I nearly always have my smartphone with me, but hardly ever use it, I prefer my 10" tablet. However I sometimes want to check calendar, mail, banking, weather radar and such. I use a belt holster and so the smaller the better.
At present I use an LG-D160 3.5" model which runs Android 4.4.2 (KitKat) and only has 2G/3G reception. The smallest modern phone I could find is the Sony Xperia Compact, but that is 4.5" and I regard that as too big. Wiko and Alcatel make 4" models but they are 3G (and slow).
Price is not a problem (within reason, say under 1000 €/$/£) and battery life of 6 hours is enough (my LG gives me 48 hours, because I hardly ever turn it on).
I prefer not to use Apple because  I don't want to pay Steve Jobs' premium (20% more quality for 80% more price) and  I am used to Android's flexibility.
Does anybody make a small, functional phone?