Note on some consequences of a problem by Dellac

Abstract

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 ω> 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∈ (a, b) , and f″(x)+ω2f(x)>0,x∈(a,b). Then the following inequality holds b− a> π/ ω, 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∈ (a, b) , and (p(x)f′(x))′+f(x)>0,x∈(a,b), where p is a continuous function on [a, b] , differentiable on (a, b) , and such that p(x)≥1ω2for x∈[a,b], for some ω> 0 , we show by using the methods of differential calculus that then it holds b− a≥ π/ ω.