restrictions; Grunsky operator; quasiconformal extension; exponentiation technique; functions without common values; homeomorphic assembling; Lebedev-Milin exponential inequality; analytic functions; finite range area; Bieberbach-Eilenberg functions; finite logarithmic area; Lebedev- Milin inequalities; Grunsky coefficients; coefficient results
UR - http://eudml.org/doc/262298
ER -

References

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  2. [2] L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137-152. Zbl0573.30014
  3. [3] P. L. Duren, Univalent Functions, Springer, Berlin 1983. 
  4. [4] A. Z. Grinshpan, An application of the area principle to Bieberbach-Eilenberg functions, Mat. Zametki 11 (1972), 609-618 (in Russian); English transl.: Math. Notes 11 (1972), 371-377. Zbl0246.30015
  5. [5] A. Z. Grinshpan, On the growth of coefficients of univalent functions with a quasiconformal extension, Sibirsk. Mat. Zh. 23 (2) (1982), 208-211 (in Russian). Zbl0494.30018
  6. [6] A. Z. Grinshpan, Coefficient inequalities for conformal mappings with homeomorphic extension, Sibirsk. Mat. Zh. 26 (1) (1985), 49-65 (in Russian); English transl.: Siberian Math. J. 26 (1) (1985), 37-50. Zbl0569.30016
  7. [7] A. Z. Grinshpan, Univalent functions and regularly measurable mappings, Sibirsk. Mat. Zh. 27 (6) (1986), 50-64 (in Russian); English transl.: Siberian Math. J. 27 (6) (1986), 825-837. Zbl0617.30016
  8. [8] A. Z. Grinshpan, Method of exponentiation for univalent functions, in: Theory of Functions and Applications (Proc. Conf. Saratov 1988), Part 2, Izdat. Saratov. Univ., Saratov 1990, 72-74 (in Russian). 
  9. [9] A. Z. Grinshpan and I. M. Milin, Simply connected domains with finite logarithmic area and Riemann mapping functions, in: Constantin Carathéodory: An International Tribute, World Scientific, Singapore, to appear. 
  10. [10] S. L. Krushkal' and R. Kühnau, Quasiconformal Mappings--New Methods and Applications, Nauka, Sibirsk. Otdel., Novosibirsk 1984 (in Russian). 
  11. [11] N. A. Lebedev, The Area Principle in the Theory of Univalent Functions, Nauka, Moscow 1975 (in Russian). Zbl0747.30015
  12. [12] I. M. Milin, Univalent Functions and Orthonormal Systems, Nauka, Moscow 1971 (in Russian); English transl.: Amer. Math. Soc., Providence 1977. 
  13. [13] Ch. Pommerenke, Univalent Functions, Vandenhoeck und Ruprecht, Göttingen 1975. 
  14. [14] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk 1962 (in Russian). 
  15. [15] O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildungen, Deutsche Math. 3 (6) (1938), 621-678. Zbl0020.23801

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