I have the following inequality $$\frac{(1+x(c-1))^2}{(1+x^{\frac{2}{m}}(c-1))^m}\geq c^{2-m}$$ where $0<x<1$, $c$ is some positive integer $\geq 1$ and $m>2$. I tried to plot the function for different values and it seems that the inequality is true however I do not know whether it is true for all possible values of the parameters. Any help in this regard will be much appreciated. Thanks in advance.