Applications of Theory of Differential Subordination for Functions with   Fixed Initial Coefficient to Univalent Functions
  By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions is evident in the results obtained. 
Sumit Nagpal, V. Ravichandran
math.CV