Pre-heat the oven to 180C (350F).
Grease a 1.5 litre (1.5 quart) loaf pan. (I spray mine with coconut oil spray and line with some baking paper.)
In a saucepan, bring the apple juice, carrot, apricots and honey to a boil. Turn off the heat and let rest for 30 minutes, until the mixture is nearly at room temperature. If you transfer the mixture into a large ceramic dish, it will cool faster.
Meanwhile, combine the flour with the coconut, baking powder, cinnamon and salt. Then spoon in the carrot mixture.
Stir in the beaten eggs and finally the banana slices.
Pour the batter into a prepared pan.
Bake the cake for 50 minutes or so, until a cake skewer inserted into the cake comes out clean.
Let it cool in the pan, then turn out onto a rack. Once cooled, the cake will be slightly firmer.