Will the father incur sin or discomfort by allocating a portion of his wealth to one of his children for investment in exchange for a share of the profit, without seeking the permission of his other children or granting them the same as he granted their brother, and will his action cause estrangement and animosity?

There is no objection to a father giving one of his children money to invest for a share of the profit, without giving the others. This is not considered the forbidden favoritism in pure gifting. If a father sells something to one of his children with a valid sale at a full price, there is no objection. However, if the sale is a ruse or a favoritism, then it is not permissible. It is preferable for a father to equalize among his children if possible. If this is not feasible due to reasons such as limited funds or differences in the children's endeavors, there is no objection. It is permissible for a son to offer his father a partnership in investment. A son must honor his father even if the father wrongs him.