Geodesic
Some inequalities for polynomials and rational functions associated with a lemniscate
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 83-99
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Inequalities for the area of a lemniscate, and inequalities for polynomials in a lemniscate containing no critical points other than zeros are given. Also, multipoint distortion estimates at the boundary of a lemniscate are proved. Some open problems are discussed. One of them associates with the well-known Smale's mean value conjecture. 
@article{ZNSL_2012_404_a4,
     author = {V. N. Dubinin},
     title = {Some inequalities for polynomials and rational functions associated with a~lemniscate},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--99},
     year = {2012},
     volume = {404},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a4/}
}
TY  - JOUR
AU  - V. N. Dubinin
TI  - Some inequalities for polynomials and rational functions associated with a lemniscate
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2012
SP  - 83
EP  - 99
VL  - 404
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a4/
LA  - ru
ID  - ZNSL_2012_404_a4
ER  - 
%0 Journal Article
%A V. N. Dubinin
%T Some inequalities for polynomials and rational functions associated with a lemniscate
%J Zapiski Nauchnykh Seminarov POMI
%D 2012
%P 83-99
%V 404
%U http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a4/
%G ru
%F ZNSL_2012_404_a4
V. N. Dubinin. Some inequalities for polynomials and rational functions associated with a lemniscate. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 83-99. http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a4/

[1] P. Borwein, T. Erdelyi, Polynomials and polynomial inequalities, Grad. Texts in Math., 161, Springer-Verlag, New York, 1995 | DOI | MR

[2] G. Polya, “Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete”, S. B. Preuss. Akad. Wiss. Berlin, Math.-phys.-tech. Kl., 1928, 280–282 | Zbl

[3] V. N. Dubinin, “Neravenstva dlya kriticheskikh znachenii polinomov”, Mat. sb., 197:8 (2006), 63–72 | DOI | MR | Zbl

[4] S. Smale, “The fundamental theorem of algebra and complexity theory”, Bull. Amer. Math. Soc., 4 (1981), 1–36 | DOI | MR | Zbl

[5] G. V. Kuzmina, “Metody geometricheskoi teorii funktsii. I”, Algebra i analiz, 9:3 (1997), 41–103 | MR | Zbl

[6] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, M., 1966 | Zbl

[7] R. B. Burckel, D. E. Marshall, D. Minda, P. Poggi-Corradini, T. J. Ransford, “Area, capacity and diameter versions of Schwarz's lemma”, Conform. Geom. Dyn., 12 (2008), 133–152 | DOI | MR | Zbl

[8] V. N. Dubinin, Emkosti kondensatorov i simmetrizatsiya v geometricheskoi teorii funktsii kompleksnogo peremennogo, Vladivostok, 2009

[9] I. P. Mityuk, “Printsip simmetrizatsii dlya mnogosvyaznykh oblastei”, Dokl. AN SSSR, 157 (1964), 268–270 | Zbl

[10] G. P. Bakhtina, “Ob odnoi ekstremalnoi zadache konformnogo otobrazheniya edinichnogo kruga na nenalegayuschie oblasti”, Ukr. mat. zh., 26 (1974), 635–637

[11] E. Fujikawa, T. Sugawa, “Geometric function theory and Smale's mean value conjecture”, Proc. Japan Acad. Ser. A, Math. Sci., 82:7 (2006), 97–100 | DOI | MR | Zbl

[12] Bl. Sendov, P. Marinov, “Verification of Smale's mean value conjecture for $n\leq10$”, C. R. Acad. Bulgare Sci., 60:11 (2007), 1151–1156 | MR | Zbl

[13] E. Crane, “A bound for Smale's mean value conjecture for complex polynomials”, Bull. London Math. Soc., 39 (2007), 781–791 | DOI | MR | Zbl

[14] Y. Wang, “Some results on Smale's mean value conjecture”, Third International Congress of Chinese Mathematicians, Part 1, 2, AMS/IP Stud. Adv. Math., 42, pt. 1, 2, Amer. Math. Soc., Providence, RI, 2008, 595–602 | MR | Zbl

[15] A. Hinkkanen, I. Kayumov, “Smale's problem for critical points on certain two rays”, J. Aust. Math. Soc., 88:2 (2010), 183–191 | DOI | MR | Zbl

[16] Yu. E. Alenitsyn, “Ob odnolistnykh funktsiyakh bez obschikh znachenii v mnogosvyaznoi oblasti”, Tr. Mat. in-ta AN SSSR, 94, 1968, 4–18 | MR | Zbl