Note on the problem of de la Vallée Poussin

A survey of results related to this problem is given. The extension to the vector case is examined. The main stimulation comes from the lecture of Á. Elbert given at the ICNO XII Conference in Cracow 1990 (see also [10]). 1. In reply to the question concerning the lower distance estimate of the consecutive zeros t1, t2 of the nontrivial oscillatory solutions x(t) of the equation x′′ + a(t)x′ + b(t)x = 0, with continuous bounded coefficients on [t1, t2], namely A := max t∈[t1,t2] |a(t)|, B := max t∈[t1,t2] |b(t)|, Ch. J. de la Vallée Poussin [24] came for h = t2 − t1 > 0 to the inequality 1 < 2Ah + 1 2 Bh. Since that time there have been stated several improvements of this result, e.g., 1 < 1 2 Ah + 1 6 Bh, by P. Hartman and A. Wintner [11] or, so far the sharpest inequality of this type, 1 ≤ 2Ah π2 + Bh π2 , Mathematics Subject Classification: 34B10, 34C10.