On Markov's and Bernstein's inequalities in the unit ball in \(\mathbf R^ k\)
Let \(B(0,1)\) be the closed unit ball in a \(k\)-dimensional Euclidean space \((k\in \mathbb{N})\). In 1927, O. D. Kellogg generalized Markov's inequality to \(B(0,1)\) by showing \[ \| \nabla P\|_{B(0,1)}\leq n^ 2\| P\|_{B(0,1)}, \] where \(\nabla\) denotes the gradient, \(P\) denotes polynomials of degree \(n\), \(\|\cdot \|_{B(0,1)}\) denotes the maximum norm on \(B(0,1)\). [See Math. Z. 27, 55-64 (1927; JFM 53.0082.03)] and he also gave a similar generalization of Bernstein's inequality. In the present note, the author refines these inequalities by a simple and elaborated method. His refinement for the generalized Markov's inequality can be stated as Theorem: Let \(P\) be a polynomial of degree \(n\), \(| x|< 1\) and \({\mathbf v}\) be a unit vector. Denote by \(2d\) the length of the chord through \(x\) in the direction \({\mathbf v}\). Then \[ | P_{{\mathbf v}}(x)|\leq {nd\over \sqrt{1- | x|^ 2}} \| P\|_{B(0,1)}, \] where \(P_{{\mathbf v}}\) denotes the directional derivative in the direction \({\mathbf v}\).
scientific article; zbMATH DE number 636065