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Abstract
In the paper we give a brief overview of the O. Bakhtin' scientific results
Keywords:
inner radius of domain, non-overlapping domains
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How to Cite
Denega, I., & Zabolotnyi, Y. (2023). A.K. Bakhtin. Scientific legacy. Proceedings of the International Geometry Center, 16(1), 1-16. https://doi.org/10.15673/tmgc.v16i1.2387
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References
1. A. Bakhtin. On the coefficients of the functions of the class S. Dokl. Akad. Nauk SSSR, 254(5):1033-1035, 1980.
2. A. Bakhtin. About some extremal problems in geometric function theory of a complex variable. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (9):7-11, 2006.
3. A. Bakhtin. A generalization of some results in the theory of schlicht functions to multidimensional complex spaces. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (3):7-11, 2011.
4. A. Bakhtin. Analytic functions of vector argument and partially conformal mappings in multidimensional complex spaces. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (2):13-18, 2012.
5. A. Bakhtin, G. Bakhtina, and R. Podvysotskiui. Inequalities for inner radii of non-overlapping domains. Dopov. Nac. akad. nauk Ukr., (9), 2009.
6. A. Bakhtin, G. Bakhtina, and Y. Zelinskiui. Topological-algebraic structures and geometric methods in complex analysis. Zb. prats of the Inst. of Math. of NASU, 2008.
7. A. Bakhtin and I. Denega. Extremal decomposition of the complex plane with free poles. J. Math. Sci. (N.Y.), 246(1):1-17, 2020. https://doi.org/10.1007/s10958-020-04718-z pathdoi:10.1007/s10958-020-04718-z.
8. A. Bakhtin and I. Denega. Extremal decomposition of the complex plane with free poles II. J. Math. Sci. (N.Y.), 246(5):602-616, 2020. https://doi.org/10.1007/s10958-020-04766-5 pathdoi:10.1007/s10958-020-04766-5.
9. A. Bakhtin and I. Denega. Generalized M. A. Lavrentiev's inequality. J. Math. Sci. (N.Y.), 262(2):138-153, 2022. https://doi.org/10.1007/s10958-022-05806-y pathdoi:10.1007/s10958-022-05806-y.
10. A. Bakhtin, I. Denega, L. Vyhivska, and I. Dvorak. Problems of extremal decomposition of a complex plane. Zb. Pr. Inst. Mat. NAN Ukr., 17(2):10-56, 2020.
11. A. Bakhtin and R. Podvysotskiui. Quadratic differentials and extremal problems on non-overlapping domains. Neliniuini Koliv., 14(1):3-6, 2011.
12. A. Bakhtin and A. Targonskiui. Extremal problems and quadratic differentials. Neliniuini Koliv., 8(3):298-303, 2005. https://doi.org/10.1007/s11072-006-0002-9 pathdoi:10.1007/s11072-006-0002-9.
13. A. Bakhtin and A. Targonskiui. Some extremal problems in the theory of non-overlapping domains with free poles on rays. Ukrain. Mat. Zh., 58(12):1715-1719, 2006. https://doi.org/10.1007/s11253-006-0179-1 pathdoi:10.1007/s11253-006-0179-1.
14. A. Bakhtin and A. Targonskiui. Generalized (n,d)-ray systems of points and inequalities for non-overlapping domains and open sets. Ukrainian Math. J., 63(7):999-1012, 2011. https://doi.org/10.1007/s11253-011-0560-6 pathdoi:10.1007/s11253-011-0560-6.
15. A. Bakhtin and L. Vygovskaya. Estimates for the inner radii of symmetric non-overlapping domains. Ukr. Mat. Visn., 15(3):298-320, 442, 2018.
16. A. Bakhtin, L. Vyhivska, and I. Denega. Inequality for the inner radii of symmetric non-overlapping domains. Bull. Soc. Sci. Lett. L odz Ser. Rech. Deform., 68(2):37-44, 2018. https://doi.org/10.26485/0459-6854 pathdoi:10.26485/0459-6854.
17. A. Bakhtin and V. V'yun. Application of a separating transformation to estimates of inner radii of open sets. Ukrainian Math. J., 59(10):1313-1321, 2007.
