Management Companies for an Apartment Unit Rental

Hello! I’m moving out of NYC and I’ve been looking for a company to rent out my apartment. More than that, I want them to manage that rental while I’m gone. This would include everything from listing it, vetting renters, handling security deposit, handling any problems with the apartment or with the tenant….all without calling me. I found only one potential company, but I’m having a bit of a problem getting terms from them (not a great sign, really). Isn’t this a big business in NY? All the companies I seem to find only do whole apartment buildings, others only do whole apartment buildings _they own_. Am I searching for the wrong terms? If anyone has any suggestions of companies or even suggestions for finding companies such as this I’d really appreciate it! Cheers! –Alec