In their beginning, the Ottomans had quite close, though not always peaceful, relations with Byzantium. Many fortified localities and their rulers simply switched sides from when Byzantine power waned, and Orhan I was actually the son-in-law of the Byzantine emperor. As the Ottomans were essentially a militant ghazi state, Christianization is out of the question. However until Selim I's conquest of the Mamluk Sultanate, the (slim) majority of the Ottomans' subjects was probably Christian. Until some time after 1453, their center of power was even in the Balkans, rather than Anatolia, and Mehmed II looked more to the west than the east. If other Turkic states like the Karamanids or the Safavids further east are more successful in challenging Ottoman domination of eastern Anatolia, a Balkan-centered Ottoman state will be very different from the Arab- and Pesian-influenced empire of the 16th century and on. However, in this timeframe it is actually more likely that it will become Slavicized rather than Hellenized: IIRC, the predominant language in the palace was Serbo-Croatian...