18. A. Bakhtin and V. V'yun. Quadratic differentials in problems of the product of inner radii of non-overlapping domains. Neliniuini Koliv., 10(4):435-442, 2007. https://doi.org/10.1007/s11072-008-0007-7 pathdoi:10.1007/s11072-008-0007-7.
19. A. Bakhtin, V. V'yun, and Y. Trokhimchuk. Inequalities for inner radii of non-overlapping domains. Dopov. Nac. akad. nauk Ukr., (8), 2007.
20. A. Bakhtin and Y. Zabolotnii. Estimation of the products of some powers of inner radii for multiconnected domains. Ukrainian Math. J., 73(9):1341-1358, 2022. https://doi.org/10.1007/s11253-022-01998-3 pathdoi:10.1007/s11253-022-01998-3.
21. A. Bakhtin, Y. Zabolotnii, and I. Dvorak. Estimates for the product of inner radii of five non-overlapping domains. Ukrainian Math. J., 69(2):304-311, 2017. https://doi.org/10.1007/s11253-017-1362-2 pathdoi:10.1007/s11253-017-1362-2.
22. A. Bakhtin and Y. Zabolotniui. Estimates for the products of the inner radii of multiply connected domains. Ukrain. Mat. Zh., 73(1):9-22, 2021. https://doi.org/10.37863/umzh.v73i1.6200 pathdoi:10.37863/umzh.v73i1.6200.
23. G. Bakhtina. The extremization of some functionals in a problem of non-overlapping domains. Ukrainian Math. J., 27(2):202-204, 285, 1975. https://doi.org/10.1007/BF01089997 pathdoi:10.1007/BF01089997.
24. L. Bieberbach. Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Preuss. Akad. Wiss., Phys.-Math. Kl., 138:940-955, 1916.
25. L. de Branges. A proof of the Bieberbach conjecture. Acta Math., 154(1-2):137-152, 1985. https://doi.org/10.1007/BF02392821 pathdoi:10.1007/BF02392821.
26. I. Denega and Y. Zabolotnii. Estimates of products of inner radii of non-overlapping domains in the complex plane. Complex Var. Elliptic Equ., 62(11):1611-1618, 2017. https://doi.org/10.1080/17476933.2016.1265952 pathdoi:10.1080/17476933.2016.1265952.
27. I. Denega and Y. Zabolotnii. Problem on extremal decomposition of the complex plane. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 27(1):61-77, 2019. https://doi.org/10.2478/auom-2019-0004 pathdoi:10.2478/auom-2019-0004.
28. V. Dubinin. Symmetrization in the geometric theory of functions of a complex variable. Uspekhi Mat. Nauk, 49(1(295)):3-76, 1994. https://doi.org/10.1070/RM1994v049n01ABEH002002 pathdoi:10.1070/RM1994v049n01ABEH002002.
29. V. Dubinin. Condenser capacities and symmetrization in geometric function theory. Birkhauser/Springer, Basel, 2014.
30. G. Goluzin. Geometric theory of functions of a complex variable. Amer. Math. Soc. Providence, R.I., 1969.
31. J. Jenkins. Univalent functions and conformal mapping. Moscow: Publishing House of Foreign Literature, 256, 1962.
32. L. Kolbina. Some extremal problems in conformal mapping. Doklady Akad. Nauk SSSR (N.S.), 84:865-868, 1952.
33. L. Kolbina. Conformal mapping of the unit circle onto mutually non-overlapping regions. Vestnik Leningrad. Univ., 10(5):37-43, 1955.
34. L. Kovalev. On the problem of extremal partitioning with free poles on the circle. Dalnevost. Mat. Sb., (2):96-98, 199, 1996.
35. G. Kuzmina. Methods of the geometric theory of functions. II. Algebra i Analiz, 9(5):1-50, 1997.
36. M. Lavrentiev. On the theory of conformal mappings. Tr. Mat. Inst. Steklova, 5:159-245, 1934. in Russian.
37. N. Lebedev. The area principle in the theory of univalent functions. Moscow, Science, 1975.
38. I. M. Milin. Univalent functions and orthonormal systems. Translations of Mathematical Monographs, Vol. 49. American Mathematical Society, Providence, R.I., 1977. Translated from the Russian.
