For nonperiodic functions \( x\in L_{\infty}^r(R) \) defined on the entire real axis, we prove analogs of the Babenko inequality. The obtained inequalities estimate the norms of derivatives \( \left\| {x_{\pm}^{(k) }} \right\|{L_{q[a,b] }} \) on an arbitrary interval [a, b] ⊂ R such that x (k)(a) = x (k)(b) = 0 via local L p -norms of the functions x and uniform nonsymmetric norms of the higher derivatives x(r) of these functions.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 636–648, May, 2012.
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Kofanov, V.A. Inequalities for derivatives of functions on an axis with nonsymmetrically bounded higher derivatives. Ukr Math J 64, 721–736 (2012). https://doi.org/10.1007/s11253-012-0674-5
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DOI: https://doi.org/10.1007/s11253-012-0674-5