Oscillatory properties of the half-linear second-order differential equation
are investigated, and these properties are compared with the oscillatory behavior of a certain associated linear second-order differential equation. At the first, essentials of the half-linear oscillation theory are recalled and also some auxiliary technical results are presented. The main results of the paper are statements where oscillatory properties of (N) are compared with those of a certain associated linear equation. As a corollary, a conjecture posed by the first author in the paper [O. Došlý, Perturbations of the half-linear Euler-Weber differential equations, J. Math. Anal. Appl. 323, 426–440 (2006; Zbl 1107.34030)] is proved and some new research are formulated.