Second-order differential subordinations for analytic functions with   fixed initial coefficient
  Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several interesting examples are given. The results obtained are applied to the classes of convex and starlike functions with fixed second coefficient. 
Rosihan M. Ali, Sumit Nagpal, V. Ravichandran
math.CV