Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Prokhorov D. V. Value Regions in Classes of Conformal Mappings. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 3, pp. 258-279. DOI: 10.18500/1816-9791-2019-19-3-258-279, EDN: VBBALE

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2019
Full text:
download
(downloads: 170)
Language: 
English
Heading: 
Mathematics
Article type: 
Article
UDC: 
517.54
DOI: 
10.18500/1816-9791-2019-19-3-258-279
EDN: 
VBBALE

Value Regions in Classes of Conformal Mappings

Autors: 
Prokhorov Dmitri Valentinovich, Saratov State University
Abstract: 

The survey is devoted to most recent results in the value region problem over different classes of holomorphic univalent functions represented by solutions to the Loewner differential equations both in the radial and chordal versions. It is important also to present classical and modern solution methods and to compare their efficiency. More details are concerned with optimization methods and the Pontryagin maximum principle, in particular. A value region is the set {f(z 0 )} of all possible values for the functional f 7→ f(z 0 ) where z 0 is a fixed point either in the upper half-plane for the chordal case or in the unit disk for the radial case, and f runs through a class of conformal mappings. Solutions to the Loewner differential equations form dense subclasses of functio families under consideration. The coefficient value regions {(a 2 ,...,a n ) : f(z) = z+P ∞n=2 a n zn }, |z| < 1, are the part of the field closely linked with extremal problems and the Bombieri conjecture about the structure of the coefficient region for the class S in a neighborhood of the point (2,...,n) corresponding to the Koebe function.

