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Kolmogorov-type inequalities for mixed derivatives of functions of many variables
Ukrainian Mathematical Journal, 2004Let γ = (γ1,...,γ d ) be a vector with positive components and let Dγ be the corresponding mixed derivative (of order γ j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that
V. F. Babenko +2 more
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On the set of extremal functions in certain Kolmogorov-type inequalities
Ukrainian Mathematical Journal, 2004Summary: We determine the sets of all extremal functions in certain Kolmogorov-type and Bohr-Favard-type inequalities.
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On Kolmogorov-type inequalities for fractional derivatives of functions of two variables
Ukrainian Mathematical Journal, 2008Summary: We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order.
Babenko, V. F., Pichugov, S. A.
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Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness
Ukrainian Mathematical Journal, 2001New unimprovable Kolmogorov-type inequalities \[ \|x^{(k)}(t)\|_{L_q}\leq K\|x(t)\|^\alpha_{L_p} \|x^{(r)}(t)\|^{1-\alpha}_{L_\infty},\quad L_p=L[0,2\pi],\tag{1} \] are obtained for differentiable periodic functions \(x(t)\in L_{\infty}^r=L_{\infty}^r[0,2\pi]\). It is proved that if \(k=1;\) \(r=2,3\); \(p\in (0,\infty]\) or \(k=2;\) \(r=3\); \(p\in (1/
Babenko, V. F. +2 more
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Russian Mathematics, 2015
The main topic of this paper is to study some sharp inequalities of Kolmogorov type (and other types) for the estimation of derivatives of a function in terms of its higher derivatives. This is done for higher dimensional setting, which is a much more difficult problem than the single-variable case.
Vakarchuk, S. B., Shvachko, A. V.
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The main topic of this paper is to study some sharp inequalities of Kolmogorov type (and other types) for the estimation of derivatives of a function in terms of its higher derivatives. This is done for higher dimensional setting, which is a much more difficult problem than the single-variable case.
Vakarchuk, S. B., Shvachko, A. V.
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One Inequality of the Landau–Kolmogorov Type for Periodic Functions of Two Variables
Ukrainian Mathematical Journal, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives
Ukrainian Mathematical Journal, 2003New exact Kolmogorov-type inequalities \[ \| x^{(k)}\| _q\leq \left(\frac{\nu(x^{(k)})}{2}\right)^{1/q} \frac{\| \varphi_{r-k}\| _q}{| | | \varphi_r| | | _p^\alpha} | | | x| | | _p^\alpha\| x^{(r)}\| _\infty^{1-\alpha},\;q\in [1,\infty],\;p\in (0,\infty], \;k,r\in {\mathbb N ...
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Pointwise Inequalities of Landau–Kolmogorov Type for Functions Defined on a Finite Segment
Ukrainian Mathematical Journal, 2001For arbitrary \(t\in [0,1]\), \(p\in [1,\infty ]\) and \(A\geq 2\) the author finds the best possible constant \(B\) in the inequality \[ |x'(t)|\leq A\|x\|_{L_\infty [0,1]}+B\|x''\|_{L_p(0,1)}. \] This leads to the precise inequality for the norms \[ \|x'\|_\infty \leq \frac{2}{h}\|x\|_\infty +\left( \frac{h}{p'+1}\right)^{1/p'}\|x''\|_p \] valid for ...
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Relationship Between the Bojanov–Naidenov Problem and the Kolmogorov-Type Inequalities
Ukrainian Mathematical JournalzbMATH Open Web Interface contents unavailable due to conflicting licenses.
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