A generalization of Liapunov's inequality
We consider the boundary value problem \(x''+p(t)x=0\), \(x(0)=x(a)=0\), \(x\not\equiv 0\), and generalize Lyapunov's classical inequality to \(\int^{a}_{0}| p(t)|^{\gamma}dt\geq k_{\gamma}/a^{2\gamma -1}\) for the cases \(\gamma =2,3\), and 3/2. We found the best possible \(k_{\gamma}\) and the optimizing coefficient functions p(t) and corresponding solutions x(t). These results are applied in demonstrating the stability of Hill's equation under suitable conditions.
scientific article; zbMATH DE number 3889924