On the zeroes and the critical points of a solution of a second order half-linear differential equation (Q1938295)

Summary: This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second-order half-linear differential equation \[ (p(x)\Phi(y'))' + q(x)\Phi(y) = 0 \] with \(p(x)\) and \(q(x)\) piecewise continuous and \(p(x) > 0\), \(\Phi(t) = |t|^{r-2}t\) and \(r\) being real such that \(r > 1\). Lower bounds (i.e., Lyapunov inequalities) for this distance are also provided and compared with other methods.