Hi, I was having similar problems with battery drain. After checking around on the Internet I found two things which helped improve my battery life significantly.
The first was to not use the music player widget. If you check your battery in "Settings" and one of the main sources of battery drain is "mediaserver", then it's probably the use of the music widget.
The second was after the 1.10 update to do a "Factory data reset" in "Backup and reset" under "Settings".
Doing these two things has improved battery life for me quite a lot, but I still think the ZX1 has an idle drainage issue which should be solved by Sony in an update. For example, if I put my LG phone in "airplane mode" it will still be at 100% even 5 or 6 hours later. I exclusively use the ZX1 in airplane mode, but it will still drain battery, even when not being used. It would be at about 92% after about 6 hours of idle in airplane mode. Surely Sony can rectify this situation, it doesn't even make sense that it happens in the first place!