Estimates of eigenfunctions for one class of boundary conditions

Requirements for boundary conditions are found which imply sharp by order estimates in L-infinity-metrics of eigenfunctions of the ordinary differential operator l(y) = D(n)y + Sigma(k-0)(n-2) Pn-k(x) D(k)y, p(n-k)(x) is an element of L-1 (-1, 1); D = 1 d/i dx, x is an element of[-1, 1]. It is shown that the inequalities parallel to y((k))(lambda)parallel to(infinity) less than or equal to c\lambda\(k/n) parallel to y(lambda)parallel to(2), c = const.c\lambda\k/n parallel to y lambda parallel to 2, c = const. for the eigenfunctions ({Y-lambda} yield certain restrictions on the spectrum.