An exact estimate of the second derivative in \(L_ p\) (Q1175471)

Apart from \(L_ p(\mathbb{R})\), we consider the spaces of \(2\pi\)-periodic functions \(f\) with finite norm \[ \| f \|=\left\{ {1 \over \pi} \int^ \pi_{-\pi} | f(x) |^ p dx \right\}^{1/p} < \infty,\quad 1 \leq p< \infty. \] \textit{L. V. Taikov} [Anal. Math. 2, 77-85 (1976; Zbl 0326.30028)] stated exact estimates of the derivative of a function in terms of the modulus of smoothness of the function and that of its second derivative. In the present note we give an exact estimate of the second derivative of a function in terms of the modulus of smoothness of the second-order derivative of the function and that of the function itself.