Branched Circle Packings and Discrete Blaschke Products

In this paper we introduce the notion of discrete Blaschke products via circle packing. We first establish necessary and sufficient conditions for the existence of finite branched circle packings. Next, discrete Blaschke products are defined as circle packing maps from univalent circle packings that properly fill <!-- MATH $D = \{ z:\left| z \right| < 1\}$ --> <img width="154" height="41" align="MIDDLE" border="0" src="images/img3.gif" alt="$ D = \{ z:\left\vert z \right\vert < 1\} $"> to the corresponding branched circle packings that properly cover . It is verified that such maps have all geometric properties of their classical counterparts. Finally, we show that any classical finite Blaschke product can be approximated uniformly on compacta of by discrete ones.