Flying alone with my toddler a couple of years ago, I fell in love with my backpack. Haven't carried a purse since. Your six-year old can also wear a backpack to carry snacks and entertainment items so yours can be diapering essentials, extra shirts, etc. (You'll probably want to carry the DVD player, but she can carry a cd wallet with movie selections.
Children tend to be especially sensitive to the lower pressure of an airplane and sleep a lot. (That is if their ears aren't causing them pain.) Have the older two chew on something during take off and landing and like the others said, nurse the baby.
Oh, and bring empty sippy cups for the kids. The cups they give you on the airplane are a joke and everyone will wind up wearing their beverage.
If you have a stroller big enough for the middle child, I would consider bringing it and gate-checking it. Wearing one child, pushing the other in the stroller, keeping the six-year old's hands on the stroller.... make the layover much easier.
Good luck to you.