A lower estimate of a minival eigenvalue of a Sturm-Liuville problem with boundary conditions of the second type

We prove that the infinum $m_\gamma$ of minimal eigenvalues fot the problem
$ -y''+qy=\lambda y$, $y'(0)=y'(1)=0$,
can be attained. Here non-negative potential $q\in L_1[0,1]$ runs through the unit sphere in the space $L_\gamma[0,1]$, where $\gamma\in (0,1)$. We prove also the equality $m_\gamma=1$ for $\gamma\leqslant 1-2\pi^{2}$ and the inequality $m_\gamma<1$in the opposite case