Geodesic
A case of an explicit solutions for the three-element problem of Carleman type for analytic functions in a circle
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 159-165.

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The article investigates the three-element Carleman boundary value problem in the class of analytic functions, continuous extension to the contour in the Holder sense, when this problem can not be reduced to a two-element boundary value problems. The unit circle is considered as the contour. To be specific, we study a case of inverse shift. In this case, the solution of the problem is reduced to solving a system of two integral equations of Fredholm second kind; thus significantly used the theory of F. D. Gakhov about Riemann boundary value problem for analytic functions. Based on this result, an algorithm for the solution of the problem is built. Then it is proved that if the boundary condition coefficients are rational functions, and shift a linear-fractional function, then the boundary value problem is solved in an explicit form (in quadrature). Then we consider a simple case of an explicit solution of the problem, when in addition to the above restrictions on the coefficients and shift function is required also analytic continuation of some functions defined on the contour, inside the area. This case is illustrated by a concrete example. 
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N. R. Perelman. A case of an explicit solutions for the three-element problem of Carleman type for analytic functions in a circle. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 159-165. http://geodesic.mathdoc.fr/item/ISU_2016_16_2_a5/

[1] Perelman N. R., Rasulov K. M., “Three-element problem of Carleman type for bianalitic functions in a circle”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 12:2 (2012), 18–26 (in Russian) | Zbl

[2] Litvinchuk G. S., Solvability theory of boundary value problems and singular integral equations with shift, Kluwer Academic Publ., Dordrecht–Boston, 2000, 378 pp. | DOI | MR | MR | Zbl

[3] Perelman N. R., Rasulov K. M., “Three-element Carleman boundary value problem with a reverce shift for analytic functions on an exceptional case”, The Bryansk State Univ. Herald, 2012, no. 4(2), 46–53 (in Russian) | MR

[4] Rasulov K. M., Conjugation method of analytic functions and some of its applications, SmolGU Publ., Smolensk, 2013, 189 pp. (in Russian)