The Fekete-Szego problem by a variational method
Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 2, pp. 133-150.

Voir la notice de l'article provenant de la source Math-Net.Ru

The article is devoted to the well-known Fekete and Szego problem. The paper investigate the problem in sufficient detail using some new observations by the classical method of internal variations, developed at the Tomsk School of Complex Analysis. One particular case is considered. We carried out complete qualitative analysis of the functional-differential equation relative boundary mapping. We completely solved the problem for the real parameter.
@article{DVMG_2021_21_2_a1,
     author = {Ya. V. Borisova and I. A. Kolesnikov and S. A. Kopanev and G. D. Sadritdinova},
     title = {The {Fekete-Szego} problem by a variational method},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {133--150},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2021_21_2_a1/}
}
TY  - JOUR
AU  - Ya. V. Borisova
AU  - I. A. Kolesnikov
AU  - S. A. Kopanev
AU  - G. D. Sadritdinova
TI  - The Fekete-Szego problem by a variational method
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2021
SP  - 133
EP  - 150
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DVMG_2021_21_2_a1/
LA  - ru
ID  - DVMG_2021_21_2_a1
ER  - 
%0 Journal Article
%A Ya. V. Borisova
%A I. A. Kolesnikov
%A S. A. Kopanev
%A G. D. Sadritdinova
%T The Fekete-Szego problem by a variational method
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2021
%P 133-150
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DVMG_2021_21_2_a1/
%G ru
%F DVMG_2021_21_2_a1
Ya. V. Borisova; I. A. Kolesnikov; S. A. Kopanev; G. D. Sadritdinova. The Fekete-Szego problem by a variational method. Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 2, pp. 133-150. http://geodesic.mathdoc.fr/item/DVMG_2021_21_2_a1/

[1] M. Fekete, G. Szegö, “Eine Bemerkung über ungerade schlichte Funktionen”, J.London Math. Soc., 8:2 (1933), 85–89 | DOI | MR

[2] P. L. Duren, Univalent functions, Springer-Verlag, 1983 | Zbl

[3] G. M. Goluzin, “Nekotorye voprosy teorii odnolistnykh funktsii”, Tr. MIAN SSSR, 27 (1949), 3–110 | MR

[4] J. A. Jenkins, “On certain coefficients of univalent functions”, Analytic Functions. Princeton University Press, 1960, 159–194 | Zbl

[5] A. Pfluger, “The Fekete-Szegö inequality for complex parameters”, Complex Variables, 7 (1986), 149–160 | Zbl

[6] I. A. Aleksandrov, “Ekstremalnye svoistva klassa $S(w_0)$”, Tr. Tomskogo un-ta., 169 (1963), 24–58 | Zbl

[7] A. Pfluger, “The Fekete-Szegö inequality by a variational method”, Ann. Acad. Sci. Fenn., Ser. A.I. Math., 10 (1985), 447–454 | DOI | MR | Zbl

[8] A. Pfluger, “On the Functional $a_3-\lambda a_2^2$ in the Class $S$”, Complex Variables, 10 (1988), 83–95 | Zbl

[9] H. Siejka, O. Tammi, “On maximizing a homogeneous functional in the class of bounded univalent functions”, Ann. Acad. Sci. Fenn. Ser. A.I. Math., 6 (1981), 273–288 | DOI | MR | Zbl

[10] J. A. Hummel, “Extremal problems in the class of starlike functions”, Proc. Amer. Math. Soc., 11:5 (1960), 741–749 | DOI | MR | Zbl

[11] R. R. London, “Fekete-Szegö inequalities for close-to-convex functions”, Proc. Amer. Math. Soc., 117 (1993), 947–950 | MR | Zbl

[12] B. S. Mehrok, Singh H., “A Coefficient Inequality for a Certain Class of Analytic Functions”, Int. Journal of Math. Analysis, 5:7 (2011), 311–318 | MR | Zbl

[13] B. Bhowmik, S. Ponnusamy, K.-J. Wirths, “On the Fekete–Szegö problem for concave univalent functions”, J Math. Anal. Appl., 373 (2011), 432–438 | DOI | MR | Zbl

[14] Q. Xu, T. Liu, X. Liu, “Fekete and Szegö problem in one and higher dimensions”, Sci. China Math., 61 (2018), 1775–1788 | DOI | MR | Zbl

[15] N. A. Lebedev, Ob oblastyakh znachenii funktsionalov, zadannykh na klassakh analiticheskikh funktsii, Doktorskaya dissertatsiya, Leningradskii universitet, 1955

[16] Ya. V. Borisova, I. A. Kolesnikov, S. A. Kopanev, “O malykh variatsionnykh formulakh”, Vestn. Tomsk. gos. un–ta. Matem. i mekh., 2017, no. 49, 5–15

[17] I. A. Aleksandrov, I. A. Kolesnikov, S. A. Kopanev, L. S. Kopaneva, Metod vnutrennikh variatsii v teorii odnolistnykh otobrazhenii, Izd-vo Tom. un-ta, Tomsk, 2017

[18] B. A. Fuks, V. I. Levin, Funktsii kompleksnogo peremennogo i nekotorye ikh prilozheniya. Spetsialnye glavy., Izd-vo tekhniko-teoreticheskoi lit-ry, Moskva, 1951