On approximating the quasi-arithmetic mean
In this article, we prove that the double inequalities α 1 [ 7 C ( a , b ) 16 + 9 H ( a , b ) 16 ] + ( 1 − α 1 ) [ 3 A ( a , b ) 4 + G ( a , b ) 4 ] < E ( a , b ) < β 1 [ 7 C ( a , b ) 16 + 9 H ( a , b ) 16 ] + ( 1 − β 1 ) [ 3 A ( a , b ) 4 + G ( a , b ) 4 ] , [ 7 C ( a , b ) 16 + 9 H ( a , b...
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Veröffentlicht in: | Journal of inequalities and applications 2019-02, Vol.2019 (1), p.1-12, Article 42 |
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Hauptverfasser: | , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
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Zusammenfassung: | In this article, we prove that the double inequalities
α
1
[
7
C
(
a
,
b
)
16
+
9
H
(
a
,
b
)
16
]
+
(
1
−
α
1
)
[
3
A
(
a
,
b
)
4
+
G
(
a
,
b
)
4
]
<
E
(
a
,
b
)
<
β
1
[
7
C
(
a
,
b
)
16
+
9
H
(
a
,
b
)
16
]
+
(
1
−
β
1
)
[
3
A
(
a
,
b
)
4
+
G
(
a
,
b
)
4
]
,
[
7
C
(
a
,
b
)
16
+
9
H
(
a
,
b
)
16
]
α
2
[
3
A
(
a
,
b
)
4
+
G
(
a
,
b
)
4
]
1
−
α
2
<
E
(
a
,
b
)
<
[
7
C
(
a
,
b
)
16
+
9
H
(
a
,
b
)
16
]
β
2
[
3
A
(
a
,
b
)
4
+
G
(
a
,
b
)
4
]
1
−
β
2
hold for all
a
,
b
>
0
with
a
≠
b
if and only if
α
1
≤
3
/
16
=
0.1875
,
β
1
≥
64
/
π
2
−
6
=
0.484555
…
,
α
2
≤
3
/
16
=
0.1875
and
β
2
≥
(
5
log
2
−
log
3
−
2
log
π
)
/
(
log
7
−
log
6
)
=
0.503817
…
, where
E
(
a
,
b
)
=
(
2
π
∫
0
π
/
2
a
cos
2
θ
+
b
sin
2
θ
d
θ
)
2
,
H
(
a
,
b
)
=
2
a
b
/
(
a
+
b
)
,
G
(
a
,
b
)
=
a
b
,
A
(
a
,
b
)
=
(
a
+
b
)
/
2
and
C
(
a
,
b
)
=
(
a
2
+
b
2
)
/
(
a
+
b
)
are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of
a
and
b
, respectively. |
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ISSN: | 1029-242X 1025-5834 1029-242X |
DOI: | 10.1186/s13660-019-1991-0 |