On the estimates for the minimum eigenvalue of the Sturm-Liouville problem with integral condition (Q941997)

Consider the Sturm-Liouville problem \[ y''-Q(x)y(x)+\lambda y(x)=0, \quad y(0)=y(1)=0, \] where \(Q\) is a nonnegative bounded function on \([0,1]\) satisfying the condition \(\int_0^1Q^{\alpha}(x)\,dx=1\), \(\alpha\neq0\). A detailed study of the possible range for the minimal eigenvalue \(\lambda_1\) of this problem for various values of \(\alpha\) is given.