May I recommend that you look at a thread -
The Power of the Luxury Slider - which for some reason I couldn't find in the list of 'must read' threads at the top of this page...
My synopsis: as long as you are up to date on developing your tiles with workers (roads on every tile you are using as a minimum) then it is
almost always better to let cities grow and use the luxury slider to control them than to limit their growth using specialists (btw since when you make a specialist, an unhappy worker is always turned into a content specialist, you usually do better to make taxmen or scientists than entertainers). There is a slight balancing act between having excess luxuries produced in a city that doesn't need them vs using population as specialists in other cities to keep them happy, but even in despotism it is better to have size 6 cities and the luxury slider at 50% than size 3 cities with the luxury slider at 0 (worst case scenario: you get the same cash flow, but have the population ready for when you get some temples built or an extra source of luxuries connected - and usually you get extra shields from the extra population as well).
Size 3 city - 3 gold (or science), 3 shields.
Size 6 city (luxury slider at 50%) 3 gold/sci, 3 luxuries, 6 shields.
Under republic or democracy it is even more advantageous to have big cities.
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