On the proof of Erdős’ inequality
Tóm tắt
Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality
$${\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}$$
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.
Tài liệu tham khảo
N. C. Ankeny, T. J. Rivlin: On a theorem of S. Bernstern. Pac. J. Math., Suppl. II 5 (1955), 849–852.
P. Borwein, T. Erdélyi: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics 161, Springer, New York, 1995.
R. A. DeVore, G. G. Lorentz: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften 303, Springer, Berlin, 1993.
T. Erdélyi: Inequalities for Lorentz polynomials. J. Approx. Theory 192 (2015), 297–305.
P. Erdős: On extremal properties of the derivatives of polynomials. Ann. of Math. (2) 41 (1940), 310–313.
N. K. Govil: On the derivative of a polynomial. Proc. Am. Math. Soc. 41 (1973), 543–546.
P. D. Lax: Proof of a conjecture of P. Erdős on the derivative of a polynomial. Bull. Am. Math. Soc. 50 (1944), 509–513.
M. A. Malik: On the derivative of a polynomial. J. Lond. Math. Soc., II. Ser. 1 (1969), 57–60.