We obtain some simple criteria for the boundedness and compactness of the product of differentiation and composition operator CφD on Bloch type spaces.

The main purpose of this paper is to discuss Hardy type spaces, Bloch spaces and the composition operators complex-valued harmonic functions. We first establish a sharp estimate Lipschitz continuity functions in with respect pseudo-hyperbolic metric, which gives an answer open problem. Then some classes on will be investigated. obtained results improve extend corresponding known results.

The boundedness of the weighted differentiation composition operator from the logarithmic Bloch space to the weighted-type space is characterized in terms of the sequence (zn)n∈N0 . An asymptotic estimate of the essential norm of the operator is also given in terms of the sequence, which gives a characterization for the compactness of the operator.