Sharp Estimates for Derivatives of Functions in the Sobolev Classes (W)over-circle2r(-1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f((k))(x), x epsilon (-1, 1), kappa epsilon {0, 1, ... , r - 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to <= r - 1 at the points +/- 1 and are such that parallel to f((r))parallel to(L2(- 1, 1)) <= 1. As a corollary, it is shown that the first eigenvalue lambda(1,r) of the operator (-D-2)(r) with these boundary conditions is root 2(2r)! (1+O(1/r)), r -> infinity.