Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Opening a barbershop is permissible if it meets the following conditions:
1) The shaving should be limited only to what is permitted in Shariah, such as shaving head, mustache, etc.
2) The hair should not be shaved or cut in to the style of Kuffars or imitating women.
3) Shaving some part of the head and leaving other parts is forbidden because the Prophet (Sallallahu Alaihi wa Sallam) forbade "Qaz'a". Nafi'a a narrator of the Hadith was asked about the meaning of Qaz'a, he said: "to shave a part of a child's head and to leave another" .
Allah knows best.