Is it permissible for a father to give his children money on condition that it be repaid from their share of the inheritance, while exempting one of them due to his dutifulness?

It is not clear what is meant by the payment being part of the children's share of the inheritance. If the father wishes to give money to his children and favors the younger one with an increase due to his piety, this is impermissible. The father must equalize between his children in gifting, unless there is a need to particularize some or with the consent of all children to favor the younger one.