The September Sisters by Jillian Cantor
Abigail Reed and her younger sister, Becky, are always at each other's throats. Their mother calls them the September Sisters, because their birthdays are only a day apart, and pretends that they're best friends. But really, they delight in making each other miserable. Then Becky disappears in the middle of the night, and a torn gold chain with a sapphire heart charm is the only clue to the mystery of her kidnapping.