Abstract
In this paper, we study A-analytic functions. We provide main fundamental theorems of the theory of A-analytic functions and prove analogs of the Schwartz inequality, the Schwartz formula, and the Poisson formula for A-analytic functions.
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References
L. Ahlfors, Lectures on Quasiconformal Mappings, Springer, Toronto–New York–London (1966).
B. V. Boyarskiy, “Homeomorphic solutions of Beltrami systems”, Dokl. AN SSSR, 102, No. 4, 661–664 (1955).
B. V. Boyarskiy, “Generalized solutions of a first-order system of differential equations of elliptic type with discontinuous coefficients”, Mat. Sb., 43, No. 85, 451–503 (1957).
A. L. Bukhgeym, Inversion Formulas in Inverse Problems. Addition to the Book: M. M. Lavrentyev, L. Ya. Savelyev, Linear Operators and Ill-Posed Problems [in Russian], Nauka, Moscow (1991).
A. L. Bukhgeym and S. G. Kazantsev, “Elliptic systems of Beltrami type and problems of tomograpy”, Dokl. AN SSSR, 315, No. 1, 15–19 (1990).
V. Gutlyanski, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation. A Geometric Approach, Springer, Berlin (2011).
A. N. Kondrashov, “Beltrami equations degenerating on an arc”, Vestn. Volgograd. Gos. Un-ta. Ser. 1. Mat. Fiz., 24, No. 5, 24–39 (2014).
D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, and R. R. Salimov, “Boundary behavior and the Dirichlet problem for the Beltrami equations”, Algebra i Analiz, 25, No. 4, 101–124 (2013).
M. A. Lavrent’ev and B. V. Shabat, Method of the Theory of Functions of Complex Variable [in Russian], Fizmatgiz, Moscow (1958).
A. Sadullaev and N. M. Jabborov, “On a class of A-analytic functions,” J. Sib. Fed. Univ. Maths. Phys., 9, No. 3, 374–383 (2016).
U. Srebro and E. Yakubov, “μ-Homeomorphisms,” Contemp. Math., 211, 473–479 (1997).
I. N. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow (1988).
L. I. Volkovyskiy, “Some questions of the theory of quasiconformal mappings”, In: Nekotorye Problemy Matematiki i Mekhaniki. K Semidesyatiletiyu M. A. Lavrent’eva, Nauka, Leningrad, pp. 128–134 (1970).
E. Kh. Yakubov, “On solutions of the Beltrami equations with degeneration”, Dokl. AN SSSR, 243, No. 5, 1148-1149 (1978).
N. M. Zhabborov, “Morer’s theorem and functional series in the class of A-analytic functions,” J. Sib. Fed. Univ. Maths. Phys., 9, No. 3, 374–383 (2018).
N. M. Zhabborov and Kh. Kh. Imomnazarov, Some Initial Boundary-Value Problems of Mechanics of Two-Velocity Media [in Russian], Izd-vo NUUz, Tashkent (2012).
N. M. Zhabborov and T. U. Otaboev, “The Cauchy theorem for A(z)-analytic functions”, Uzb. Mat. Zh. [Uzb. Math. J.], No. 1, 15–18 (2014).
N. M. Zhabborov and T. U. Otaboev, “An analog of the Cauchy integral formula for A-analytic functions”, Uzb. Mat. Zh., No. 4, 50–59 (2016).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 64, No. 4, Contemporary Problems of Mathematics and Physics, 2018.
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Zhabborov, N.M., Otaboev, T.U. & Khursanov, S.Y. The Schwartz Inequality and the Schwartz Formula for A-Analytic Functions. J Math Sci 264, 703–714 (2022). https://doi.org/10.1007/s10958-022-06031-3
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DOI: https://doi.org/10.1007/s10958-022-06031-3