OnePlus One ($229)
Easily the most affordable phone on the list, you still get a 5.5-inch display with 1080 x 1920 resolution, and Android 4.2.2 OS. The only major problem from a games perspective is the lack of expandable storage options, but for the price tag this is a fine choice.
Choosing between these phones is hard, but the LG G3 has unbeatable specs for those who can afford it.
Source: Black Girl Nerds
The Note 4 is an excellent gaming phone. I haven't found the weight to be an issue, but then again I don't spend hours holding my phone lol