The authors prove that the equation \[ [x(t)-x(t-\tau)]'+q(t)f[x(t-\sigma)]=0,\quad t\geq 0 \] has an eventually positive (positive for all large \(t\)) solution iff such a solution satisfies the differential inequality \[ [x(t)-x(t-\tau)]'+q(t)f[x(t-\sigma)]\leq 0,\quad t\geq 0. \] This equation for a small \(\tau\) is related to the so-called Emden-Fowler equation \(x''(y)+q(t)x^{\gamma}(t)=0\). Similar results are proved for analogous equations.