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In fact it is a solvability condition for naturally arising in applications system of differential equations. I first heard it from Prof. Alexandr Soldatov many years ago. He originally proposed that the set $D$ is empty, but careful calculations lead to the above version. And - can anybody help with an accurate graph of the set $D$, please.
a remark. Please note that one case $2k=n,n+1$ is not covered by the cited paper's inequalities with its strict conditions on $a,b$ and has to be considered separately. But it is easy, fortunately.
your solution seems to be clever and good. Unfortunately I am too silly for understanding it, but it is my problem... But - it is just an estimate for an integral, let me believe it has a simple calculus solution.
what I do not understand properly. For $k=n$ we may take $c=1/e$. If the reverse induction works the inequality must be true with this constant, $c=1/e$. But it is not so.
Is the next generalization valid with two pivoted elements: let $(a_1,a_2,a_3,\dots)$ be a line of a determinant with $a_1+a_2>\sum_{k=3}^n a_k$. Is the Ostrowski inequality still valid with a product of \prod(a_1+a_2-\sum_{k=3}^n a_k)$ -???
@VladimirZolotov - may you give more details, with proper references on connection of curvature with value of radius? I do not know much of diffgeometry...