  For $n$-dimensional weighted Riemannian manifolds, lower
$m$-Bakry-\'{E}mery-Ricci curvature bounds with $\varepsilon$-range, introduced
by Lu-Minguzzi-Ohta, integrate constant lower bounds and certain variable lower
bounds in terms of weight functions. In this paper, we prove a Cheng type
inequality and a local Sobolev inequality under lower $m$-Bakry-\'{E}mery-Ricci
curvature bounds with $\varepsilon$-range. These generalize those inequalities
under constant curvature bounds for $m \in (n,\infty)$ to
$m\in(-\infty,1]\cup\{\infty\}$.
