Analogs of the Markov and Schaeffer-Duffin inequalities for convex bodies (Q5930782)

There are presented estimations for Fréchet derivatives of polynomial mappings of a real Banach space \(X\) into \(\mathbb{R}\) on some convex bodies of \(X\). Results are expressed in dependence on the degree \(n\) of the polynomial mapping \( P_{n}:X\rightarrow \mathbb{R}\) and on certain geometrical characterizations of the corresponding convex body \(V\subset X\). These results generalize inequalities of \textit{A. C. Schaeffer} and \textit{R. I. Duffin} [Bull. Am. Math. Soc. 44, 289-297 (1938; Zbl 0018.39503)].