What is this
This is bougainvillea. It may be small in these photos, but it can grow to massive sizes. SInce you are in Sarasota, make a trip to Fort Myers to visit the Edison and Ford Winter Estates--it's an amazing place that you don't want to miss. The reason I bring this up is that the largest bougainvillea in the world grows there, and it's the size of a small house. That being said, bougainvilleas do well if frequently pruned. Here's some additional information about it.