Small size, internet(browsing) speed, and most important battery life

Well, the HTC One has a much fast processor so if the general speed of the phone is important to you, that's something to keep in mind. The One's screen is 4.3 inches, so it's a little bigger than what  you want, but that may be useful for web browsing if that's your key phone activity, especially since the resolution is also a bit higher on the One than the Desire. Between those two phones I'd probably go for the One since it just generally seems like it has more features, including a better camera. Both phones have similar battery capacity, which is decent but not exceptional. If you want another phone option that has incredible battery and is still small (4.3 inches), take a look at Droid Razr Maxx: http://sortable.com/phone/Motorola-Droid-RAZR-MAXX

this phone is good for casual users. It has better battery life as compared to one s.