39. Z. Nehari. Some inequalities in the theory of functions. Trans. Amer. Math. Soc., 75:256-286, 1953. https://doi.org/10.2307/1990732 pathdoi:10.2307/1990732.
40. P. Tamrazov. Extremal conformal mappings and poles of quadratic differentials. Izv. Akad. Nauk SSSR Ser. Mat., 32(5):1033-1043, 1968.
41. A. Targonskiui. Extremal problems on the generalized (n,d)-equiangular system of points. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 22(2):239-251, 2014. https://doi.org/10.2478/auom-2014-0044 pathdoi:10.2478/auom-2014-0044.
42. A. Targonskiui. Extremal problem on (2n,2m-1)-system points on the rays. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 24(2):283-299, 2016. https://doi.org/10.1515/auom-2016-0043 pathdoi:10.1515/auom-2016-0043.
43. A. Targonskiui. An extremal problem for the non-overlapping domains. Ukr. Mat. Visn., 14(1):126-134, 2017.
44. A. Targonskiui and I. Targonskaya. An extremal problem for partially non-overlapping domains on a Riemann sphere. Ukr. Mat. Visn., 15(1):94-102, 150-151, 2018.
45. A. Targonskiui and I. Targonskaya. Extreme problem for a mosaic system of points. J. Math. Sci. (N.Y.), 252(4):558-565, 2021. https://doi.org/10.1007/s10958-020-05180-7 pathdoi:10.1007/s10958-020-05180-7.
46. Y. Zelinskiui and I. Vygovskaya. A criterion for the convexity of a domain in Euclidean space. Ukrain. Mat. Zh., 60(5):709-711, 2008. https://doi.org/10.1007/s11253-008-0086-8 pathdoi:10.1007/s11253-008-0086-8.
47. Y. Zelinskiui, I. Vygovskaya, and H. Dakhil. Shadow problem and related questions. Proc. Int. Geom. Cent., 9(3-4):50-58, 2016. in Russian. https://doi.org/10.15673/tmgc.v9i3-4.319 pathdoi:10.15673/tmgc.v9i3-4.319.
48. Y. Zelinskiui, I. Vygovskaya, and H. Dakhil. The problem of shadow for balls with fixed radius. J. Math. Sci., 224:604-606, 2017. https://doi.org/10.1007/s10958-017-3438-9 pathdoi:10.1007/s10958-017-3438-9
2. A. Bakhtin. About some extremal problems in geometric function theory of a complex variable. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (9):7-11, 2006.
3. A. Bakhtin. A generalization of some results in the theory of schlicht functions to multidimensional complex spaces. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (3):7-11, 2011.
4. A. Bakhtin. Analytic functions of vector argument and partially conformal mappings in multidimensional complex spaces. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (2):13-18, 2012.
5. A. Bakhtin, G. Bakhtina, and R. Podvysotskiui. Inequalities for inner radii of non-overlapping domains. Dopov. Nac. akad. nauk Ukr., (9), 2009.
6. A. Bakhtin, G. Bakhtina, and Y. Zelinskiui. Topological-algebraic structures and geometric methods in complex analysis. Zb. prats of the Inst. of Math. of NASU, 2008.
7. A. Bakhtin and I. Denega. Extremal decomposition of the complex plane with free poles. J. Math. Sci. (N.Y.), 246(1):1-17, 2020. https://doi.org/10.1007/s10958-020-04718-z pathdoi:10.1007/s10958-020-04718-z.
8. A. Bakhtin and I. Denega. Extremal decomposition of the complex plane with free poles II. J. Math. Sci. (N.Y.), 246(5):602-616, 2020. https://doi.org/10.1007/s10958-020-04766-5 pathdoi:10.1007/s10958-020-04766-5.
9. A. Bakhtin and I. Denega. Generalized M. A. Lavrentiev's inequality. J. Math. Sci. (N.Y.), 262(2):138-153, 2022. https://doi.org/10.1007/s10958-022-05806-y pathdoi:10.1007/s10958-022-05806-y.
10. A. Bakhtin, I. Denega, L. Vyhivska, and I. Dvorak. Problems of extremal decomposition of a complex plane. Zb. Pr. Inst. Mat. NAN Ukr., 17(2):10-56, 2020.
