A NEW ESSENTIAL NORM ESTIMATE OF COMPOSITION OPERATORS FROM α-BLOCH SPACES INTO μ-BLOCH SPACES

Let mu be any weight function defined on the unit disk D and let phi be an analytic self-map of D. In this paper, we show that the essential norm of composition operator C-phi mapping from the alpha-Bloch space, with alpha > 0, to mu-Bloch space B-mu is comparable to lim sup(vertical bar a vertical bar -> 1-) parallel to sigma a circle phi parallel to(B mu), where, for a is an element of D, sa is a certain special function in alpha-Bloch space. As a consequence of our estimate, we extend recent results, about the compactness of composition operators, due to Tjani in [Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355(11) (2003) 4683-4698] and Malave Ramirez and Ramos-Fernandez in [On a criterion for continuity and compactness of composition operators acting on alpha-Bloch spaces, C. R. Math. Acad. Sci. Paris 351 (2013) 23-26, http://dx.doi.org/10.1016/j.crma.2012.11.013].