There constructed a new method of solving extremal problems on the class of univalent functions based on the parametric Loewner representation and optimization principles. This allowed us to solve some known and new problems. The Jakubowski conjecture was proved stating that the Pick functions are extremal in the problem of estimating even coefficients in the class of univalent functions $f$, $|f(z)|<M$ for sufficiently large $M$. We proved the conjecture that the maximum of the product $|a_2a_n|$ in the class of univalent functions close to identity is<br> attained for the Pick functions. We solved the Hengartner–Shober–Goodman–Saff–Brannan problem on the radius of a disk such that every function convex in a direction maps this disk onto a domain convex in another given direction.
Biography
Graduated from Department of Mathematics and Mechanics of Saratov State University in 1968. Ph.D. thesis was defended in 1974. D.Sci. thesis was defended in 1990. A list of my works contains more than 80 titles.
A corresponding member of the Russian Academy of Natural Sciences.
Main publications:
Michalska M., Prokhorov D. V., Szynal J. The composition of hyperbolic triangle mappings // Complex Variable. 2000, 43, 179–186.
Jakubwski Z., Prokhorov D. V., Szynal J. Proof of a coefficient product conjecture for bounded univalent functions // Complex Variable, 2000, 43, 241–258.
D. V. Prokhorov, A. M. Zakharov, A. V. Zherdev, “Solutions of the Loewner equation with combined driving functions”, Izv. Saratov Univ. Math. Mech. Inform., 21:3 (2021), 317–325
D. V. Prokhorov, “Value regions in classes of conformal mappings”, Izv. Saratov Univ. Math. Mech. Inform., 19:3 (2019), 258–279
3.
V. G. Gordienko, D. V. Prokhorov, “The Bombieri Problem for Bounded Univalent Functions”, Mat. Zametki, 105:3 (2019), 364–374; Math. Notes, 105:3 (2019), 342–350
D. V. Prokhorov, “Extension of starlike functions to a finitely punctured plane”, Probl. Anal. Issues Anal., 6(24):1 (2017), 58–67
2015
5.
D. V. Prokhorov, D. V. Ukrainskii, “Asymptotic ratio of harmonic measures of slit sides”, Izv. Saratov Univ. Math. Mech. Inform., 15:2 (2015), 160–167
2014
6.
M. V. Pavlov, D. V. Prokhorov, A. Yu. Vasiliev, A. M. Zaharov, “Löwner evolution and finite-dimensional reductions of integrable systems”, TMF, 181:1 (2014), 155–172; Theoret. and Math. Phys., 181:1 (2014), 1263–1278
D. V. Prokhorov, K. A. Samsonova, “Integrals of the Loewner equation with exponential driving function”, Izv. Saratov Univ. Math. Mech. Inform., 13:4(2) (2013), 98–108
D. V. Prokhorov, V. G. Gordienko, “Definition of a boundary in the local Charzyński-Tammi conjecture”, Izv. Vyssh. Uchebn. Zaved. Mat., 2008, no. 9, 59–68; Russian Math. (Iz. VUZ), 52:9 (2008), 51–59
D. V. Prokhorov, S. V. Romanova, “Local extremal problems for bounded analytic functions without zeros”, Izv. RAN. Ser. Mat., 70:4 (2006), 209–224; Izv. Math., 70:4 (2006), 841–856
D. V. Prokhorov, A. A. Nikulin, “Asymptotic estimates for the coefficients of bounded univalent functions”, Sibirsk. Mat. Zh., 47:5 (2006), 1128–1138; Siberian Math. J., 47:5 (2006), 924–933
1998
12.
Yu. E. Hohlov, D. V. Prokhorov, A. Yu. Vasil'ev, “On geometrical properties of free boundaries in the Hele–Shaw flows moving boundary problem”, Lobachevskii J. Math., 1 (1998), 3–12
D. V. Prokhorov, “The sum of coefficients of bounded univalent functions”, Mat. Zametki, 61:5 (1997), 728–733; Math. Notes, 61:5 (1997), 609–613
1993
14.
D. V. Prokhorov, A. M. Zaharov, “The set of values of a function and its derivative in the class of univalent mappings of a half-plane”, Izv. Vyssh. Uchebn. Zaved. Mat., 1993, no. 2, 33–37; Russian Math. (Iz. VUZ), 37:2 (1993), 31–35
D. V. Prokhorov, “Set of values of the initial coefficients of bounded univalent typically real-valued functions”, Sibirsk. Mat. Zh., 32:5 (1991), 132–141; Siberian Math. J., 32:5 (1991), 840–848
1990
16.
