generalized weighted composition operators from logarithmic bloch type spaces to n'th weighted type spaces

Let h(d) denote the space of analytic functions on the open unit disc d. for a weight μ and a nonnegative integer n, the n'th weighted type space w^(n)μ is the space of all f∈h(d) such that supz∈dμ(z)∣∣f^(n)(z)∣∣<∞. endowed with the norm∥f∥w(n)μ=∑j=0n−1∣∣f(j)(0)∣∣+supz∈dμ(z)∣∣f(n)(z)∣∣, the n'th weighted type space is a banach space. in this paper, we characterize the boundedness of generalized weighted composition operators d^mφ,u from logarithmic bloch type spaces b^αlog^β to n'th weighted type spaces w^(n)μ, where u and φ are analytic functions on d and φ(d)⊆d. we also provide an estimation for the essential norm of these operators.