Pointwise inequalities of Landau-Kolmogorov type for functions defined on a finite segment (Q2754798)

For arbitrary \(t\in [0,1]\), \(p\in [1,\infty ]\) and \(A\geq 2\) the author finds the best possible constant \(B\) in the inequality NEWLINE\[NEWLINE |x'(t)|\leq A\|x\|_{L_\infty [0,1]}+B\|x''\|_{L_p(0,1)}. NEWLINE\]NEWLINE This leads to the precise inequality for the norms NEWLINE\[NEWLINE \|x'\|_\infty \leq \frac{2}{h}\|x\|_\infty +\left( \frac{h}{p'+1}\right)^{1/p'}\|x''\|_p NEWLINE\]NEWLINE valid for any \(x\in L_\infty^2\), \(0<h\leq 1\).