Inequalities for the first and second derivatives of algebraic polynomials on an ellipse. (English) Zbl 1499.41021

Nikiforova, Tatiana M.

Math. Pannonica (N.S.) 27, No. 2, 233-240 (2021).

S. Bernstein [Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné. Belg. Mem. in \(4^\circ\) (2) 4, 104 S. (1912; JFM 45.0633.03)] proved the well-known inequalities for derivatives of algebraic polynomials on the unit circle and on the real axis. A similar problem for an ellipse was considered by W. E. Sewell [Am. Math. Mon. 44, 577–578 (1937; Zbl 0018.01303)]. Some generalization of Sewell’s result is given.
Let \[ E_r=\left\{ z=\frac{1}{2}\left( rw+\frac{1}{rw} \right) ,\ |w|=1\right\}, \] which is an ellipse for \(r\in (0,1]\) and is the segment \([-1,1]\) for \(r=1\). For some functions \(M_n\), defined on \(E_r\), \(r\in (0,1]\), it is proved that if some polynomial \(p_n(z)\) of order \(n\) satisfies the condition \(|p_n(z)|\leq |M_n(z)|\), \(z\in E_r\), \(r\in (0,1]\), then for the derivatives of the first and second orders of \(p_n(z)\) and \(M_n(z)\), the inequalities \(|p_n^{(k)}(z)|\leq |M_n^{(k)}(z)|\), \(k=1,2\), are true.

Reviewer: D. K. Ugulava (Tbilisi)

MSC:

41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
30A10	Inequalities in the complex plane
30C10	Polynomials and rational functions of one complex variable

Keywords:

polynomial; derivative; ellipse; majorant; Bernstein inequality; Duffin-Schaeffer inequality; Vidensky inequality

Citations:

Zbl 0018.01303; JFM 45.0633.03