  We study mass bounds of Maxwell fields in RN black holes and genuine hair in
Einstein-Born-Infeld black holes with various cosmological constants. It shows
that the Maxwell field serves as a good probe to disclose the hair distribution
described with the event horizon and the photonsphere. And we find that the
Hod's lower bound obtained in asymptotically flat space also holds in the
asymptotically dS Einstein-Born-Infeld hairy black holes. In contrast, the
Hod's lower bound can be invaded in the asymptotically AdS Einstein-Born-Infeld
hairy black holes. It implies that the AdS boundary could make the Born-Infeld
hair easier to condense in the near horizon area.
