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Equivalence of Markov’s and Schur’s inequalities on compact subsets of the complex plane. (English) Zbl 0930.41014

The equivalence of the inequalities of Markov \[ \|P'\|_{L^q(E)}\leq C_1(\text{deg} P)^{m_1}\|P\|_{L^q(E)} \] and Schur \[ \|P\|_{L^q(E)}\leq C_2(\text{deg} P)^{m_2}\|(x-x_0)P(x)\|_{L^q(E)} \] (\(x\in{\mathbb C}\); \(C_1, C_2, m_1, m_2\) positive constants depending on the compact subset \(E\subset {\mathbb C}\) and \(q\in [1,\infty]\)) is well known by now.
In this short note the author adds two inequalities that are equivalent to Markov and Schur (where either all polynomials \(P, R\) and constants \(a, b, c\) are real or all are complex): 1. For any triple \(a, b, c\): \[ \|(ax^2+bx+c)P'(x)\|_{L^q(E)}\leq C_3(\text{deg} P)^{m_3}\|(ax^2+bx+c)P\|_{L^q(E)} \] 2. For any polynomial \(R\): \[ \|RP'\|_{L^q(E)}\leq C_4(\text{deg} P+\text{deg} R)^{m_4}\|P\|_{L^q(E)} . \]

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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