We present criteria of Hille-Nehari-type for the linear dynamic equation ( r ( t ) y Δ ) Δ + p ( t ) y σ = 0 , that is, the criteria in terms of the limit behavior of ( ∫ a t 1 / r ( s ) Δ s ) ∫ t ∞ p ( s ) Δ s as t → ∞ . As a particular important case, we get that there is a (sharp) critical constant in those criteria which belongs to the interval [ 0 , 1 / 4 ] , and its value depends on the graininess μ and the coefficient r . Also we offer some applications, for example, criteria for strong (non-) oscillation and Kneser-type criteria, comparison with existing results (our theorems turn out to be new even in the discrete case as well as in many other situations), and comments with examples.