An analog of the Men’shov – Trokhimchuk theorem for monogenic functions in a three-dimensional commutative algebra

  • M. V. Tkachuk Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv
  • S. A. Plaksa Institute of Mathematics of the Natianal Academy of Sciences of Ukraine

Abstract

UDC 517.54

The aim of this work is to weaken the conditions of monogeneity for functions that take values in a given three-dimensional commutative algebra over the field of complex numbers.
The monogeneity of the function is understood as a combination of its continuity and the existence of the Gateaux derivative.

References

E. Goursat, Cours d’analyse mathematique, vol. 2, Gauthier-Villars, Paris (1910).

H. Bohr, Uber streckentreue und konforme Abbildung, Math. Z., 1, 403 – 420 (1918), https://doi.org/10.1007/BF01465097 DOI: https://doi.org/10.1007/BF01465097

H. Rademacher, Uber streckentreue und winkeltreue Abbildung ¨ , Math. Z., 4, 131 – 138 (1919), https://doi.org/10.1007/BF01203392 DOI: https://doi.org/10.1007/BF01203392

D. Menchov, Sur les differentielles totales des fonctions univalentes, Math. Ann., 105, 75 – 85 (1931), https://doi.org/10.1007/BF01455809 DOI: https://doi.org/10.1007/BF01455809

D. Menchov, Sur les fonctions monogenes, Bull. Soc. Math. France, 59, 141 – 182 (1931). DOI: https://doi.org/10.24033/bsmf.1178

D. Menchov, Les conditions de monogeneite, Act. Sci. Ind., № 329 (1936).

V. S. Fedorov, O monogenny`kh funkcziyakh, Mat. sb., 42, № 4, 485 – 500 (1935).

G. P. Tolstov, O krivolinejnom i povtornom integrale, Tr. Mat. in-ta AN SSSR, 35, 3 – 101 (1950).

Yu. Yu. Trokhimchuk, Neprery`vny`e otobrazheniya i usloviya monogennosti, Fizmatiz, Moskva (1963).

Yu. Yu. Trokhimchuk, Differenczirovanie, vnutrennie otobrazheniya i kriterii analitichnosti, Praczi In-t matematiki NAN Ukrayini, 70 (2007).

G. Kh. Sindalovskij, O differencziruemosti i analitichnosti odnolistny`kh otobrazhenij, Dokl. AN SSSR, 249, № 6, 1325 – 1327 (1979).

G. Kh. Sindalovskij, Ob usloviyakh Koshi – Rimana v klasse funkczij s summiruemy`m modulem i nekotory`kh granichny`kh svojstvakh analiticheskikh funkczij, Mat. sb., 128(170), № 3(11), 364 – 382 (1985).

D. S. Telyakovskij, Obobshhenie odnoj teoremy` Men`shova o monogenny`kh funkcziyakh, Izv. AN SSSR, ser. mat., 53, № 4, 886 – 896 (1989).

D. S. Telyakovskij, O golomorfnosti funkczij, kotory`e zadayut otobrazheniya, sokhranyayushhie ugly`, Mat. zametki, 56, № 5, 149 – 154 (1994).

D. S. Telyakovskij, Ob oslablenii usloviya asimptoticheskoj monogennosti, Mat. zametki,60, № 6, 902 – 911 (1996). DOI: https://doi.org/10.4213/mzm1908

D. S. Telyakovskij, Obobshhenie teoremy` Men`shova o funkcziyakh, udovletvoryayushhikh usloviyu $K''$, Mat. zametki, 76, № 4, 578 – 591 (2004). DOI: https://doi.org/10.4213/mzm133

E. P. Dolzhenko, Raboty` D. E. Men`shova po teorii analiticheskikh funkczij i sovremennoe sostoyanie teorii monogennosti, Uspekhi mat. nauk, 47, № 5, 67 – 96 (1992).

