In my research I've stumbled upon a certain class of functions. I would like to show they are all Lipschitz continuous, but I am stuck. The function class is
$$f_{\omega}(x) = \frac{1}{(1 + e^x)([1 + e^{-x}]^{\omega} - 1)}, \quad -\infty < x < \infty, \,\, 0 < \omega < 1.$$
I would like to that $f_{\omega}$ is Lipschitz continuous for any $w \in (0, 1)$.
I have made plots of $f_{\omega}$ and $f'_{\omega}$ for several values of $\omega$. For example, the graph of $f_{0.7}$ is
and the graph of $f'_{0.7}$ is
Based on these graphs I am confident that $f_{\omega}$ is in fact Lipschitz. I am aware that one technique would be to show that the derivative is bounded, however, the derivative is quite unfriendly, taking the form below:
$$f'_{\omega} (x) = \frac{\omega (-(e^x ((1 + e^{-x}) - 1)) + \omega(1 + e^{-x})^{\omega}}{(1 + e^x)^2 ((1 + e^{-x})^{\omega} - 1)^2}$$
Quite frankly I can't imagine how to show that this is bounded. I have racked my brain trying to think of other functions that bound it from above but I am at a loss. Any help would be appreciated!