If you're able, try charging the phone with it turned off.
On a website I came across devoted to mobile battery life (tablets, cellphones, laptops) it strongly that when you first get the device to run it down until it turns off, leave it turned off, and charge it to 100% undisturbed. It also recommends doing a complete discharge (and power-off recharge) every 2 months or so, but other then that to do quick "top-up" charges when you're on the go - trying to keep the battery above 60% was the rough goal. Aparently these practices keep the "fuel gauge" (how full the battery actually is) properly calibrated and is also the best for your battery's overall lifespan (long term).
Hope this info helps. Good luck with solving your battery problems, although personally I wouldn't worry if it it would drop a point or 2 fairly quickly but then stabilize for the rest of the day. If it continues to drop fairly rapidly then there is much greater cause for concern.