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Note on some consequences of a problem by Dellac. (English) Zbl 1485.34080

Summary: By using a solvability method for differential equations, we present an elegant and straightforward/direct proof of the following slight generalization of a classical result: Let \(\omega >0\) and \(f\) be a two times continuously-differentiable function on the closed non-degenerated interval \([a,b]\) such that \(f(a)=f(b)=0\), \(f(x)>0, x\in (a,b)\), and \[ f''(x)+\omega^2f(x)>0,\quad x\in (a,b). \] Then the following inequality holds \(b-a>\pi /\omega \), which is strict. For the case when \(f\) is continuous on \([a,b]\) and two times differentiable function on \((a,b)\), such that \(f(a)=f(b)=0\), \(f(x)>0\), \(x\in (a,b)\), and \[ \bigl(p(x)f'(x)\bigr)'+f(x)>0,\quad x\in (a,b), \] where \(p\) is a continuous function on \([a,b]\), differentiable on \((a,b)\), and such that \[ p(x)\geq \frac{1}{\omega^2}\quad \text{for }x\in [a,b], \] for some \(\omega >0\), we show by using the methods of differential calculus that then it holds \(b-a\geq \pi /\omega\).

MSC:

34A40 Differential inequalities involving functions of a single real variable
34A30 Linear ordinary differential equations and systems
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