A note on disfocality (Q1106379)

Consider the initial value problem \[ (*):\quad (g(s)p'(s))'+f(s)p(s)=0,\quad p(0)=1,\quad p'(0)=0,\quad s\in [0,\infty[, \] where f and g are given continuous functions and \(g(s)>0\) for all s. If the solution p has a zero, then it has a smallest zero, which is denoted by \(\xi(*)\). The following assertions are proved: Let f(s)g(s)\(\leq k\) 2 for some \(k\in {\mathbb{R}}\) and all \(s\in [0,c]\). 1. If Q(s)\(\leq \pi /4\) for all \(s\in [0,c]\), where Q is defined in terms of f,g and k, then the initial value problem (*) has \(\xi(*)>c\). 2. If \(\int^{c}_{0}k^{-1}f(t)dt\geq \pi/2,\) then \(\xi(*)\leq c\).