Lisa's answers are excellent. There's one more option. If you want to emphasise that he's still jealous you could say: "Dave admitted he is jealous of his younger brother"

It depends which part of the sentence happened yesterday. Dave made the admission yesterday, and the jealousy is ongoing: "Dave admitted to being jealous of his younger brother." Dave made the admission just now, but the jealousy occurred yesterday: "Dave admits he was jealous of his younger brother." I hope that helps!