11. A. Bakhtin and R. Podvysotskiui. Quadratic differentials and extremal problems on non-overlapping domains. Neliniuini Koliv., 14(1):3-6, 2011.
12. A. Bakhtin and A. Targonskiui. Extremal problems and quadratic differentials. Neliniuini Koliv., 8(3):298-303, 2005. https://doi.org/10.1007/s11072-006-0002-9 pathdoi:10.1007/s11072-006-0002-9.
13. A. Bakhtin and A. Targonskiui. Some extremal problems in the theory of non-overlapping domains with free poles on rays. Ukrain. Mat. Zh., 58(12):1715-1719, 2006. https://doi.org/10.1007/s11253-006-0179-1 pathdoi:10.1007/s11253-006-0179-1.
14. A. Bakhtin and A. Targonskiui. Generalized (n,d)-ray systems of points and inequalities for non-overlapping domains and open sets. Ukrainian Math. J., 63(7):999-1012, 2011. https://doi.org/10.1007/s11253-011-0560-6 pathdoi:10.1007/s11253-011-0560-6.
15. A. Bakhtin and L. Vygovskaya. Estimates for the inner radii of symmetric non-overlapping domains. Ukr. Mat. Visn., 15(3):298-320, 442, 2018.
16. A. Bakhtin, L. Vyhivska, and I. Denega. Inequality for the inner radii of symmetric non-overlapping domains. Bull. Soc. Sci. Lett. L odz Ser. Rech. Deform., 68(2):37-44, 2018. https://doi.org/10.26485/0459-6854 pathdoi:10.26485/0459-6854.
17. A. Bakhtin and V. V'yun. Application of a separating transformation to estimates of inner radii of open sets. Ukrainian Math. J., 59(10):1313-1321, 2007.
18. A. Bakhtin and V. V'yun. Quadratic differentials in problems of the product of inner radii of non-overlapping domains. Neliniuini Koliv., 10(4):435-442, 2007. https://doi.org/10.1007/s11072-008-0007-7 pathdoi:10.1007/s11072-008-0007-7.
19. A. Bakhtin, V. V'yun, and Y. Trokhimchuk. Inequalities for inner radii of non-overlapping domains. Dopov. Nac. akad. nauk Ukr., (8), 2007.
20. A. Bakhtin and Y. Zabolotnii. Estimation of the products of some powers of inner radii for multiconnected domains. Ukrainian Math. J., 73(9):1341-1358, 2022. https://doi.org/10.1007/s11253-022-01998-3 pathdoi:10.1007/s11253-022-01998-3.
21. A. Bakhtin, Y. Zabolotnii, and I. Dvorak. Estimates for the product of inner radii of five non-overlapping domains. Ukrainian Math. J., 69(2):304-311, 2017. https://doi.org/10.1007/s11253-017-1362-2 pathdoi:10.1007/s11253-017-1362-2.
22. A. Bakhtin and Y. Zabolotniui. Estimates for the products of the inner radii of multiply connected domains. Ukrain. Mat. Zh., 73(1):9-22, 2021. https://doi.org/10.37863/umzh.v73i1.6200 pathdoi:10.37863/umzh.v73i1.6200.
23. G. Bakhtina. The extremization of some functionals in a problem of non-overlapping domains. Ukrainian Math. J., 27(2):202-204, 285, 1975. https://doi.org/10.1007/BF01089997 pathdoi:10.1007/BF01089997.
24. L. Bieberbach. Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Preuss. Akad. Wiss., Phys.-Math. Kl., 138:940-955, 1916.
25. L. de Branges. A proof of the Bieberbach conjecture. Acta Math., 154(1-2):137-152, 1985. https://doi.org/10.1007/BF02392821 pathdoi:10.1007/BF02392821.
26. I. Denega and Y. Zabolotnii. Estimates of products of inner radii of non-overlapping domains in the complex plane. Complex Var. Elliptic Equ., 62(11):1611-1618, 2017. https://doi.org/10.1080/17476933.2016.1265952 pathdoi:10.1080/17476933.2016.1265952.
27. I. Denega and Y. Zabolotnii. Problem on extremal decomposition of the complex plane. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 27(1):61-77, 2019. https://doi.org/10.2478/auom-2019-0004 pathdoi:10.2478/auom-2019-0004.
