the paper, we find sharp constants in a Kolmogorov-type inequality between the L-2r-norm of the derivative, the L-r-norm (quasinorm for r < 1) of the function, and the asymmetric L-infinity-norm (including the norm of the nonnegative part) of the second derivative for any 1/2 <= r < infinity on the real line and on the period.
the Ramsey number R(K-1,K-s, P-t) one means the least positive integer n such that, for every n-vertex graph G, the following condition holds: either G contains a vertex of degree at least s or the complement of G contains a simple t-path. In this paper, we find precise values of R(K-1,K-s, P-t) for certain values s > t.
