We obtain comparison theorems for the second-order half-linear
dynamic equation $\big[r(t)\Phi \big(y^{\Delta}\big)\big]^{\Delta}+p(t)\Phi\big(y\sig\big)=0,$ , where
$\Phi(x)=|x|^{\alpha-1}\mathrm{sgn} x$ with $\alpha>1$ . In particular,
it is shown that the
nonoscillation of the previous dynamic equation is preserved if
we multiply the coefficient $p(t)$ by a suitable function $q(t)$
and lower the exponent $\alpha$ in the nonlinearity $\Phi$ , under
certain assumptions. Moreover, we give a generalization of
Hille-Wintner comparison theorem. In addition to the aspect of
unification and extension, our theorems provide some new results
even in the continuous and the discrete case.