Author:
Wen Jiajin,Han Tianyong,Cheng Sui Sun
Abstract
Abstract
Let p be a real density function defined on a compact subset Ω of
R
m
, and let
E
(
f
,
p
)
=
∫
Ω
p
f
d
ω
be the expectation of f with respect to the density function p. In this paper, we define a one-parameter extension
Var
γ
(
f
,
p
)
of the usual variance
Var
(
f
,
p
)
=
E
(
f
2
,
p
)
−
E
2
(
f
,
p
)
of a positive continuous function f defined on Ω. By means of this extension, a two-parameter mean
V
r
,
s
(
f
,
p
)
, called the Dresher variance mean, is then defined. Their properties are then discussed. In particular, we establish a Dresher variance mean inequality
min
t
∈
Ω
{
f
(
t
)
}
≤
V
r
,
s
(
f
,
p
)
≤
max
t
∈
Ω
{
f
(
t
)
}
, that is to say, the Dresher variance mean
V
r
,
s
(
f
,
p
)
is a true mean of f. We also establish a Dresher-type inequality
V
r
,
s
(
f
,
p
)
≥
V
r
∗
,
s
∗
(
f
,
p
)
under appropriate conditions on r, s,
r
∗
,
s
∗
; and finally, a V-E inequality
V
r
,
s
(
f
,
p
)
≥
(
s
r
)
1
/
(
r
−
s
)
E
(
f
,
p
)
that shows that
V
r
,
s
(
f
,
p
)
can be compared with
E
(
f
,
p
)
. We are also able to illustrate the uses of these results in space science.
MSC:26D15, 26E60, 62J10.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
Reference28 articles.
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