On a lower a priori estimate of minimal eigenvalue of one Sturm-Liouville problem with second-type boundary conditions

It is proved that for class $A_\gamma=\{q\in L_1[0,1]: q\geq 0, \int_0^1 q^\gamma\,dx=1\}$, where $\gamma\in (0,1)$, there exists a potential $q_*\in A_\gamma$ such that minimal eigenvalue $\lambda_1(q_*)$ of boundary problem $$ -y"+q_*y=\lambda y, y'(0)=y'(1)=0 $$ is equal to $m_\gamma=\inf_{q\in A_\gamma}\lambda_1(q)$. The equality $m_\gamma=1$ for $\gamma\leq 1-2\pi^{-2}$ and the inequality $m_\gamma<1$ for $\gamma>1-2\pi^{-2}$ are also obtained.