M. T. Brodovich, Ob otobrazheniyakh prostranstvennoj oblasti, sokhranyayushhikh ugly` i rastyazheniya vdol` sistemy` luchej, Sib. mat. zhurn., 38, № 2, 260 – 262 (1997). DOI: https://doi.org/10.1007/BF02674619

A. V. Bondar`, Mnogomernoe obobshhenie odnoj teoremy` D. E. Men`shova, Ukr. mat. zhurn., 30, № 4, 435 – 443 (1978).

A. V. Bondar`, Lokal`ny`e geometricheskie kharakteristiki golomorfny`kh otobrazhenij, Nauk. dumka, Kiev (1992).

V. I. Siry`k, Nekotory`e kriterii golomorfnosti neprery`vny`kh otobrazhenij, Ukr. mat. zhurn., 37, № 6, 751 – 756 (1985).

O. S. Grecz`kij, Pro $C$ -diferenczijovnist` vidobrazhen` banakhovikh prostoriv, Ukr. mat. zhurn., 46, № 10, 1336 – 1342 (1994).

E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc., Providence, R. I. (1957).

S. A. Plaksa, R. P. Pukhtaievych, Monogenic functions in a finite-dimensional semi-simple commutative algebra, An. ¸Stiin¸t. Univ. “Ovidius” Constan¸ta, Ser. Mat., 22, № 1, 221 – 235 (2014). https://doi.org/10.2478/auom-2014-0018 DOI: https://doi.org/10.2478/auom-2014-0018

V. Shpakivskyi, Constructive description of monogenic functions in a finite-dimensional commutative associative algebra, Adv. Pure and Appl. Math., 7, № 1, 63 – 75 (2016), https://doi.org/10.1515/apam-2015-0022 DOI: https://doi.org/10.1515/apam-2015-0022

I. P. Mel`nichenko, S. A. Plaksa, Kommutativny`e algebry` i prostranstvenny`e polya, Praczi In-t matematiki NAN Ukrayini, 71 (2008).

S. A. Plaksa, V. S. Shpakovskii, Constructive description of monogenic functions in a harmonic algebra of the third rank, Ukr. Math. J., 62, № 8, 1251 – 1266 (2011), https://doi.org/10.1007/s11253-011-0427-x DOI: https://doi.org/10.1007/s11253-011-0427-x

P. W. Ketchum, Analytic functions of hypercomplex variables, Trans. Amer. Math. Soc., 30, 641 – 667 (1928), https://doi.org/10.2307/1989440 DOI: https://doi.org/10.1090/S0002-9947-1928-1501452-7

I. P. Mel’nichenko, The representation of harmonic mappings by monogenic functions, Ukr. Math. J., 27, № 5, 499 – 505 (1975). DOI: https://doi.org/10.1007/BF01089142

G. Scheffers, Verallgemeinerung der Grundlagen der gewohnlich complexen Funktionen, I, II ¨ , Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Phys. Kl., 45, 828 – 848 (1893); 46, 120 – 134 (1894).

E. R. Lorch, The theory of analytic function in normed abelian vector rings, Trans. Amer. Math. Soc., 54, 414 – 425 (1943), https://doi.org/10.2307/1990255 DOI: https://doi.org/10.1090/S0002-9947-1943-0009090-0

S. A. Plaksa, Commutative algebras associated with classic equations of mathematical physics, Adv. Appl. Anal., Trends Math., 177 – 223 (2012), https://doi.org/10.1007/978-3-0348-0417-2_5 DOI: https://doi.org/10.1007/978-3-0348-0417-2_5

S. A. Plaksa, Monogenic functions in commutative algebras associated with classical equations of mathematical physics, J. Math. Sci., 242, № 3, 432 – 456 (2019). DOI: https://doi.org/10.1007/s10958-019-04488-3

S. A. Plaksa, On differentiable and monogenic functions in a harmonic algebra, Zb. pracz` In-tu matematiki NAN Ukrayini, 14, № 1, 210 – 221 (2017).

M. V. Tkachuk, S. A. Plaksa, Analog teoremi Men`shova – Trokhimchuka dlya monogennikh funkczij v trivimirnij komutativnij algebri, e-print: arXiv:2006.12492v1 [math.CA], 2020.

Published
18.08.2021
How to Cite
TkachukM. V., and PlaksaS. A. “An Analog of the Men’shov – Trokhimchuk Theorem for Monogenic Functions in a Three-Dimensional Commutative Algebra”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 8, Aug. 2021, pp. 1120 -28, doi:10.37863/umzh.v73i8.6658.
Section
Research articles