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JOURNALS // Izvestiya of Saratov University. Mathematics. Mechanics. Informatics

Izv. Saratov Univ. Math. Mech. Inform., 2019, Volume 19, Issue 3, Pages 258–279 (Mi isu806)

Value regions in classes of conformal mappings
D. V. Prokhorov

References

1. Nauka, M., 1966, 628 pp.  mathscinet  zmath
2. Aleksandrov I. A., Parametric continuations in the theory of univalent functions, Nauka, M., 1976, 344 pp. (in Russian)  mathscinet  zmath
3. Duren P. L., Univalent functions, Springer Verlag, New York, 1983, 382 pp.  mathscinet  zmath
4. Pommerenke Ch., Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975, 376 pp.  mathscinet  zmath
5. Rogosinski W., “Zum Schwarzen Lemma”, Jahresber. Deutsche Math-Verein., 44 (1934), 258–261
6. Grunsky H., “Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche”, Schr. Math. Inst. u Inst. Angew. Math. Univ. Berlin, 1 (1932), 95–140
7. Popov V. I., “Value domain of a system of functionals on the class $S$”, Proceedings of Tomsk University. Issues of the geometric function theory, 182:3 (1965), 106–132 (in Russian)  mathscinet  zmath
8. Gutlyanskii V. Ya., “Parametric representation of univalent functions”, Soviet Math. Dokl., 11:5 (1970), 1273–1276  mathscinet
9. Schaeffer A. C., Spencer D. C., Coefficient regions for schlicht functions, Coll. Publ., 35, Amer. Math. Soc., New York, 1950, 325 pp.  mathscinet  zmath
10. Loewner K., “Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I”, Math. Ann., 89:1–2 (1923), 103–121  mathscinet  zmath
11. Kufarev P. P., “On one-parameter families of analytic functions”, Mat. Sbornik, N. S., 13:1(55) (1943), 87–118 (in Russian)  mathnet  mathscinet
12. Pommerenke Ch., “Über die Subordination analytischer Funktionen”, J. Reine Angew. Math., 218 (1965), 159–173  mathscinet  zmath
13. Kufarev P. P., “A remark on integrals of the Loewner equation”, Doklady Akad. Nauk SSSR, 57:7 (1947), 655–656 (in Russian)  mathscinet  zmath
14. Goryainov V. V., Gutlyanskii V. Ya., “On extremal problems in the class $S_M$”, Matematicheskii sbornik, Naukova dumka, Kiev, 1976, 242–246 (in Russian)  mathnet  mathscinet
15. Roth O., Schleissinger S., “Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation”, Bull. London Math. Soc., 46:5 (2014), 1099–1109  crossref  mathscinet  zmath
16. Prokhorov D., Samsonova K., “Value range of solutions to the chordal Loewner equation”, J. Math. Anal Appl., 428:2 (2015), 910–919  crossref  mathscinet  zmath  elib  scopus
17. Zherdev A., “Value range of solutions to the chordal Loewner equation with restriction on the driving function”, Probl. Anal. Issues Anal., 8(26):2 (2019), 92–104  mathnet  crossref  mathscinet  scopus
18. Koch J. D., Value regions for schlicht functions, Dissertationsschrift zur Erlangung des naturwissenschaftlichen Doktorgrades der Julius-Maximilians-Universität Würzburg, Würzburg, 2016, 93 pp.
19. Goodman G. S., Univalent functions and optimal control, Ph.D. Thesis, Stanford University, Ann Arbor, MI; ProQuest LLC, 1967  mathscinet
20. Friedland S., Schiffer M., “Global results in control theory with applications to univalent functions”, Bull. Amer. Math. Soc., 82:6 (1976), 913–915  mathscinet  zmath
21. Friedland S., Schiffer M., “On coefficient regions of univalent functions”, J. Anal. Math., 31 (1977), 125–168  mathscinet  zmath
22. Roth O., Control Theory in $\mathcal H(\mathbb D)$, Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilians-Universität Würzburg, Würzburg, 1998, 178 pp.
23. Math. USSR-Sb., 71:2 (1992), 499–516  mathnet  mathscinet  zmath  zmath
24. Prokhorov D., Reachable set methods in extremal problems for univalent functions, Saratov Univ. Press, Saratov, 1993, 228 pp.  mathscinet  zmath
25. Schiffer M., “Sur l'équation différentielle de M. Löwner”, C. R. Acad. Sci. Paris, 221 (1945), 369–371  mathscinet
26. Prokhorov D., Vasil'ev A., “Univalent functions and integrable systems”, Commun. Math. Phys., 262:2 (2006), 393–410  crossref  mathscinet  zmath  adsnasa  elib  scopus
27. Roth O., Is there a Teichmüller principle in higher dimension?, Geometric function theory in higher dimension, Springer INdAM Series, 26, ed. F. Bracci, Cham, 2017, 87–105  crossref  mathscinet  zmath  scopus
28. Popov V. I., “L. S. Pontryagin's maximum principle in the theory of univalent functions”, Soviet Math. Dokl., 10 (1969), 1161–1164  mathscinet  zmath
29. Prokhorov D., “The method of optimal control in an extremal problem on a class of univalent functions”, Soviet Math. Dokl., 29 (1984), 301–303  mathscinet  zmath
30. Prokhorov D., “Bounded univalent functions”, Handbook of complex analysis: Geometric function theory, v. 1, North Holland, Amsterdam, 2002, 207–228  mathscinet  zmath
31. Koch J., Schleissinger S., “Value ranges of univalent self-mappings of the unit disc”, J. Math. Anal. Appl., 433:2 (2016), 1772–1789  crossref  mathscinet  zmath  elib  scopus
