Equioscillatory property of the Laguerre polynomials

We show that the function ((x−dm)(x−dM))1/4xα/2e−x/2Lk(α)(x) is almost equioscillating with the amplitude 2/π provided kk and αα are large enough. Here Lk(α)(x) is the orthonormal Laguerre polynomial of degree kk and dm,dMdm,dM are some approximations for the extreme zeros. As a corollary we obtain a very explicit, uniform in kk and αα, sharp upper bound on the Laguerre polynomials.