Key words: 
value region
Loewner equation
reachable set
boundary curve
References: 
  1. Goluzin G. M. Geometric theory of functions of a complex variable. New York, Amer.Math. Soc., 1969. 684 p. (Russ. ed.: Moscow, Nauka, 1966. 628 p.).
  2. Aleksandrov I. A. Parametrichrskie prodolzheniya v teorii odnolistnykh funktsij [Parametric continuations in the theory of univalent functions]. Moscow, Nauka, 1976. 344 p. (in Russian).
  3. Duren P. L. Univalent functions. New York, Springer Verlag, 1983. 382 p.
  4. Pommerenke Ch. Univalent functions. Göttingen, Vandenhoeck & Ruprecht, 1975. 376 p.
  5. Rogosinski W. Zum Schwarzen Lemma. Jahresber. Deutsche Math-Verein., 1934, vol. 44, pp. 258–261.
  6. Grunsky H. Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche. Schr. Math. Inst. u Inst. Angew. Math. Univ. Berlin, 1932, vol. 1, pp. 95–140.
  7. Popov V. I. Value domain of a system of functionals on the class S. Trudy Tomskogo universiteta. Voprosy geometricheskoj teorii funktsij [Proceedings of Tomsk University. Issues of the geometric function theory], 1965, vol. 182, iss. 3, pp. 106–132 (in Russian).
  8. Gutlyanskii V. Ya. Parametric representation of univalent functions. Soviet Math. Dokl., 1970, vol. 11, no. 5, pp. 1273–1276.
  9. Schaeffer A. C., Spencer D. C. Coefficient regions for schlicht functions. Coll. Publ., vol. 35. New York, Amer. Math. Soc., 1950. 325 p.
  10. Loewner K. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann., 1923, vol. 89, no. 1–2, pp. 103–121.
  11. Kufarev P. P. On one-parameter families of analytic functions. Rec. Math. [Mat. Sbornik] N.S., 1943, vol. 13(55), no. 1, pp. 87–118 (in Russian).
  12. Pommerenke Ch.¨Uber die Subordination analytischer Funktionen. J. Reine Angew. Math., 1965, vol. 218, pp. 159–173.
  13. Kufarev P. P. A remark on integrals of the Loewner equation. Doklady Akad. Nauk SSSR, 1947, vol. 57, no. 7, pp. 655–656 (in Russian).
  14. Goryainov V. V., Gutlyanskii V. Ya. On extremal problems in the class S M . In: Matematicheskii sbornik. Kiev, Naukova dumka, 1976, pp. 242–246 (in Russian).
  15. Roth O., Schleissinger S. Rogosinski’s lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation. Bull. London Math. Soc., 2014, vol. 46, no. 5, pp. 1099–1109. DOI: https://doi.org/10.1112/blms/bdu054
  16. Prokhorov D., Samsonova K. Value range of solutions to the chordal Loewner equation. J. Math. Anal Appl., 2015, vol. 428, no. 2, pp. 910–919. DOI: https://doi.org/10.1016/j.jmaa.2015.03.065
  17. Zherdev A. Value range of solutions to the chordal Loewner equation with restriction on the driving function. Probl. Anal. Issues Anal, 2019, vol. 8(26), no. 2, pp. 92–104. DOI: https://doi.org/10.15393/j3.art.2019.6270
  18. Koch J. D. Value regions for schlicht functions. Dissertationsschrift zur Erlangung des naturwissenschaftlichen Doktorgrades der Julius-Maximilians-Universität Würzburg. Würzburg, 2016. 93 p.
  19. Goodman G. S. Univalent functions and optimal control. Ph.D. Thesis, Stanford University, 1967. ProQuest LLC, Ann Arbor, MI.
  20. Friedland S., Schiffer M. Global results in control theory with applications to univalent functions. Bull. Amer. Math. Soc., 1976, vol. 82, no. 6, pp. 913–915.
  21. Friedland S., Schiffer M. On coefficient regions of univalent functions. J. Anal. Math., 1977, vol. 31, pp. 125–168.
  22. Roth O. Control Theory in H (D). Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilians-Universität Würzburg. Würzburg, 1998. 178 p.
  23. Prokhorov D. Sets of values of systems of functionals in classes of univalent functions. Mat. Sb., 1990, vol. 181, no. 12, pp. 1659–1677; Math. USSR-Sb., 1992, vol. 71, no. 2, pp. 499–516.
  24. Prokhorov D. Reachable set methods in extremal problems for univalent functions. Saratov, Saratov Univ. Press, 1993. 228 p.
  25. Schiffer M. Sur l’équation différentielle de M. Löwner. C. R. Acad. Sci. Paris, 1945, vol. 221, pp. 369–371.
  26. Prokhorov D., Vasil’ev A. Univalent functions and integrable systems. Commun. Math. Phys., 2006, vol. 262, no. 2, pp. 393–410. DOI: https://doi.org/10.1007/s00220-005-1499-y
  27. Roth O. Is there a Teichmüller principle in higher dimension? Geometric function theory in higher dimension. F. Bracci (ed.). Cham, Springer INdAM Series 26, 2017, pp. 87–105. DOI: https://doi.org/10.1007/978-3-319-73126-1_7
  28. Popov V. I. L. S. Pontryagin’s maximum principle in the theory of univalent functions. Soviet Math. Dokl., 1969, vol. 10, pp. 1161–1164.
  29. Prokhorov D. The method of optimal control in an extremal problem on a class of univalent functions. Soviet Math. Dokl., 1984, vol. 29, pp. 301–303.
  30. Prokhorov D. Bounded univalent functions. Handbook of complex analysis: Geometric function theory, vol. 1. Amsterdam, North Holland, 2002, pp. 207–228.
  31. Koch J., Schleissinger S. Value ranges of univalent self-mappings of the unit disc. J. Math. Anal. Appl., 2016, vol. 433, no. 2, pp. 1772–1789. DOI: https://doi.org/10.1016/j.jmaa.2015.08.068
  32. Koch J., Schleissinger S. Three value ranges for symmetric self-mappings of theunit disk. Proc. Amer. Math. Soc., 2017, vol. 145, no. 4, pp. 1747–1761. DOI: https://doi.org/10.1090/proc/13350
