On an a priori majorant of the least eigenvalues of the Sturm–Liouville problem

We examine the exact a priori majorant $M_\gamma\rightleftharpoons\sup\limits_{q\in A_\gamma}\lambda_0(q)$ of the least eigenvalue of the Sturm–Liouville problem $-y''+qy=\lambda y$, $y(0)=y(1)=0$, with a potential $q\in C[0,1]$ of the class $A_\gamma$ determined by the conditions $q\le 0$ and $\int\limits_0^1|q|^\gamma dx=1$, where $\gamma\in(0,1/2)$. For this majorant, we prove the strict estimate $M_\gamma<\pi^2$. The last estimate was known earlier in the case where $\gamma<1/3$.