  We prove the nonlinear inviscid damping for a class of monotone shear flows
with non-constant background density for the two-dimensional ideal
inhomogeneous fluids in $\mathbb{T}\times [0,1]$ when the initial perturbation
is in Gevrey-$\frac{1}{s}$ ($\frac{1}{2}<s<1$) class with compact support.
