Let
α
\alpha
be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates
α
i
{\alpha _i}
, are confined to a sector
|
arg
z
|
≤
θ
|\arg z| \leq \theta
. We compute the greatest lower bound
c
(
θ
)
c(\theta )
of the absolute Mahler measure
(
∏
i
=
1
d
max
(
1
,
|
α
i
|
)
)
1
/
d
(\prod \nolimits _{i = 1}^d {\max (1,|{\alpha _i}|){)^{1/d}}}
of
α
\alpha
, for
θ
\theta
belonging to nine subintervals of
[
0
,
2
π
/
3
]
[0,2\pi /3]
. In particular, we show that
c
(
π
/
2
)
=
1.12933793
c(\pi /2) = 1.12933793
, from which it follows that any integer
α
≠
1
\alpha \ne 1
and
α
≠
e
±
i
π
/
3
\alpha \ne {e^{ \pm i\pi /3}}
all of whose conjugates have positive real part has absolute Mahler measure at least
c
(
π
/
2
)
c(\pi /2)
. This value is achieved for
α
\alpha
satisfying
α
+
1
/
α
=
β
0
2
\alpha + 1/\alpha = \beta _0^2
, where
β
0
=
1.3247
…
{\beta _0} = 1.3247 \ldots
is the smallest Pisot number (the real root of
β
0
3
=
β
0
+
1
\beta _0^3 = {\beta _0} + 1
).