Covering results and perturbed Roper--Suffridge operators
  This work is devoted to the advanced study of Roper--Suffridge type extension operators. For a given non-normalized spirallike function (with respect to an interior or boundary point) on the open unit disk of the complex plane, we construct perturbed extension operators in a certain class of Banach spaces and prove that these operators preserve the spirallikeness property. In addition, we present an extension operator for semigroup generators.   We use a new geometric approach based on the connection between spirallike mappings and one-parameter continuous semigroups. It turns out that the new one-dimensional covering results established below are crucial for our investigation. 
Mark Elin and Marina Levenshtein
math.CV