Denote by B the class of transcendental meromorphic functions for which the set of finite critical and asymptotic values is bounded, and by S that class of functions for which this set is finite. We give some conditions on transcendental deficient functions of members of the class B, including an improvement of a result of Langley and Zheng. This leads to corresponding results for the class S. It is also proved that no derivative of a finite lower order periodic function has non-zero finite deficient values. We show by example that the finite lower order condition is necessary here
