I won't speak to Persia, but I remember that Chris Wickham's "The Inheritance of Rome," which despite focusing on Europe discusses the Umayyads and Abbasids extensively, claiming that the 10th century was the rough tipping point for Egypt and Syria becoming more Muslim than Christian. The Umayyads were not big on encouraging conversion (converts no longer pay the jizya, or tax on nonbelievers, after all!).

From the Holy Mountain is a fantastic book, it is a travelogue by William Dalrymple (who is one of my favorite historians) of his trip through the former lands of Eastern Christendom around 1990 looking for what remained of it today, and he does a fantastic job describing the arc of Christianity in various places from Anatolia to Syria to Lebanon to Palestine to Egypt.