How to compute the main lobe width of generalized Hamming windows (i.e. Hann, Hamming, etc.)?
(This used to be a partial answer, but I have edited it to include the requested result at the bottom.)

First of all, you can view a Hamming family window as the sum of a Hann window and a rectangular window: If $W_N$ is the fourier transform of a Hann window, then $W_H(\omega)=(\alpha-2\beta)W_R(\omega)+4\beta W_N(\omega)$. If, for simplicity, we assume unit amplitude ($\beta=\frac{1-\alpha}{2}$), then $x:=4\beta=2-2\alpha$ determines the weighting of the Hann window against the rectangular window ($\alpha-2\beta=1-x$ in that case). In the general case, you just need to factor out the coefficient $\alpha+2\beta$ first and let $x:=\frac{4\beta}{\alpha+2\beta}$.

For example, here is a plot of $W_H$ (green) and its two components for $x=0.5$ (assuming $M=2$, I guess, so that $W_R(\omega)=\frac{\sin \omega}{\omega}$):



Obviously, giving the Hann window more weight by increasing $x$ also increases the main lobe width. However, plotting this relationship reveals an interesting effect:



Since $W_R(2\Omega_M)=W_N(2\Omega_M)=0$, as soon as the Hann window component completely cancels out the first side lobe of the rectangular window component, the main lobe width stays constant at $4\Omega_M$. By calculating the derivatives at this point, it is easy to find out that this is the case for $x\geq\frac{6}{7}$ (equivalently, $\alpha\leq\frac{4}{7}$ in the unit amplitude case, or $3\alpha\leq 8\beta$ in the general case).

Unfortunately, from the plot, I haven't successfully guessed anything about the remaining $\frac{6}{7}$. ;-) Hopefully someone else will figure this out, if it is possible at all.

Edit: Don't know why I didn't go for the simplest route first. Maxima (open-source CAS) gives $\Omega_M \sqrt{2\frac{x-2}{x-1}}$ for the remaining part. So, in general, the main lobe width is:

$\begin{cases}
2\Omega_M \sqrt{\frac{\alpha}{\alpha-2\beta}} &amp; \text{if } 8\beta&lt;3\alpha\\
4\Omega_M &amp; \text{otherwise}
\end{cases}$