28. V. Dubinin. Symmetrization in the geometric theory of functions of a complex variable. Uspekhi Mat. Nauk, 49(1(295)):3-76, 1994. https://doi.org/10.1070/RM1994v049n01ABEH002002 pathdoi:10.1070/RM1994v049n01ABEH002002.
29. V. Dubinin. Condenser capacities and symmetrization in geometric function theory. Birkhauser/Springer, Basel, 2014.
30. G. Goluzin. Geometric theory of functions of a complex variable. Amer. Math. Soc. Providence, R.I., 1969.
31. J. Jenkins. Univalent functions and conformal mapping. Moscow: Publishing House of Foreign Literature, 256, 1962.
32. L. Kolbina. Some extremal problems in conformal mapping. Doklady Akad. Nauk SSSR (N.S.), 84:865-868, 1952.
33. L. Kolbina. Conformal mapping of the unit circle onto mutually non-overlapping regions. Vestnik Leningrad. Univ., 10(5):37-43, 1955.
34. L. Kovalev. On the problem of extremal partitioning with free poles on the circle. Dalnevost. Mat. Sb., (2):96-98, 199, 1996.
35. G. Kuzmina. Methods of the geometric theory of functions. II. Algebra i Analiz, 9(5):1-50, 1997.
36. M. Lavrentiev. On the theory of conformal mappings. Tr. Mat. Inst. Steklova, 5:159-245, 1934. in Russian.
37. N. Lebedev. The area principle in the theory of univalent functions. Moscow, Science, 1975.
38. I. M. Milin. Univalent functions and orthonormal systems. Translations of Mathematical Monographs, Vol. 49. American Mathematical Society, Providence, R.I., 1977. Translated from the Russian.
39. Z. Nehari. Some inequalities in the theory of functions. Trans. Amer. Math. Soc., 75:256-286, 1953. https://doi.org/10.2307/1990732 pathdoi:10.2307/1990732.
40. P. Tamrazov. Extremal conformal mappings and poles of quadratic differentials. Izv. Akad. Nauk SSSR Ser. Mat., 32(5):1033-1043, 1968.
41. A. Targonskiui. Extremal problems on the generalized (n,d)-equiangular system of points. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 22(2):239-251, 2014. https://doi.org/10.2478/auom-2014-0044 pathdoi:10.2478/auom-2014-0044.
42. A. Targonskiui. Extremal problem on (2n,2m-1)-system points on the rays. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 24(2):283-299, 2016. https://doi.org/10.1515/auom-2016-0043 pathdoi:10.1515/auom-2016-0043.
43. A. Targonskiui. An extremal problem for the non-overlapping domains. Ukr. Mat. Visn., 14(1):126-134, 2017.
44. A. Targonskiui and I. Targonskaya. An extremal problem for partially non-overlapping domains on a Riemann sphere. Ukr. Mat. Visn., 15(1):94-102, 150-151, 2018.
45. A. Targonskiui and I. Targonskaya. Extreme problem for a mosaic system of points. J. Math. Sci. (N.Y.), 252(4):558-565, 2021. https://doi.org/10.1007/s10958-020-05180-7 pathdoi:10.1007/s10958-020-05180-7.
46. Y. Zelinskiui and I. Vygovskaya. A criterion for the convexity of a domain in Euclidean space. Ukrain. Mat. Zh., 60(5):709-711, 2008. https://doi.org/10.1007/s11253-008-0086-8 pathdoi:10.1007/s11253-008-0086-8.
47. Y. Zelinskiui, I. Vygovskaya, and H. Dakhil. Shadow problem and related questions. Proc. Int. Geom. Cent., 9(3-4):50-58, 2016. in Russian. https://doi.org/10.15673/tmgc.v9i3-4.319 pathdoi:10.15673/tmgc.v9i3-4.319.
48. Y. Zelinskiui, I. Vygovskaya, and H. Dakhil. The problem of shadow for balls with fixed radius. J. Math. Sci., 224:604-606, 2017. https://doi.org/10.1007/s10958-017-3438-9 pathdoi:10.1007/s10958-017-3438-9