Sharp two parameter bounds for logarithmic and arithmetic-geometric means
Abstract
For fixed
s≥1
and
t
1
,
t
2
∈(0,1/2)
we prove that the inequalities
G
s
(
t
1
a+(1−
t
1
)b,
t
1
b+(1−
t
1
)a)
A
1−s
(a,b)>AG(a,b)
and
G
s
(
t
2
a+(1−
t
2
)b,
t
2
b+(1−
t
2
)a)
A
1−s
(a,b)>L(a,b)
hold for all
a,b>0
with
a≠b
if and only if
t
1
≥1/2−
2s
−
−
√
/(4s)
and
t
2
≥1/2−
6s
−
−
√
/(6s)
. Here
G(a,b)
,
L(a,b)
,
AG(a,b)
and
A(a,b)
are the geometric, logarithmic, arithmetic-geometric and arithmetic means of
a
and
b
, respectively.