D. V. Prokhorov, “Sets of values of systems of functionals in classes of univalent functions”, Mat. Sb., 181:12 (1990), 1659–1677; Math. USSR-Sb., 71:2 (1992), 499–516
D. V. Prokhorov, “Ranges of values of systems of functionals and integration of univalent functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1986, no. 10, 33–39; Soviet Math. (Iz. VUZ), 30:10 (1986), 44–53
D. V. Prokhorov, “The maximum principle in the solution of an extremal problem on the class of univalent functions”, Sibirsk. Mat. Zh., 27:1 (1986), 186–190
1985
20.
D. V. Prokhorov, “Relation of measures of boundary sets under a mapping of a disk onto a plane with cuts”, Izv. Vyssh. Uchebn. Zaved. Mat., 1985, no. 4, 53–55; Soviet Math. (Iz. VUZ), 29:4 (1985), 66–69
D. V. Prokhorov, “Combined criteria for univalence of functions analytic in the disk”, Izv. Vyssh. Uchebn. Zaved. Mat., 1983, no. 8, 76–77; Soviet Math. (Iz. VUZ), 27:8 (1983), 94–97
1980
23.
D. V. Prokhorov, “Integral transformations in some classes of univalent functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1980, no. 12, 45–49; Soviet Math. (Iz. VUZ), 24:12 (1980), 53–58
D. V. Prokhorov, “Linearly invariant extensions of families of analytic functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1979, no. 9, 41–47; Soviet Math. (Iz. VUZ), 23:9 (1979), 40–47
1978
25.
D. V. Prokhorov, “Integrals from univalent functions”, Mat. Zametki, 24:5 (1978), 671–678; Math. Notes, 24:5 (1978), 858–862
1976
26.
D. V. Prokhorov, B. N. Rakhmanov, “Integral representation of a class of univalent functions”, Mat. Zametki, 19:1 (1976), 41–48; Math. Notes, 19:1 (1976), 24–28
D. V. Prokhorov, “A geometric property of functions starlike of order $\alpha$”, Mat. Zametki, 10:3 (1971), 287–293; Math. Notes, 10:3 (1971), 597–600
2018
32.
L. A. Aksent'ev, A. I. Aptekarev, A. M. Bikchentaev, V. V. Goryainov, V. N. Dubinin, A. M. Elizarov, I. R. Kayumov, A. Laptev, S. R. Nasyrov, D. V. Prokhorov, A. G. Sergeev, V. D. Stepanov, “Farit Gabidinovich Avkhadiev (on his 70th birthday)”, Uspekhi Mat. Nauk, 73:1(439) (2018), 187–190; Russian Math. Surveys, 73:1 (2018), 181–185
2017
33.
D. V. Prokhorov, S. I. Dudov, A. M. Zakharov, V. B. Poplavskii, V. V. Rozen, S. P. Sidorov, “In memory of Alexandr Yu. Vasiliev”, Izv. Saratov Univ. Math. Mech. Inform., 17:1 (2017), 117–121
2010
34.
D. V. Prokhorov, A. M. Zaharov, S. I. Dudov, “Avgust Petrovich Khromov (on his 75th birthday)”, Izv. Saratov Univ. Math. Mech. Inform., 10:3 (2010), 92–93
2007
35.
D. V. Prokhorov, A. M. Zaharov, S. I. Dudov, “Avgust Petrovich Khromov”, Izv. Saratov Univ. Math. Mech. Inform., 7:2 (2007), 82–84
36.
S. I. Dudov, A. M. Zaharov, D. V. Prokhorov, A. P. Khromov, “Petr Lavrentievich Ulianov”, Izv. Saratov Univ. Math. Mech. Inform., 7:1 (2007), 89–93
1995
37.
L. D. Kudryavtsev, S. M. Nikol'skii, D. V. Prokhorov, S. B. Stechkin, S. A. Telyakovskii, P. L. Ul'yanov, A. P. Khromov, V. A. Yurko, “Nikolai Petrovich Kuptsov (obituary)”, Uspekhi Mat. Nauk, 50:4(304) (1995), 71–72; Russian Math. Surveys, 50:4 (1995), 727–729
A. M. Bogomolov, M. K. Potapov, A. A. Privalov, D. V. Prokhorov, “Четвертая Саратовская зимняя школа по теории функций и приближений”, Izv. Vyssh. Uchebn. Zaved. Mat., 1988, no. 6, 84–88
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