A non-commutative Julia Inequality
  We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in $\mathbb{C}^d$. We look at differentiability at a boundary point for functions that have a certain regularity there. 
John E. McCarthy and James E. Pascoe
math.CV math.FA