Does the bandwidth of an NBFM modulation change depending with the modulation index?
I assume that your quantity beta is the maximum momentary frequency offset per the maximum baseband signal frequency fm. That beta is called &quot;modulation index&quot; and it exists in the approximated formula of the needed transmission bandwidth which is the uppermost of your bandwidth formula options.
In narrow band FM the modulation index is much smaller than 1. It makes the needed transmission bandwidth same as for AM and your both bandwidth formula options make quite the same result.
Use the 2nd option if the beta is say 0,1  or even less. But if you know the beta you can as well use the 1st option because it's valid for small and large beta.