Geodesic
A problem of Mahler on the zeros of a polynomial and its derivative
Sbornik. Mathematics, Tome 187 (1996) no. 5, pp. 735-744 
È. A. Storozhenko. A problem of Mahler on the zeros of a polynomial and its derivative. Sbornik. Mathematics, Tome 187 (1996) no. 5, pp. 735-744. http://geodesic.mathdoc.fr/item/SM_1996_187_5_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Mahler has obtained an inequality for the products of zeros of an algebraic polynomial and its derivative lying outside the unit disc. In this paper a converse inequality with best possible constant is established.

[1] Mahler K., “On the zeros of the derivative of a polinomial”, Proc. Roy. Soc. London Ser. A, 264:1317 (1961), 145–154 | DOI | MR | Zbl

[2] Bernshtein S. N., Sobranie sochinenii, T. I, II, Izd-vo AN SSSR, M., 1952–1959

[3] Zigmund A., Trigonometricheskie ryady, T. I, II, Mir, M., 1965 | MR

[4] Storozhenko E. A., Krotov V. G., Osvald P., “Pryamye i obratnye teoremy tipa Dzheksona v prostranstvakh $L_p$, $0

1$”, Matem. sb., 98 (140):3 (11) (1975), 395–415 | MR | Zbl

[5] Ivanov V. I., “Nekotorye neravenstva dlya trigonometricheskikh polinomov i ikh proizvodnykh v raznykh metrikakh”, Matem. zametki, 18:4 (1975), 489–498 | MR | Zbl

[6] Arestov V. V., “Ob integralnykh neravenstvakh dlya trigonometricheskikh polinomov i ikh proizvodnykh”, Izv. AN SSSR. Ser. matem., 45:1 (1981), 3–22 | MR | Zbl

[7] Khardi G., Littlvud D., Polia G., Neravenstva, IL, M., 1948

[8] Korneichuk N. P., Babenko V. F., Ligun A. A., Ekstremalnye svoistva polinomov i splainov, Naukova dumka, Kiev, 1992 | MR

[9] Polia G., Sege G., Zadachi i teoremy iz analiza, T. I, II, Nauka, M., 1978

[10] De Bruijn N. J., Springer T. A., “On the zeros of composition-polynomials”, Jndag. Math., 9 (1947), 406–414

[11] Arestov V. V., “Integralnye neravenstva dlya algebraicheskikh mnogochlenov na edinichnoi okruzhnosti”, Matem. zametki, 48:4 (1990), 7–18 | MR

[12] Arestov V. V., “Neravenstvo Sege dlya proizvodnykh sopryazhennogo trigonometricheskogo polinoma v $L_0$”, Matem. zametki, 56:6 (1994), 10–26 | MR | Zbl

[13] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, GITTL, M., 1964