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Analogs of the Markov and Schaeffer-Duffin inequalities for convex bodies. (English. Russian original) Zbl 1009.46028
Math. Notes 68, No. 1, 130-134 (2000); translation from Mat. Zametki 68, No. 1, 146-150 (2000).
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 A. C. Schaeffer and R. I. Duffin [Bull. Am. Math. Soc. 44, 289-297 (1938; Zbl 0018.39503)].

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 52A40 Inequalities and extremum problems involving convexity in convex geometry 46T20 Continuous and differentiable maps in nonlinear functional analysis 46G25 (Spaces of) multilinear mappings, polynomials