In my short tenure as a RE investor, inherited tenants have been my biggest "X factor" when it comes to trying to stabilize a property with regards to cash flow and return.
My advice to you would be professional but firm with them, as the initial interactions will set the perception in their minds whether you are "fly by night" pushover landlord, or someone who is running a business. Did they once have a security deposit and it was used up with late fees, etc.? I think their response to your request for a security deposit will speak volumes as to how they are as a tenant.
I would definitely get a security deposit and as a courtesy, incrementally increase their rent over time (you decide an appropriate pace) until it reaches what you think is fair market rent based on the unit's condition. They will shop around and realize you are still giving them a fair price.
I can understand your need to be "fair" but you also have to protect your investment. Be careful about setting precedents that are unhealthy for your business relationship with your tenants.
Good luck!