Lyapunov-type inequalities for self-adjoint differential equations of second order

We present a general method that yields Patula, Hartman-Wintner, and Lyapunov-type inequalities for the second-order differential equation \begin{document}$ -(\alpha(x)u'(x))'+\beta(x)u(x) = \gamma(x)u(x), \quad x\in I, $\end{document} where $ I $ is an interval of $ \mathbb{R} $, $ \alpha\in C^1(I) $, $ \alpha > 0 $, $ \beta, \gamma\in C(I) $, and $ \beta\geq 0 $. Applications cover the generalized radial Schrödinger equation and the modified Bessel equation and include an improved (sharpened) form of Bargmann's inequality.