  33. Fedorov S. I. The moduli of certain families of curves and the range of {f(ζ 0 )} in the class of univalent functions with real coefficients. J. Soviet Math., 1987, vol. 36, pp. 282–291.
  34. Jenkins J.A. On univalent functions with real coefficients. Ann. Math., 1960, vol. 71, pp. 1–15.
  35. Prokhorov D., Samsonova K. A description method in the value region problem. Complex Anal. Oper. Theory, 2017, vol. 11, no. 7, pp. 1613–1622. DOI: https://doi.org/10.1007/s11785-016-0551-6
  36. Cowen C. C., Pommerenke Ch., Inequalities for the angular derivative of an analytic function in the unit disk. J. London Math. Soc., 1982, vol. 26, no. 2, pp. 271–289. DOI: https://doi.org/10.1112/jlms/s2-26.2.271
  37. Goryainov V. V. Fractional iterates of functions that are analytic in the unit disk with given fixed points. Math. USSR-Sb., 1993, vol. 74, no. 1, pp. 29–46. DOI: https://doi.org/10.1070/SM1993v074n01ABEH003332
  38. Gumenyuk P., Prokhorov D. Value regions of univalent self-maps with two boundary fixed points. Ann. Acad. Sci. Fenn. Math., 2018, vol. 43, no. 1, pp. 451–462. DOI: https://doi.org/10.5186/aasfm.2018.4321
  39. Frolova A., Levenshtein M., Shoikhet D., Vasil’ev A. Boundary distortion estimates for holomorphic maps. Complex Anal. Oper. Theory, 2014, vol. 8, no. 5, pp. 1129–1149. DOI: https://doi.org/10.1007/s11785-013-0345-z
  40. Goryainov V. V., Kudryavtseva O. S. One-parameter semigroups of analytic functions, fixed points and the Koenigs function. Sb. Math., 2011, vol. 202, no. 7–8, pp. 971–1000. DOI: https://doi.org/10.1070/SM2011v202n07ABEH004173
  41. Goryainov V. V. Evolution families of conformal mappings with fixed points and the Loewner–Kufarev equation. Sb. Math., 2015, vol. 206, no. 1–2, pp. 33–60. DOI: https://doi.org/10.1070/SM2015v206n01ABEH004445
  42. Goryainov V. V. Holomorphic mappings of the unit disc into itself with two fixed points. Sb. Math., 2017, vol. 208, no. 3, pp. 360–376. DOI: https://doi.org/10.1070/SM8802
  43. Babenko K. I. The theory of extremal problems for univalent functions of class S. Providence, RI, Amer. Math. Soc., 1975. 320 p.
  44. Markina I., Prokhorov D., Vasil’ev A. Sub-Riemannian geometry of the coefficients of univalent functions. J. Funct. Anal., 2007, vol. 245, no. 2, pp. 475–492. DOI: https://doi.org/10.1016/j.jfa.2006.09.013
  45. Takebe T., Teo L.-P., Zabrodin A. Löwner equation and dispersionless hierarchies. J. Phys. A: Math. Theor., 2006, vol. 39, no. 37, pp. 11479–11501. DOI: https://doi.org/10.1088/0305-4470/39/37/010
  46. Pavlov M. V., Prokhorov D. V., Vasil’ev A. Yu., Zakharov A. M. Löwner evolution andfinite dimensional reductions of integrable systems. Theor. Math. Phys., 2014, vol. 181, no. 1, pp. 1263–1278. DOI: https://doi.org/10.1007/s11232-014-0211-9
  47. Bombieri E. On the local maximum property of the Koebe function. Invent. Math., 1967, vol. 4, pp. 26–67.
  48. Prokhorov D., Roth O. On the local extremum property of the Koebe function. Math. Proc. Cambr. Phil. Soc., 2004, vol. 136, no. 2, pp. 301–312. DOI: https://doi.org/10.1017/S030500410300714X
  49. Greiner R., Roth O. On support points of univalent functions and a disproof of a conjecture of Bombieri. Proc. Amer. Math. Soc., 2001, vol. 129, no. 12, pp. 3657–3664. DOI: https://doi.org/10.1090/S0002-9939-01-05994-9
  50. Gordienko V., Prokhorov D. Analogy of Bombieri’s number for bounded univalent functions. Lobachevskii J. Math., 2017, vol. 38, no. 3, pp. 429–434. DOI: https://doi.org/10.1134/S1995080217030118
  51. Gordienko V., Prokhorov D. The Bombieri Problem for Bounded Univalent Functions. Math. Notes, 2019, vol. 105, no. 3–4, pp. 442–450. DOI: https://doi.org/10.1134/S0001434619030040
  52. Bshouty D., Hengartner W. A variation of the Koebe mapping in a dense subclass of S. Canad J. Math., 1987, vol. 39, no. 1, pp. 54–73. DOI: https://doi.org/10.4153/CJM-1987-004-8
  53. Prokhorov D., Vasil’ev A. Optimal control in Bombieri’s and Tammi’s conjectures. Georgian Math. J., 2005, vol. 12, no. 4, pp. 743–761. DOI: https://doi.org/10.1515/GMJ.2005.743
  54. Aharonov D., Bshouty D. A problem of Bombieri on univalent functions. Comput. Methods Funct. Theory, 2016, vol. 16, iss. 4, pp. 677–688. DOI: https://doi.org/10.1007/s40315-016-0165-z
  55. Leung Y.-J. On the Bombieri numbers for the class S. J. Anal., 2016, vol. 24, no. 2, pp. 229–250. DOI: https://doi.org/10.1007/s41478-016-0017-2
  56. Efraimidis I. On the failure of Bombieri’s conjecture for univalent functions. Comput. Methods Funct. Theory, 2018, vol. 18, iss. 3, pp. 427–438. DOI: https://doi.org/10.1007/s40315-017-0222-2
  57. Efraimidis I., Pastor C. Some more counterexamples for Bombieri’s conjecture on univalent functions. https://arxiv.org:1710.10426v1[math.CV] (28 Oct. 2017).
  58. Prokhorov D. Necessary criteria for the Bombieri conjecture. Anal. Math. Phys., 2018, vol. 18, no. 4, pp. 679–690. DOI: https://doi.org/10.1007/s13324-018-0248-2
Received: 
07.04.2018
Accepted: 
12.05.2019
Published: 
31.08.2019
  • 1256 reads