Geometric properties of A-harmonic functions

Authors

Shokhrukh Khursanov, National University of UzbekistanFollow

Abstract

This paper is devoted to geometric properties of A(z)-harmonic functions and the corresponding Laplace operator Δ _A(u). It is proved that the generalized A(z)-harmonic function is generated by the usual A(z)-harmonic function.

First Page

236

Last Page

245

References

1. Ahlfors L. Lectures on quasiconformal mappings, Toronto-New York-London, (1966).

2. Bojarsky B. Homeomorphic solution of Beltrami systems, Report ACUSSR, Vol. 102, №4, 661–664 (1955).

3. Bojarski B. Generalized solutions of a system of differential equations of the first order of the elliptic type with discontinuous coefficients. Math. comp., 1957, Vol. 43(85), 451–503 (1957).

4. Sadullaev A. Theory Plurepotentials. Applications. Palmarium academic publishing (2012).

5. Sadullaev A. Dirichlet problem to Monge-Ampere equation. DAN USSR, Vol. 267, No.3, 563–566 (1982).

6. Vekua I.N. Generalized analytical functions. M., "Science" (1988).

7. Volkovysky L.I. Quasiconformal mappings. Lviv (1954).

8. Gutlyanski V., Ryazanov V., Srebro U. and Yakubov E. The Beltrami equation: a geometric approach. Springer (2011).

9. Zhabborov N.M., Otaboev T.U. Cauchy's theorem for A-analytical functions. Uzbek Mathematical Journal, No. 1, 15–18 (2014).

10. Zhabborov N.M., Otaboev T.U. An analogue of the Cauchy integral formula for analytic functions. Uzbek Mathematical Journal, No. 4, 50–59 (2016).

11. Sadullaev A., Zhabborov N.M. On a class of A-analitic functions. Sibirian Federal University, Maths and Physics, Vol. 9(3), 374–383 (2016).

12. Zhabborov N.M. Morer’s theorem and functional series in the class of A-analytic functions. Siberian Federal University, Maths and Physics, v. 11(1), 50–59 (2018).

13. Zhabborov N.M., Otaboev T.U. and Khursanov Sh.Ya. Schwartz Inequality and Schwartz Formula for -analytical Functions. Contemporary Mathematics. Fundamental Directions, Vol. 64, No.4, 637–649 (2018).

14. Khursanov Sh.Ya. A(z)- harmonic function. VESTNIK NUUz, 2/2, 5–8 (2017).

15. Bukhheim A.L., Kazantsev S.G. Beltrami-type elliptic systems and tomography problems. Report. USSR Academy of Sciences, T. 315, No. 1, 15–19 (1990).

16. Zhabborov N.M., Imomnazarov H.Kh. Some initial boundary value problems in the mechanics of two-speed media. Monograph (2012).

17. Solomentsev E.D. Harmonic and subharmonic functions and their generalizations. Itogi Nauki. Ser. Mat. anal Theor Probably Reg., 83–100 (1964).

18. Landkof N.S. Foundations of Modern Potential Theory. Springer, (1972).

19. Axler P.S., Bourdon R.W. Harmonic Function Theory (2nd edition). Springer-Verlag, ISBN 0-387-95218-7 (2001).

20. Helms L.L. Introduction to potential theory. R.E. Krieger ISBN 0-88275-224-3 (1975).

Recommended Citation

Khursanov, Shokhrukh (2020) "Geometric properties of A-harmonic functions," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 3: Iss. 2, Article 12.
DOI: https://doi.org/10.56017/2181-1318.1099