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Journal Article
Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters
2017
Overview
In the article, we present the best possible parameters
λ
=
λ
(
p
)
and
μ
=
μ
(
p
)
on the interval
[
0
,
1
/
2
]
such that the double inequality
G
p
[
λ
a
+
(
1
−
λ
)
b
,
λ
b
+
(
1
−
λ
)
a
]
A
1
−
p
(
a
,
b
)
<
E
(
a
,
b
)
<
G
p
[
μ
a
+
(
1
−
μ
)
b
,
μ
b
+
(
1
−
μ
)
a
]
A
1
−
p
(
a
,
b
)
holds for any
p
∈
[
1
,
∞
)
and all
a
,
b
>
0
with
a
≠
b
, where
A
(
a
,
b
)
=
(
a
+
b
)
/
2
,
G
(
a
,
b
)
=
a
b
and
E
(
a
,
b
)
=
[
2
∫
0
π
/
2
a
cos
2
θ
+
b
sin
2
θ
d
θ
/
π
]
2
are the arithmetic, geometric and special quasi-arithmetic means of
a
and
b
, respectively.
Publisher
Springer International Publishing,Springer Nature B.V,SpringerOpen
