A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed
p
p
and
d
d
, there exists a number
ϵ
(
d
,
p
)
>
0
\epsilon (d,p) > 0
such that for any nonregular unmixed ring
R
R
its Hilbert-Kunz multiplicity is at least
1
+
ϵ
(
d
,
p
)
1+\epsilon (d,p)
. We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and
F
F
-rational.