Harry Turtledove wrote an AH collection of short stories a while back that addresses this very issue. It was published in book form under the title, "Agent of Byzantium".
The premise of the book, as you might surmise from the title, is that the Turkish conquest of 1453 never happened, because for undisclosed reasons (possibly because the Empire & the Orthodox religion penetrated deeper into Arabia than in OTL), Mohammed early in life was converted to Orthodoxy, became one of the holiest of the Greek Orthodox saints, & spent the rest of his life composing hymns to Jesus. Yeah-- literally.
Saint Mahmud, as I believe he was called, was the favorite saint of the protagonist of the series, whose name I do not recall. (Basil Argyros. That memory of mine. Just as I was posting this is came to me. I had to go back & edit it in.)
No jihad. No conquest of the Christian Middle East or North Africa. No Islamicization of the Turks, who become good Christians & a bulwark of the Empire against the Mongols. This enables the Byzantines to focus their efforts at regaining their hegemony in Europe, at which they do a middling good job.
Pretty nice world, as AH speculation goes.