On some a priori majorant of eigenvalues of Sturm--Liouville problems

Abstract

Let $M_\gamma$ be precise a priori majorant of first eigenvalues of Sturm--Liouville problems $-y"+qy=\lambda y,\quad y(0)=y(1)=0$, where $q\leqslant 0$ and $\int_0^1 |q|^\gamma\,dx=1$, $\gamma\in (0,1/2)$. It is shown that the inequality $M_\gamma<\pi^2$ is true.