Exact constants in Kolmogorov-type inequalities (Q2714711)

Let \(m>2\) be a positive integer. The aim of the authors is to find the best constant \(K\) satisfying the inequality NEWLINE\[NEWLINE\|x\|_{L_\infty(I)}\leq K\|x\|^\alpha_{L_2(I)} \|x^{(m)}\|^\beta_{L_\infty(I)},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEI=\mathbb{R}_+,\quad \alpha=\frac{2m}{2m+1},\quad \beta=\frac{1}{2m+1} .NEWLINE\]NEWLINE With this end in view they investigate the minimization problem NEWLINE\[NEWLINE\int^{\infty}_{0}x^2 dt\to \inf,\text{ subject to }x(0)=1,\quad |x^{(m)}(t)|\leq 1.\tag{2}NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThey prove that \(K_m = (S_m)^{-\frac{m}{2m+1}}\) is the best constant satisfying (1), where \(S_m\) denotes the optimal value of the objective functional in (2). By applying this result they find that \(K_3\cong 1.90488\) and give an asymptotic estimate of \(K_m\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].