On inner functions with and derivatives in the unit ball of

Abstract: Let be the unit ball in . If f is a bounded holomorphic function, we say that f is inner provided that where is the unit sphere and σ is normalized surface measure on . If and then denotes the weighted Bergman space of all holomorphic functions weighted by . For , set and if let denote the usual Hardy space of holomorphic functions on the ball. In this paper, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, , will be considered in the spaces for and will be related to membership in weighted Dirichlet spaces, weighted Bergman spaces for , and to the spaces for . Moreover, it will be shown that if f is an inner function, , and either , , or then f must be constant. [Copyright &y& Elsevier]