Let $$f(x)=\left(1-\frac{2x}{x+c}\right)^{-n}-\left(1+\frac{x}{c}\right)^{2n}$$ and $$g(x)=\left(1-\frac{2x}{c}\right)^{-n}-\left(1-\frac{x}{c}\right)^{-2n}$$ where $c>0$ and $n>0$ are constants.
I am wondering if $f(x)\leq g(x)$ for $0\leq x <c/2$ (I am actually interested in small positive $x$).
Clearly, this holds with equality when $x=0$ and I think the inequality holds for $0< x <c/2$. This is based on the numerical evaluations as well as the series expansion in Mathematica of $h(x)=g(x)-f(x)$ which yields:
$$h(x)=\frac{2 n x^3}{c^3}+\frac{3 \left(2 n^2+n\right) x^4}{c^4}+\frac{2 \left(4 n^3+5 n^2+3 n\right) x^5}{c^5}+\ldots$$
However, I am having trouble actually proving this. Any help?