Third party battery makers can claim what ever they want. There is no way that you will get more power out of a battery using the same chemistry without increasing the size and there is NO additional room inside the iPhone case.
Just think about it, if it could be done, Apple would have done it. Do not be suckered into these lies!
you can buy a powerbank for charging on the go.. or an iphone case with backup battery upto 5,000mah..