On the upper estimates for the first eigenvalue of a Sturm–Liouville problem with a weighted integral condition

In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the condition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm–Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved.