Strategy for inheriting 2 (long term) underpaying tenants?

[David VanWinkle]

I'm about to close on my first multifamily property in 2 weeks! I am buying the property because it is a growing area and current rents in 2 of the 3 units are much lower than they should be, but the catch is that all 3 tenants have been living at the property for 4+ years and don't want to leave. All are month to month on their leases. The main floor is Section 8 and very close to market rate so I plan to keep them. The other two are both playing between $300 and $400 under market value for each unit. 

What do you think is the best strategy is to handle this situation? My plan was to give them until the end of February at their current rent before signing new leases on March 1st with increased rent if they want to stay. Is this being being too lenient or harsh? What could go wrong with my plan that I'm not thinking about because of my inexperience? 

Thanks for your help!

[Rivy S.]

@David VanWinkle I have the exact same situation.  I have to do about 15k in renovations before I can raise rent, but should be able to raise by about 300 per side (duplex).  However I definitely don't want a winter time vacancy, both because of marketplace consideration and also vacancy cost (have to winterize property, make sure pipes are running, pay for heat, etc)  so I'm going to let them stay at the current rate until April or May, and then raise it probably 150 if they stay as is, otherwise they'll leave and I'll do renovation and hopefully have it ready to re-lease in time for the more competitive spring/summer market. 
Do you have to do any work to bring the rents up? Or can it be rented for 3-400 more as is?