Re: Pureview on Microsoft phones.... don't hold your breath it is NOT the same!
The OS use on a modern smartphone is so small, it's not really relevant. The main point of designing custom OSs for mobile or even just for smartphones was that, simply put, early smartphones were not capable of running a PC-class operating system. The same issues were true for PDAs of the day, which is of course why so many early smartphones were based on PDA operating systems (and in fact, the Windows 7 Phones still are... they're running WinCE, not WinNT).
Back when you had a monochrome, transflective LCD screen, a CPU, a tiny bit of memory, and a phone modem as pretty much the whole phone, the OS could well have been a significant part of the power consumption. Not anymore, and not for a long time. The main impact from the OS on battery life isn't even the OS itself, it's the hardware's ability to manage power (clock speed control, shutting off unused resources, etc) and the OS's power management support for that, too. When it comes to power hogs, it's primarily the screen, if you have a good signal, or the radio (3G, 4G, doesn't much matter) when you don't have a good signal.
In the case of Android, yeah, that's Linux. UNIX began as an OS for minicomputers in fact, not microcomputers. But that was the 1960s, and nothing particularly relevant to today. Linux is fully capable of function on modern low power microprocessors, and under the Android project, modern power management was brought in as well (that was part of the original Linux fork Android took, which was merged back earlier this year). Linux is also one of the most popular embedded operating systems today -- it's seen plenty of use in low power applications, even before Android.
iOS is derived from MacOS, of course, which is based on CMU's Mach kernel and BSD UNIX -- no Linux in there. On MacOS, Apple has delivered some of the best power management on any OS... MacOS PCs typically run longer than most similar Windows PCs, even these days given identical hardware... most of that's due to MacOS's well tuned power management. Which is also on the iPhone.
Your battery life depends quite a bit on what you're making those batteries do. Certainly, if your OS is doing more things, it's going to use more battery power. If it's running on higher class processors, it may (though not always) use more power. Larger and higher definition screens also suck down much more power... an example: the current 9.7" iPad screen takes 2.5x as much power as last year's identically sized iPad screen. The only differences are due to pixel density. SymbianOS phones have typically had much lower resolution screens than their iOS or Android counterparts, which is a significant contributer to battery life. They also tend to have slower CPUs and fewer hardware features. That's not a SymbianOS limit, that's just the reality of Nokia not focusing on SymbianOS phones for the last several years, as they try to kill the market.