Convexity of the first eigenfunctions of Sturm--Liouville eigenvalue problems. (Q1426078)

The author deals with the eigenvalue problem \((py')'(x)-\rho y(x)+\mu q(x)=0\), \(y(a)=y(b)=0\), where \(p,\rho,q\) are continuous in \([a,b]\) with \(p,q\) positive, \(\rho\) nonnegative and \(\rho/q\not\equiv\)\,constant. By means of a variational method, he investigates the (positive) eigenfunction \(u\) corresponding to the first eigenvalue. Conditions are given guaranteeing that this eigenfunction is convex somewhere or that the number of its peaks is greater than one. Criteria for \(u\) being concave are also given.