About 6-18" away from the trunk depending on the size of your palm you can first dig about 6 inches down and mix in some peat moss and top soil. Then add a few inches of mulch, some palm time released fertilizer and water the area 1-2 X a week during dry spells. They will look full in a couple of months into the summer.

Is there grass around them or is it bare?

The Bottle Palm is growing in a patch of stone mulch about 6 X 12 feet, with the driveway on one side and Hawaiian Sea Grapes on the other. The Foxtail is growing in a 5 foot wide ornamental border with the pool deck on one side and my neighbor's lawn on the other.