An estinatiion for the essential norm of Compositiion operators acting on Bloch-type spaces

Let \mu be any weight function defined on the unit disk $\mathbb{D}$ and let $\varphi$ be an analytic self-map of $\mathbb{D}$. In the present paper we show that the essential norm of composition operator $C_{\varphi }$ mapping from the $\alpha$-Bloch space, with $\alpha >0$, to $\mu$-Bloch space ${\mathcal{B}}^{\mu }$ is comparable to $$\underset{\mathrm{\mid }a\mathrm{\mid }\to 1^{-}}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\parallel }{\sigma }_a\circ \varphi {\mathrm{\parallel }}_{{\mathcal{B}}^{\mu }},$$ where, for $a\in \mathbb{D}$, ${\sigma }_a$ is a certain special function in $\alpha$-Bloch space.