Consider buying pre-owned. Flagship phones from 2014 should be hitting the market as contracts end and you should be able to afford an S5, HTC One M8 or Sony Xperia Z3 fairly easily. All have displays around 5inches (so not too much smaller than your target, but all have resolutions at least 1080p (1440p in the S5) which is better than the 720p you would find in most new phones at this price bracket. They also have Snapdragon 801's which is faster than most phones at this price as well.