Czechoslovak Mathematical Journal, online first, 13 pp.

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou

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

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:,

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 2010): 41A17, 26D05, 42A05

DOI: 10.21136/CMJ.2017.0256-16

Full text available as PDF.

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