A Markov-type inequality for the derivatives of constrained polynomials
The magnitude of \[ \sup_{p\in S^ m_ n(r)}(\max_{-1\leq x\leq 1}| p'(x)| /\max_{-1\leq x\leq 1}| p(x)|) \] is examined, where \(S^ m_ n(r)\) (0\(\leq m\leq n\), \(0<r\leq 1)\) denotes the set of those real algebraic polynomials of degree at most n which have all but at most m zeroes outside the circle \(K(r)=\{z\in {\mathbb{C}}:| z| <r\}\). It is proved that \[ c_ 1(m)(n+(1-r)n^ 2)\leq \sup_{p\in S^ m_ n(r)}(\max_{-1\leq x\leq 1}| p'(x)| /\max_{-1\leq x\leq 1}| p(x)|)\leq c_ 2(m)(n+(1-r)n^ 2) \] holds for every \(0<r\leq 1\) and \(0\leq m\leq n\), where \(c_ 1(m)\) and \(c_ 2(m)\) depend only on m. This estimate is a generalization of Markov's inequality \((r=0\), \(m=0)\) and a result due to Lorentz \((r=1\), \(m=0)\).
scientific article; zbMATH DE number 4190350
scientific article