A second order quadratic integral inequality associated with regular problems

In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all f is an element of D, integral(b)(a) r|f '' |(2) + p|f'|(2) + q|f|(2) >= mu(0) integral(b)(a) |f|(2). Here r, p, q are real-valued coefficient functions on the compact interval [a, b] with r(x) > 0. D is a linear manifold in the Hilbert function space L-2(a, b) such that all integrals of the above inequality are finite and mu(0) is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in L-2(a, b). The inequality is the best possible, i.e., the number mu(0) cannot be increased. f is a complex-valued function in D.