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<p/> <p>Using inequalities for a certain function appearing in the half-linear version of Picone's identity, we show that oscillatory properties of the half-linear second-order differential equation <inline-formula><graphic file="1029-242X-2005-658372-i1.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-658372-i2.gif"/></inline-formula>, can be investigated via oscillatory properties of a certain associated second-order <it>linear</it> differential equation. This linear equation plays the role of a Sturmian majorant, in a certain sense, if <inline-formula><graphic file="1029-242X-2005-658372-i3.gif"/></inline-formula>, and the role of a minorant if <inline-formula><graphic file="1029-242X-2005-658372-i4.gif"/></inline-formula>.</p>