A note on plurisubharmonic defining functions in C^2
  Let D be a smoothly bounded domain in C^2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis. 
John Erik Fornaess (University of Michigan), Anne-Katrin Herbig   (University of Michigan)