A second order quadratic integral inequality with an application to ordinary differential operators

In this paper, we establish the following integral inequalities: For all f is an element of D, integral(b)(a) |q||f|(2) <= epsilon integral(b)(a) a|f"|(2) + epsilon integral(b)(a) p|f'|(2) + A(epsilon) integral(b)(a) w|f|(2) and integral(b)(a) |p||f'|(2) + |q||f|(2) <= epsilon integral(b)(a) r|f"|(2) +A'(epsilon) integral(b)(a) w{|f|(2) + |f'|(2) } Here r, p, q and w are given real valued coefficients with r and w non negative on the compact interval [a, b] and p is assumed non negative on [a, b] only for the first inequality. D is a linear manifold of complex-valued functions so determined that all integrals of the above two inequalities are finite. Here epsilon is an arbitrary small positive number and A( epsilon), A'( epsilon) are two different positive numbers depending on epsilon and the coefficients r, p, q and w, which in general become large with small epsilon.