Tangential Lipschitz Gain for Holomorphic Functions on Domains of Finite   Type
  Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining function near points of finite type in the boundary. 
Sivaguru Ravisankar
math.CV