Some functional inequalities under lower Bakry-Émery-Ricci curvature bounds with $\epsilon$-range

For $n$-dimensional weighted Riemannian manifolds, lower $m$-Bakry-Émery-Ricci curvature bounds with $\epsilon$-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-Émery-Ricci curvature bounds with $\epsilon$-range. These generalize those inequalities under constant curvature bounds for $m\in (n,\mathrm{\infty })$ to $m\in (-\mathrm{\infty },1]\cup \{\mathrm{\infty }\}$.