My iPhone did not show up the beatsStudio wireless when both of them were turn on bluetooth

What do you mean you pair your iPhone with your Mac? Do you mean to allow tethering for internet connection.

If you have previously paired your headphones with your Mac they are probably auto connecting to it. Temporarily turn off BT on your Mac, put the Beats in pairing mode and go to Settings > Bluetooth on your iPhone to connect them.

Have you used the headphones with another device? Did you pair them with the iPhone?