Variations on a theme of the Marden and Smirnov operators, differential inequalities for polynomials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:476

Article

Ganenkova, E. G.¹  Starkov, V. V.¹ 

[1] Petrozavodsk State Univ, Dept Math & Informat Technol, Petrozavodsk 185910, Karelia, Russia

关键词: Polynomial;    Inequality;    Differential operator;    Zeros of polynomial;   

DOI  :  10.1016/j.jmaa.2019.04.006

来源: Elsevier

【 摘 要 】

In 1949 Marden in his book introduced and studied the differential operator B[f](z) = lambda(0)f (z) + lambda(1) nz/2 f' (z) + lambda(2) (nz/2)(2) f ''(z)/2!, defined on polynomials of degree at most n. Here lambda(0), lambda(1), and lambda(2) are constants such that the polynomial u(z) = lambda(0) + C-n(1) lambda(1)z + C-n(2)lambda(2)z(2) has all roots in the half-plane Re z <= n/4. In the book of 2002 Rahman and Schmeisser proved that under simple additional conditions on a couple of polynomials f and F the differential equality vertical bar B[f](z)vertical bar <= vertical bar B[F](z)vertical bar holds for vertical bar z vertical bar >= 1. In this article, in particular, we describe the set of parameters lambda(k) for which the inequality of Rahman and Schmeisser takes place for vertical bar z vertical bar >= R, R > 1. As it turns out, in this case the roots of u(z) should belong to the complement of the open disc with diameter (nR/2(R+1), nR/2(R-1)). We show that this set can not be extended. This fact does the inequality of Rahman and Schmeisser more applicable for vertical bar z vertical bar > 1. Also for a polynomial f we obtain a sharp estimation of vertical bar f ''(z)vertical bar in terms of vertical bar f(z)vertical bar and M-r = max(vertical bar z vertical bar=r) vertical bar f(z)vertical bar. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free