Instead of making all these arguments, let's have some data points.
Samsung makes smartphones of various sizes and of the same generation (all released around mid-late last year):
- Samsung Galaxy S3 Mini (4")
- Samsung Galaxy S3 (4.8")
- Samsung Galaxy Note II (5.5")
GSMarena has done battery tests on all 3. I'll ignore the talk time one as it does not involve the screens being on:
Nothing conclusive, but the trend is that larger screens have better battery life although the anomaly here is that the S3 has worse battery life than the s3 mini on web browsing.
The above definitely does not support the argument that smaller screen=battery battery life, at least for Samsung phones.
First you need to normalize the results for battery size, but you also aren't controlling for the SoC, screen type and resolution, and other possible differences that could impact the results.
The S3 mini has a 5.6 Whr battery, the S3 at 7.8 Whr battery, and the Note 2 an 11.78 Whr battery. So the normalized results should be:
Browsing:
S3 Mini: 1.36 hours / Whr
S3: .83 hours / Whr
Note 2: .75 hours / Whr
Video:
S3 Mini: 1.38 hours / Whr
S3: 1.21 hours / Whr
Note 2: .97 hours / Whr
However, the S3 mini is only using a dual core chip at roughly 2/3 the clock speed, so it's going to draw less power in general compared to either the S3 or Note 2; and the S3 uses a Krait, whereas the Note 2 uses an Exynos with slightly faster clocks. We also need to factor in the screens. The S3 and S3 mini are pentile, so they're going to draw less power. I'm also unsure how much brightness differences between the screens factor in, so that might also have an effect.
The best test is probably the video test as it doesn't need to consider the radio or baseband chip differences, and the SoC isn't an issue as all of them should be using dedicated decoding hardware and there shouldn't be too much of a difference there.
In that case, a larger screen is less efficient assuming all other variables are being controlled for. What would really prove this point is if any of Samsung's tablets used SAMOLED screens, in which case we'd have another good data point, but unfortunately they don't.