Estimates of essential norms of weighted composition operator from Bloch type spaces to Zygmund type spaces

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Let u be a holomorphic function and φ a holomorphic self-map of the open unit disk D in the complex plane. We give some new characterizations for the boundedness of the weighted composition operators uCφ from Bloch type spaces to Zygmund type spaces in D in terms of u, φ, their derivatives and the n-th power φⁿ of φ. Moreover, we obtain some similar estimates for their essential norms. From which the sufficient and necessary conditions of compactness of the operators uCφ follows immediately.