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

$$-(\alpha (x)u^{\prime }(x){)}^{\prime }+\beta (x)u(x)=\gamma (x)u(x),x\in I,$$

where $I$ is an interval of $\mathbb{R}$, $\alpha \in C^1(I)$, $\alpha >0$, $\beta ,\gamma \in C(I)$, and $\beta \ge 0$. Applications cover the generalized radial Schrödinger equation and the modified Bessel equation and include an improved (sharpened) form of Bargmann's inequality.