## 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)

### Keywords:

Markov inequality; Schur inequality
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