Geodesic
Sharp Inequalities for Rational Functions on a Circle
Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 29-36.

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

For rational functions with prescribed poles lying outside the unit circle $|z|=1$, sharp inequalities are established at points $z$, $|z|=1$. In contrast to the known results, the location of the specified poles on either side of the circle $|z|=1$ is allowed.
Keywords: polynomials, rational functions, Bernstein inequalities, rotation theorems. 
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V. N. Dubinin. Sharp Inequalities for Rational Functions on a Circle. Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 29-36. http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a2/

[1] P. B. Borwein, T. Erdelyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995 | MR

[2] P. B. Borwein, T. Erdelyi, “Sharp extensions of Bernstein's inequality to rational spaces”, Mathematika, 43:2 (1996), 413–423 | DOI | MR

[3] X. Li, R. N. Mohapatra, R. S. Rodriguez, “Bernstein-type inequalities for rational functions with prescribed poles”, J. London Math. Soc., 51:3 (1995), 523–531 | DOI | MR

[4] A. Aziz, W. M. Shah, “Some refinements of Bernstein-type inequalities for rational functions”, Glas. Mat. Ser. III, 32 (52):1 (1997), 29–37 | MR

[5] G. Min, “Inequalities for rational functions with prescribed poles”, Canad. J. Math., 50:1 (1998), 152–166 | DOI | MR

[6] N. K. Govil, R. N. Mohapatra, “Inequalities for maximum modulus of rational functions with prescribed poles”, Approximation Theory, Monogr. Textbooks Pure Appl. Math., 212, Dekker, New York, 1998, 255–263 | MR

[7] V. N. Dubinin, “O primenenii konformnykh otobrazhenii v neravenstvakh dlya ratsionalnykh funktsii”, Izv. RAN. Ser. matem., 66:2 (2002), 67–80 | DOI | MR | Zbl

[8] Xin Li, “A comparison inequality for rational functions”, Proc. Amer. Math. Soc., 139 (2011), 1659–1665 | DOI | MR

[9] S. I. Kalmykov, “Neravenstva dlya modulei ratsionalnykh funktsii”, Dalnevost. matem. zhurn., 12:2 (2012), 231–236

[10] S. L. Wali, W. M. Shah, “Applications of the Schwarz lemma to inequalities for rational functions with prescribed poles”, J. Anal., 25:1 (2017), 43–53 | DOI | MR

[11] S. L. Wali, W. M. Shah, “Some applications of Dubinin's lemma to rational functions with prescribed poles”, J. Math. Anal. Appl., 450:1 (2017), 769–779 | DOI | MR

[12] S. Hans, D. Tripathi, A. A. Mogbademu, B. Tyagi, “Inequalities for rational functions with prescribed poles”, J. Interdisciplinary Math., 21:1 (2018), 157–169 | DOI

[13] S. I. Kalmykov, “O mnogotochechnykh teoremakh iskazheniya dlya ratsionalnykh funktsii”, Sib. matem. zhurn., 61:1 (2020), 107–119 | DOI

[14] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[15] V. N. Dubinin, “Metody geometricheskoi teorii funktsii v klassicheskikh i sovremennykh zadachakh dlya polinomov”, UMN, 67:4 (406) (2012), 3–88 | DOI | MR | Zbl

[16] S. I. Kalmykov, “O polinomakh i ratsionalnykh funktsiyakh, normirovannykh na dugakh okruzhnosti”, Analiticheskaya teoriya chisel i teoriya funktsii. 28, Zap. nauchn. sem. POMI, 418, POMI, SPb., 2013, 105–120

[17] V. N. Dubinin, “Ekstremalnaya zadacha dlya proizvodnoi ratsionalnoi funktsii”, Matem. zametki, 100:5 (2016), 732–738 | DOI | MR

[18] P. Turan, “Über die Ableitung von polynomen”, Compositio Math., 7 (1939), 89–95 | MR

[19] N. K. Govil, P. Kumar, “On sharpening of an inequality of Tuŕan”, Appl. Anal. Discrete Math., 13:3 (2019), 711–720 | DOI | MR

[20] S. Kalmykov, B. Nagy, V. Totik, “Bernstein- and Markov-type inequalities for rational functions”, Acta Math., 219:1 (2017), 21–63 | DOI | MR