For all totally positive algebraic numbers
α
\alpha
except a finite number of explicit exceptions, the following inequality holds:
\[
1
d
(
α
1
+
⋯
+
α
d
)
>
max
(
1.780022
,
1.66
+
α
1
)
,
\frac {1}{d}\,(\alpha _1+\dots +\alpha _d)>\max (1.780022,1.66+\alpha _1),
\]
where
d
d
is the degree of
α
\alpha
and
0
>
α
1
>
⋯
>
α
d
0>\alpha _1>\dots >\alpha _d
its conjugates. This improves previous results of Smyth, Flammang and Rhin.