Should I be worried if my child has dark circles under his eyes?

No. Dark circles under the eyes don't require worry, but may inspire a trip to your pediatrician to have things checked out, especially if the circles are a recent development. Some children, some families, have such dark circles and it isn't a sign of illness. But, they sometimes can be associated with certain illnesses or lack of good sleep. Again, if they are a recent development, get 'em checked.
No. Probably not! dark circles are common, and are often due to congestion in the nasal area. Some children have hereditary darkness of the skin around the eyes as well. Although often associated with sleeplessness, this is not usually the case. If you think your baby may have chronic nasal allergies, see your pediatrician.