It may sound back to front, but the airport you are flying into is a good first filter here. Belek is close to Antalya Airport (AYT): about 33 km, a drive of around 40 minutes. Alanya is much further from Antalya, about 125 km or roughly two hours, but only about 40 km, around 50 minutes, from Gazipaşa-Alanya Airport (GZP).
So if your tickets are into Antalya and you want the shortest arrival, Belek has the edge. If you are landing at Gazipaşa, Alanya is the quick one. Plenty of people still choose Alanya with an Antalya flight and simply accept a longer first afternoon. Prices are shown live on the route pages: Antalya Airport to Belek, Antalya Airport to Alanya and Gazipaşa to Alanya.