On some extremal problems of different metrics for differentiable functions on the axis.

For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A r, A0, and p > 0, we solve the extremal problem on the set of all functions x∈ L r ∞ such that ∥ x( r)∥∞ ≤ A r and L( x) p ≤ A0, where In the case where p = ∞ and k ≥ 1, this problem was solved earlier by Bojanov and Naidenov. [ABSTRACT FROM AUTHOR]