On constants in some inequalities for intermediate derivatives on a finite interval
Denoting by \(L_q\) the space of functions \(f\) measurable on \(I=[-1,1]\) and integrable to power \(q\) and by \(AC\) the set of all functions absolutely continuous on \(I\), for \(n\in\mathbb N\), \(q\in[1, +\infty)\), the author set \[ W_{n,q}=\left\{f; f^{(n-1)}\in AC, f^{(n)}\in L_q\right\}. \] The problem of accuracy of constants \(A, B\) in the inequalities \[ \begin{aligned} & \| f^{(m)}\| _q\leq A\| f\| _p+B\| f^{(m+k+1)}\| _r,\\ & m\in\mathbb{N}, k\in W,\; p,q,r\in[1,+\infty], f\in W_{n+k+1,r} \end{aligned} \] is considered. The author used the following notations \[ \begin{aligned} & A_{m+k+1, m} (p,q,r)=\inf\limits_A A\\ & B_{m+k+1,m} (p,q,r)=\inf\limits_{B}\\ & B:A=A_{m+k+1,m}(p,q,r) \end{aligned} \] Explicit expressions for \(A_{m+k+1,m}(2,\infty,r)\) \((m\in\mathbb N,\; k\in W, 1\leq r\leq\infty)\) and estimates for \(A_{m+k+1,m}(p,q,r)\) are established.
scientific article; zbMATH DE number 2086007