Preheat oven to 350 degrees F (175 degrees C). Lightly grease a 9×5 inch loaf pan.
In a large bowl, combine flour, baking soda and salt. In a separate bowl, mix together applesauce and honey. Stir in eggs and mashed bananas until well blended. Stir banana mixture into flour mixture; stir just to moisten. Pour batter into prepared loaf pan.
Bake in preheated oven for 60 to 65 minutes, until a toothpick inserted into center of the loaf comes out clean. Let bread cool in pan for 10 minutes, then turn out onto a wire rack.