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 $A_gamma={qin L_1[0,1]: qgeq 0, int_0^1 q^gamma,dx=1}$, where $gammain (0,1)$, 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 $m_gamma=inf_{qin A_gamma}lambda_1(q)$. The equality $m_gamma=1$ for $gammaleq 1-2pi^{-2}$ and the inequality $m_gamma<1$ for $gamma>1-2pi^{-2}$ are also obtained.