We prove some coupled fixed point theorems for mappings satisfying different contractive conditions on complete cone metric spaces.

We investigate the boundedness of weighted composition operator uCφ mapping the Zygmund-type space Z α into the Bloch-type space B β . Then we give essential norm estimates of such an operator in terms of u and φ .

We study boundedness of weighted differentiation composition operators Dk?,u between Zygmund type spaces Z? and Bloch type spaces ?. We also give essential norm estimates of such operators in different cases of k ? N and 0 < ?,? < ?. Applying our essential norm estimates, we get necessary and sufficient conditions for the compactness of these operators.

We characterize boundedness and compactness of the classical Volterra operator Tg : H ∞ vα → H ∞ induced by a univalent function g for standard weights vα with 0 ≤ α < 1, partly answering an open problem posed by A. Anderson, M. Jovovic and W. Smith. We also study boundedness, compactness and weak compactness of the generalized Volterra operator T ϕ g mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.2010 Mathematics Subject Classification. Primary 47B38, Secondary 46B50.

Abstract. For an analytic selfmap ϕ of the open unit disc D and an analytic function g on D , the Li-Stević integral type operator C g ϕ is given byWe give essential norm estimates of the operator between Zygmund type spaces. We also apply our approach in the case of Bloch type spaces.Mathematics subject classification (2010): 47B38, 30H30.