Vegs for partial shade.

My garden has some big trees at the bottom and I have a border running alongside my new shed that I've dug the shrubs from and would like to grow a few vegs , I have a full sun border where I grow runner beans a tomatoes but the shady border only gets early morning sun for up to 4/5 hours before the trees shade it for the rest of the day.
Any ideas of what would grow there? Its not very big only 3' x 8' and I was just reading that vegs grown for leaf do better in shade than root vegs. Would carrots do ok or not?