### A role of the coefficient of the differential term in qualitative theory of half-linear equations

The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation ${\left(r\left(t\right)\Phi \left({y}^{\Delta}\right)\right)}^{\Delta}+p\left(t\right)\Phi \left({y}^{\sigma}\right)=0$, where $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ with $\alpha >1$. We discuss sign and smoothness conditions posed on $r$, (non)availability of some transformations, and mainly we show how the behavior of $r$, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati...