Address: Markusa st. 22, Vladikavkaz,
362027, RNO-A, Russia
Phone: (8672)50-18-06
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2017.1.5824
The Multidimensional Analog of the Biberbach Hypothesis for Generalized Star Functions in the Space \(\mathbb{C}^{n}\), \(n\geq 2\)
Sultygov M. G.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 1.
Abstract: The article is an addition to the fundamental results of the geometric theory of multidimensional complex analysis problems for classes of holomorphic functions. The radii parameterization of the Reinhart region boundaries enables one to built effective sufficient conditions for the generalized star functions as a multivariate analogue of the Biberbach hypothesis.
Keywords: univalent functions, multidimensional analogue of the Biberbach hypothesis, efficiency coefficients, Taylor series, radius parameterization.
For citation:
Sultygov M. G. The Multidimensional Analog Hypothesis Biberbach for Generalized Star Functions in the Space \(\mathbb{C}^{n}\), \(n\geq 2\). Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 67-71. DOI 10.23671/VNC.2017.1.5824
1. Bieberbach L. Uber die Koeffizienten derjenigen Potenzreihen,
welche eine schlichte Abbildug des Einheitskreises vermitteln. S.-B.
Preuss. Akad. Wiss, Phys.-Math. Kl., 1916. pp. 940-955.
2. Goluzin G. M. Geometricheskaja teorija funkcij kompleksnogo
peremennogo [Geometric Theory of Functions of a Complex Variable].
Nauka, M., 1966, 628 p. [in Russian].
3. De Branges L. A Proof of the Bieberbach Conjecture, Acta Math.,
1985, vol. 154, pp. 137-152.
4. Bavrin I. I. Operatornyj metod v kompleksnom analize [Operator
Method in Complex Analysis], Prometey, M., 1976, 200 p. [in
Russian].
5. Sultygov M. D. About single \(M_D\) subclass of the class of functions
of two complex variables, Moscow regional pedagogical Institute
named after N. Krupskaya, M., 1982, pp. 14. Dep in VINITI no.
828-82 [in Russian].
6. Bavrin I. I. Estimates of Taylor coefficients of functions of two
complex variables, Doklady AN SSSR [Reports of the USSR Academy of Sciences],
1960, vol. 131, no. 6, pp. 1231-1233 [in Russian].
7. Temlyakov A. A. Integral representations of functions of two
complex variables, Doklady AN SSSR [Reports of the USSR Academy of
Sciences], 1958, vol. 120, no. 5, pp. 976-979 [in Russian].
8. Bavrin I. I. Functions, Univalent Weight, Doklady AN SSSR
[Reports of the USSR Academy of Sciences], 1996, vol. 349, no. 6,
pp. 727-728 [in Russian].
9. Bavrin I. I. Classes of functions, univalent weight, Doklady AN
SSSR [Reports of the USSR Academy of Sciences], 2000, vol. 371, no.
6, pp. 727-729 [in Russian].
10. Sultygov M. D. Star-convex functions of several complex
variables in space of Reinhart, Sb. nauch. tr. IngGU [Col. of Sci.
Works of the Ingush State Univ.], 2004, no. 2, pp. 333-362 [in
Russian].
11. Bavrin I. I. Klassy golomorfnyh funkcij mnogih kompleksnyh
peremennyh i jekstremal'nye voprosy dlja jetih klassov funkcij
[Classes of Holomorphic Functions of Several Complex Variables and
Extremal Questions for these Classes of Functions]. Publ. House.
MOPI., M., 1976, 99 p. [in Russian].
12. Sultygov M. D. Integral representations of some classes of
holomorphic functions in the space of several complex variables.
Izv. Chechenskogo gos. ped. in-ta [The Herald of the Chechen state
pedagogical Institute], 2015, no. 2(10), pp. 19-23 [in Russian].
13. Sultygov M. D. Taylor Coefficients of some classes of
holomorphic functions of several complex variables. Sb. nauch. tr.
IngGU [Col. of Sci. Works of the Ingush State Univ.], 2008, no. 6,
pp. 165-173 [in Russian].
14. Goel R. M., Mehrok B. S. On the coefficients of a subclass of
starlike functions. Indian J. Pure Appl. Math., 1981, vol. 12(5),
pp. 634-647.
