Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)

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We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities

Specifically, we prove that

for all n ∈ {1, 2,…} and k ∈ {0, …, n-1}. We establish symmetry and regularity properties of the numbers An, k and study their asymptotic behavior as n → ∞ for the cases k = O(n2/3) and k/n → α ∈ (0, 1).

Similar problems were previously studied by Gabushin and Taikov.

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