On dependence between the norm of a function and norms of its derivatives of orders k, r - 2 and r, 0 < k < r - 2

Abstract

Necessary and sufficient conditions on the system of positive numbers $ M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}$, $0= k_1<k_2<k_3=r-2$, $k_4 = r$, which guarantee the existence of a function $x\in L_{\infty,\infty}^r(R)$, such that $\|x^{(k_i)}\|_{\infty}=M_{k_i},\; i=1,2,3,4, $ are found.