In this paper we revisit once again, see Shub and Sullivan (Ergod. Th. & Dynam. Sys.5 (1985), 285–289), a family of expanding circle endomorphisms. We consider a family $\{B_\theta\}$ of Blaschke products acting on the unit circle, $\mathbb{T}$, in the complex plane obtained by composing a given Blaschke product $B$ with the rotations about zero given by multiplication by $\theta \in \mathbb{T}$. While the initial map $B$ may have a fixed sink on $\mathbb{T}$, there is always an open set of $\theta$ for which $B_\theta$ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of $\int \ln|B'(z)|\,{\it dz}$.