On constants in some inequalities for intermediate derivatives on a finite interval Original Research Article

Let Lq (1⩽q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with norm|| f||q=∫−11| f(x)|q dx1q.L∞ is the space of functions measurable on I with norm|| f||∞=esssup|x|⩽1 | f(x)|<∞.We denote by AC the set of all functions absolutely continuous on I. For n∈N, q∈[1,∞] we setWn,q={f:f(n−1)∈AC, f(n)∈Lq}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f(m)||q⩽A|| f||p+B|| f(m+k+1)||r, m∈N, k∈W; p,q,r∈[1,∞], f∈Wm+k+1,r.