A WEIGHTED COMPOSITION OPERATOR ON THE LOGARITHMIC BLOCH SPACE

We characterize the boundedness and compactness of the weighted composition operator on the logarithmic Bloch space LB = {f is an element of H(D) : sup(D)(1 - vertical bar z vertical bar(2))ln(2/1-vertical bar z vertical bar)vertical bar f'(z)vertical bar < +infinity} and the little logarithmic Bloch space CB. The results generalize the known corresponding results on the composition operator and the pointwise multiplier on the logarithmic Bloch space LB and the little logarithmic Bloch space LB(0).