32. Koch J., Schleissinger S., “Three value ranges for symmetric self-mappings of the unit disk”, Proc. Amer. Math. Soc., 145:4 (2017), 1747–1761  crossref  mathscinet  zmath  elib  scopus
33. Fedorov S. I., “The moduli of certain families of curves and the range of $\{f(\zeta_0)\}$ in the class of univalent functions with real coefficients”, J. Soviet Math., 36 (1987), 282–291  mathnet  zmath
34. Jenkins J. A., “On univalent functions with real coefficients”, Ann. Math., 71 (1960), 1–15  mathscinet  zmath
35. Prokhorov D., Samsonova K., “A description method in the value region problem”, Complex Anal. Oper. Theory, 11:7 (2017), 1613–1622  crossref  mathscinet  zmath  elib  scopus
36. Cowen C. C., Pommerenke Ch., “Inequalities for the angular derivative of an analytic function in the unit disk”, J. London Math. Soc., 26:2 (1982), 271–289  crossref  mathscinet  zmath  scopus
37. Goryainov V. V., “Fractional iterates of functions that are analytic in the unit disk with given fixed points”, Math. USSR-Sb., 74:1 (1993), 29–46  mathnet  crossref  mathscinet  scopus
38. Gumenyuk P., Prokhorov D., “Value regions of univalent self-maps with two boundary fixed points”, Ann. Acad. Sci. Fenn. Math., 43:1 (2018), 451–462  crossref  mathscinet  zmath  scopus
39. Frolova A., Levenshtein M., Shoikhet D., Vasil'ev A., “Boundary distortion estimates for holomorphic maps”, Complex Anal. Oper. Theory, 8:5 (2014), 1129–1149  crossref  mathscinet  zmath  scopus
40. Goryainov V. V., Kudryavtseva O. S., “One-parameter semigroups of analytic functions, fixed points and the Koenigs function”, Sb. Math., 202:7–8 (2011), 971–1000  mathnet  crossref  mathscinet  zmath  elib  scopus
41. Goryainov V. V., “Evolution families of conformal mappings with fixed points and the Loewner – Kufarev equation”, Sb. Math., 206:1–2 (2015), 33–60  mathnet  crossref  mathscinet  zmath  elib  scopus
42. Goryainov V. V., “Holomorphic mappings of the unit disc into itself with two fixed points”, Sb. Math., 208:3 (2017), 360–376  mathnet  crossref  mathscinet  zmath  elib  scopus
43. Babenko K. I., The theory of extremal problems for univalent functions of class $S$, Amer. Math. Soc., Providence, RI, 1975, 320 pp.  mathscinet
44. Markina I., Prokhorov D., Vasil'ev A., “Sub-Riemannian geometry of the coefficients of univalent functions”, J. Funct. Anal., 245:2 (2007), 475–492  crossref  mathscinet  zmath  elib  scopus
45. Takebe T., Teo L.-P., Zabrodin A., “Löwner equation and dispersionless hierarchies”, J. Phys. A: Math. Theor., 39:37 (2006), 11479–11501  crossref  mathscinet  zmath  adsnasa  scopus
46. Pavlov M. V., Prokhorov D. V., Vasil'ev A. Yu., Zakharov A. M., “Löwner evolution and finite dimensional reductions of integrable systems”, Theor. Math. Phys., 181:1 (2014), 1263–1278  mathnet  crossref  mathscinet  zmath  elib  scopus
47. Bombieri E., “On the local maximum property of the Koebe function”, Invent. Math., 4 (1967), 26–67  mathscinet  zmath  adsnasa
48. Prokhorov D., Roth O., “On the local extremum property of the Koebe function”, Math. Proc. Cambr. Phil. Soc., 136:2 (2004), 301–312  crossref  mathscinet  zmath  scopus
49. Greiner R., Roth O., “On support points of univalent functions and a disproof of a conjecture of Bombieri”, Proc. Amer. Math. Soc., 129:12 (2001), 3657–3664  crossref  mathscinet  zmath
50. Gordienko V., Prokhorov D., “Analogy of Bombieri's number for bounded univalent functions”, Lobachevskii J. Math., 38:3 (2017), 429–434  crossref  mathscinet  zmath  elib  scopus
51. Gordienko V., Prokhorov D., “The Bombieri Problem for Bounded Univalent Functions”, Math. Notes, 105:3–4 (2019), 442–450  crossref  mathscinet  scopus
52. Bshouty D., Hengartner W., “A variation of the Koebe mapping in a dense subclass of $S$”, Canad J. Math., 39:1 (1987), 54–73  crossref  mathscinet  zmath
53. Prokhorov D., Vasil'ev A., “Optimal control in Bombieri's and Tammi's conjectures”, Georgian Math. J., 12:4 (2005), 743–761  crossref  mathscinet  zmath  elib  scopus
54. Aharonov D., Bshouty D., “A problem of Bombieri on univalent functions”, Comput. Methods Funct. Theory, 16:4 (2016), 677–688  crossref  mathscinet  zmath  elib  scopus
55. Leung Y.-J., “On the Bombieri numbers for the class $S$”, J. Anal., 24:2 (2016), 229–250  crossref  mathscinet  zmath
56. Efraimidis I., “On the failure of Bombieri's conjecture for univalent functions”, Comput. Methods Funct. Theory, 18:3 (2018), 427–438  crossref  mathscinet  zmath  scopus
57. Efraimidis I., Pastor C., Some more counterexamples for Bombieri's conjecture on univalent functions, 28 Oct. 2017, arXiv: 1710.10426v1 [math.CV]  mathscinet  zmath
58. Prokhorov D., “Necessary criteria for the Bombieri conjecture”, Anal. Math. Phys., 18:4 (2018), 679–690  crossref  mathscinet  scopus


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