Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 967-979, 2017

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou

Received May 24, 2016.  First published March 1, 2017.

Abstract:  Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $\|p'\|_{[-1,1]}\leq\frac12\|p\|_{[-1,1]}$ for a constrained polynomial $p$ of degree at most $n$, initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(-1,1)$ and establish a new asymptotically sharp inequality.
Keywords:  polynomial; Erdős' inequality; undergraduate calculus; monotone polynomial
Classification MSC:  41A17, 26D05, 42A05
DOI:  10.21136/CMJ.2017.0256-16

[1] N. C. Ankeny, T. J. Rivlin: On a theorem of S. Bernstern. Pac. J. Math., Suppl. II 5 (1955), 849-852. DOI 10.2140/pjm.1955.5.849 | MR 0076020 | Zbl 0067.01001
[2] P. Borwein, T. Erdélyi: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics 161, Springer, New York (1995). DOI 10.1007/978-1-4612-0793-1 | MR 1367960 | Zbl 0840.26002
[3] R. A. DeVore, G. G. Lorentz: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften 303, Springer, Berlin (1993). DOI 10.1007/978-3-662-02888-9 | MR 1261635 | Zbl 0797.41016
[4] T. Erdélyi: Inequalities for Lorentz polynomials. J. Approx. Theory 192 (2015), 297-305. DOI 10.1016/j.jat.2014.12.012 | MR 3313486 | Zbl 1330.26003
[5] P. Erdős: On extremal properties of the derivatives of polynomials. Ann. of Math. (2) 41 (1940), 310-313. DOI 10.2307/1969005 | MR 0001945 | Zbl 0024.00403
[6] N. K. Govil: On the derivative of a polynomial. Proc. Am. Math. Soc. 41 (1973), 543-546. DOI 10.1090/S0002-9939-1973-0325932-8 | MR 0325932 | Zbl 0279.30004
[7] P. D. Lax: Proof of a conjecture of P. Erdős on the derivative of a polynomial. Bull. Am. Math. Soc. 50 (1944), 509-513. DOI 10.1090/S0002-9904-1944-08177-9 | MR 0010731 | Zbl 0061.01802
[8] M. A. Malik: On the derivative of a polynomial. J. Lond. Math. Soc., II. Ser. 1 (1969), 57-60. DOI 10.1112/jlms/s2-1.1.57 | MR 0249583 | Zbl 0179.37901

Affiliations:   Lai-Yi Zhu, Da-Peng Zhou (corresponding author), School of Information, Renmin University of China, No. 59 Zhongguancun Street, Beijing, 100872, Haidian, P. R. China, e-mail:,

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