Analyse et géométrie complexes
On the Erdős–Lax Inequality
Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1081-1085.

The Erdős–Lax Theorem states that if P(z)= ν=1 n a ν z ν is a polynomial of degree n having no zeros in |z|<1, then

max |z|=1 |P (z)|n 2max |z|=1 |P(z)|.

In this paper, we prove a sharpening of the above inequality (1). In order to prove our result we prove a sharpened form of the well-known Theorem of Laguerre on polynomials, which itself could be of independent interest.

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DOI : 10.5802/crmath.141
Classification : 30A10
Kumar, Prasanna 1

1 Department of Mathematics, Birla Institute of Technology and Science Pilani, K K Birla Goa Campus, Goa, India 403726
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Kumar, Prasanna. On the Erdős–Lax Inequality. Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1081-1085. doi : 10.5802/crmath.141. http://www.numdam.org/articles/10.5802/crmath.141/

[1] Ankeny, Nesmith C.; Rivlin, Theodore J. On a theorem of S. Bernstein, Pac. J. Math., Volume 5 (1955), pp. 849-852 | DOI | Zbl

[2] Bernshteĭn, Sergeĭ N. Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle, Collection de monographies sur la théorie des fonctions, Gauthier-Villars, 1926

[3] Erdős, Pál On extremal properties of derivatives of polynomials, Ann. Math., Volume 41 (1940), pp. 310-313 | DOI | MR | Zbl

[4] Govil, Narenda K.; Rahman, Qazi I. Functions of exponential type not vanishing in a halfplane and related polynomials, Trans. Am. Math. Soc., Volume 137 (1969), pp. 501-517 | DOI | Zbl

[5] Laguerre, Edmond Oeuvres Vol. 1, Nouv. Ann. Math., Volume 17 (1878) no. 2, pp. 20-25

[6] Lax, Peter D. Proof of a conjecture of P. Erdős on the derivative of a polynomial, Bull. Am. Math. Soc., Volume 50 (1944), pp. 509-513 | Zbl

[7] Malik, M. A. On the derivative of a polynomial, J. Lond. Math. Soc., Volume 1 (1969), pp. 57-60 | DOI | Zbl

[8] Pólya, George; Szegő, Gábor Aufgaben und Lehrsätze aus der Analysis, Grundlehren der Mathematischen Wissenschaften, 19, Springer, 1925

[9] Schaeffer, Albert C. Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Am. Math. Soc., Volume 47 (1941), pp. 565-579 | DOI | MR | Zbl

[10] Szegő, Gábor Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z., Volume 13 (1922), pp. 28-55 | DOI | Zbl

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