Markov and Bernstein's inequalities in Banach spaces (Q1594068)

The classical inequality for the derivatives of a polynomial \(P_n\) of one variable with \(\deg P_n\leq n\) \[ \|P^{(k)}_n\|_{[a, b]}\leq {2^k n^2(n^2- 1^2)\cdot\cdots\cdot (n^2- (k-1)^2)\over(b- a)^k (2k- 1)!!} \|P_n\|_{[a, b]} \] was proved by A. A. Markov in the case \(k=1\) and in the general case by his brother V. A. Markov. Another important polynomial inequality is Bernstein's inequality for \(P_n'(x)\) at an interior point \(x\) of the interval \([a,b]\): \[ |P_n'(x)|\leq {n\|P_n\|_{[a, b]}\over \sqrt{(x- a)(b-x)}}. \] In this paper there are announced some results concerning Markov and Bernstein type inequalities in Banach spaces. Now, the interval \([a,b]\) is replaced by a convex bounded closed body \(K\) in a Banach space and there are given estimates for the derivatives of a polynomial \(P\) at points of \(K\). These estimates are claimed to be sharp for the first derivative of \(P\), at least for a ball. In particular, there is announced the inequality \[ \|\nabla P\|_K\leq {c\over \omega_K} (\deg P)^2\|P\|_K \] with an effective universal constant \(c<4\). Here \(\omega_K\) is the width of \(K\). Such an inequality with \(c=4\) was earlier proved by \textit{D. R. Wilhelmsen} [J. Approximation Theory 11, 216-220 (1974; Zbl 0287.26016)]. The above results are claimed to hold for higher derivatives of \(P\) too, but now the obtained constants are rather far from being optimal. The author also gives bounds for the derivatives of homogeneous polynomials. In the case of a ball, it was earlier done by \textit{Y. Sarantopoulos} [Math. Proc. Camb. Phil. Soc. 110, No. 2, 307-312 (1991; Zbl 0761.46035)]. No proofs are given in the paper.