Lower bounds on the projective heights of algebraic points

If ${\alpha }_1,\dots ,{\alpha }_r$ are algebraic numbers such that $$N=\sum _{i=1}^r{\alpha }_i\mathrm{\ne }\sum _{i=1}^r{\alpha }_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum _{i=1}^r\log h({\alpha }_i)$$ where $h$ denotes the Weil Height. We will extend this result to allow $N$ to be any totally real algebraic number. Our generalization includes a consequence of a theorem of Schinzel which bounds the height of a totally real algebraic integer.