Is it permissible for a mother to give money to her younger children to start their lives, without giving to her eldest son who has a high income?

The preponderant opinion is that it is obligatory to equalize among children in gifts, unless there is a valid excuse for preferring some of them, such as illness, a large family, or engagement in seeking knowledge. So, if these two children have a need that necessitates preferring them over their older brother, then there is no harm; otherwise, not.