Introduction. Quoting from a well-known American mathematician Lipman Bers  “It would be tempting to rewrite history and to claim that quasiconformal transformations have been discovered in connection with gas-dynamical problems. As a matter of fact, however, the concept of quasiconformality was arrived at by Grotzsch  and Ahlfors  from the point of view of function theory”. The present work is devoted to the theory of analytic solutions of the Beltrami equation which directly related to the quasi-conformal mappings. The function is, in general, assumed to be measurable with almost everywhere in the domain under consideration. Solutions of equation (1) are often referred to as analytic functions in the literature. Research methods. The solutions of equation (1), as well as quasi-conformal homeomorphisms in the complex plane have been studied in suffient details. The purpose of this paper is to study analytic functions in a particular case, when the function is anti-holomorphic in a considered domain . As we can see below, in this spesial case the solution of (1) possesses many properties of analytic functions, has an integral in the norm is a function of the Hardy class and this class is generalized. Results and discussions. The aim of this paper is to investigate analytic functions in special case when the function is an anti-analytic function in a domain. Also, in paper introduces some classes for analytic functions. Nevanlinn's theorem for analytic functions is proved and its results are given. Examples of functions belonging to these classes in different cases are given. The theorems of Riesz and Smirnov for analytic functions are proved. Conclusion. The theory of boundary properties made considerable advances in the first third of the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of harmonic functions, the theory of conformal mapping, boundary value problems of analytic function theory. The theory of boundary properties of analytic functions is closely connected with various fields of application of mathematics by way of boundary value problems. The theory of boundary properties of analytical functions, which grew out of the works of the Moscow Mathematical School (V.V. Golubev, N.N. Luzin, I.I. Privalov), was developed in the further works of I.I. Privalov, as well as in the works of A.Ya. Khinchin, A.I. Plesner, G.M. Fikhtengolts, V.I. Smirnov, M.V. Keldysh, M.A. Lavrentiev and other Russian scientists. We will extend this class by constructing a Hardy class for the class of analytical functions. In general, we extend the theory of classical functions. Not everything goes exactly without an analog. In such cases, calculations are carried out in other ways.
1. Bers L. Mathematical aspects of subsonic and transonic gas dynamics, Wiley, New York, 1958.
2. GroЁtzsch H. UЁber die Verzerrung bei schlichten nichtkonformen Abbildungen und uЁber eine damit zusammenhaЁngende Erweiterung des Picardschen Satzes, Ber. Verh. SaЁchs Akad. Wiss., 80(1928), 503–507.
3. Ahlfors L. Lectures on quasiconformal mappings, Princeton, N.J., Van Nostrand, 1966.
4. Belinskiy P.P. General properties of quasiconformal mappings, Moscow, Nauka, 1974, (in Russian).
5. Bojarskiy B. Homeomorphic solutions of Beltrami systems, Doklady Akademii Nauk USSR, 102(1955), no. 4, 661–664 (in Russian).
6. Bojarski B. Generalized solutions of a system of diﬀerential equations of the ﬁrst order of the elliptic type with discontinuous coeﬃcients, USSR Sbornik Mathematics, 43(85) (1957), 451-503.
7. Jabborov N.M., Imomnazarov Kh.Kh. Some initial-boundary value problems in the mechanics of two-speed media, Tashkent, The National University Uzbekistan, 2012 (in Russian).
8. Vekua I.N. Generalized analytical functions, Moscow, Nauka, 1988 (in Russian).
9. Gutlyanskiy V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation: a geometric approach, J. Math. Sci., 175(2011), 413–449.
10. Volkovisskiy L.I. Quasiconformal mappings, L’viv, 1954 (in Russian).
11. Bers L. An outline of the theory of pseudoanalytic functions, Bull AMS, 62(1956), №. 4, 291–331.
12. Lavrentiev M.A., Shabat B.V. Methods of complex variable functions theory, Moscow, Fizmatgiz, 1958 (in Russian).
13. S. L. Krushkal, R. Kyunau, Quasiconformal mappings — new methods and applications, Moscow, Nauka, 1984 (in Russian).
14. Bukhgeym A.L. Inversion formulas in inverse problems. Addition to the book of M.M. Lavrentiev and L.Ya. Savelyev "Linear operators and ill-posed problems" Moscow, Nauka, 1991, (in Russian).
15. Privalov I.I. Boundary properties of analytical functions, Moscow, State Publishing House of Technical and Theoretical Literature, 1950, (in Russian).
16. Duren P.L. Theory of Hp Spaces, Academic Press, New York and London, 1970.
17. Aizenberg L. Carleman’s Formulas in Complex Analysis, Novosibirsk “Nauka” department of Siberia, 1990.
18. Koosis P. Introduction to Hp Spaces, Cambridge University Press, 1998.
19. Sadullayev A., Jabborov N.M. On a class of A-analytic functions. J. Sib. Fed. Univ. Math. Phys., Volume 9. Issue 3. 2016, 374-383 p.
"GENERALIZATION OF THE HARDY CLASS FOR
Scientific reports of Bukhara State University: Vol. 5
, Article 3.
Available at: https://uzjournals.edu.uz/buxdu/vol5/iss4/3