2017
Том 69
№ 9

All Issues

Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

Babenko V. F., Kofanov V. A., Pichugov S. A.

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Abstract

We obtain new exact inequalities of the form $$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$ for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$ for functions defined on the axis $R$ in the case where $$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$ and for functions of constant sign on $R$ or $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$ and in the case where $$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$

English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 3, pp 325-339.

Citation Example: Babenko V. F., Kofanov V. A., Pichugov S. A. Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis // Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 291–302.

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