On maximum principles in the class of oscillating functions

Summary.

We investigate the problem $a(x){u}\ifmmode{''}\else$''$\fi(x) + b(x){u}\ifmmode{'}\else$'$\fi(x) + c(x)u(x) = 0$ on a bounded interval [a, b] and find conditions under which the solution u achieves its extrema at the boundary points of [a, b]. That leads to a kind of maximum principles for a second order operator whose solutions can oscillate in the interior of the domain. The presented technique is applied to compute supremum norms of some known special functions. In particular we recover some classical results on orthogonal polynomials.