Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES FOR WEIGHTED POWER PSEUDO MEANS
VASILE MIHESAN
Department of Mathematics
Technical University of Cluj-Napoca
Str. C.Daicoviciu nr.15
400020 Cluj-Napoca
Romania.
volume 6, issue 3, article 75,
2005.
Received 24 November, 2004;
accepted 31 May, 2005.
Communicated by: F. Qi
EMail: Vasile.Mihesan@math.utcluj.ro
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
227-04
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Abstract
[r]
In this paper we denote by mn the following expression, which is closely con[r]
nected to the weighted power means of order r, Mn .
Let n ≥ 2 be a fixed integer and

1

 Pn xr − 1 Pn pi xr r , r 6= 0
i
i=2
p1 1
p
(x ∈ Rr ),
m[r]
.Q1
n (x; p) =

P
/p
p
/p
n
n
i 1
x 1
,
r=0
1
i=2 xi
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
P
where Pn = ni=1 pi and Rr denotes the set of the vectors x = (x1 , x2 , . . . , xn )
for which xi > 0 (i = P
1, 2, . . . , n), p = (p1 , p2 , . . . , pn ), p1 > 0, pi ≥ 0 (i =
r
1, 2, . . . , n) and Pn x1 > ni=2 pi xri .
[r]
Three inequalities are presented for mn . The first is a comparison theorem. The second and the third is Rado type inequalities. The proofs show that
the above inequalities are consequences of some well-known inequalities for
weighted power means.
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2000 Mathematics Subject Classification: 26D15, 26E60.
Key words: Weighted power pseudo means, Inequalities.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Comparison Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Rado Type Inequalities for Weighted Power Pseudo Means . .
References
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1.
Introduction
Let y = (y1 , y2 , . . . , yn ) and q = (q1 , q2 , . . . , qn ) be positive n-tuples, then the
arithmetic and geometric means of y with weights q are defined by
! Q1
n
n
n
Y
1 X
qi yi and Gn (y; q) =
yiqi
,
An (y; q) =
Qn i=1
i=1
where
Qn =
n
X
qi .
i=1
If r is a real number, then the r-th power means of y with weights q,
is defined by

r1
n
P

1

r

, r 6= 0;
qi y i
 Qn
i=1
[r]
(1.1)
Mn (y; q) = Q1

n
n
Q

qi


y
,
r = 0.
[r]
Mn (y; q)
i
i=1
If r, s ∈ R, r ≤ s then [11]
(1.2)
Mn[r] (y; q) ≤ Mn[s] (y, q)
is valid for all positive real numbers yi and qi (i = 1, 2, . . . , n). For r = 0 and
s = 1 we obtain the clasical inequality between the weighted arithmetic and
geometric means
(1.3)
Gn = Gn (y; q) =
n
Y
i=1
q /Q
yi i n
n
1 X
qi yi = An (y, q) = An .
≤
Qn i=1
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
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[r]
In this paper we denote by mn (y; q) the following expression which is
[r]
closely connected to Mn (y; q).
Let n ≥ 2 be an integer (considered fixed throughout the paper) and define

r1
n

P

P
1

pi xri
, r 6= 0
 p1n xr1 − p1
[r]
i=2
(1.4)
mn (x; p) =
(x ∈ Rr )
n

Q

P
/p
p
/p

r=0
xi i 1 ,
x1 n 1
i=2
Inequalities for Weighted Power
Pseudo Means
Pn
where Pn = i=1 pi and Rr denotes the set of the vectors x = (x1 , x2 , . . . , xn )
for which xi > 0 (i = 1, 2,P
. . . , n), p = (p1 , p2 , . . . , pn ), p1 > 0, pi ≥ 0
(i = 2, 3, . . . , n) and Pn xr1 > ni=2 pi xri .
Although there is no general agreement in literature about what constitutes
a mean value most authors consider the intermediate property as the main fea[r]
ture. Since mn (x; p) do not satisfy this condition, this means that the double
inequalities
min xi ≤ m[r]
n (x; p) ≤ max xi
1≤i≤n
1≤i≤n
[r]
are not true for all positive xi , we call mn the weighted power pseudo means
of order r.
For r = 1 we obtain by (1.4) the pseudo arithmetic means an (x, p) for r = 0
the pseudo geometric means, gn (x, p), see [2]. In 1990, H. Alzer [2] published
the following companion of inequality (1.3):
(1.5)
an (x; p) ≤ gn (x; p).
Vasile Mihesan
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For the special case p1 = p2 = · · · = pn the inequality (1.5) was proved by
S. Iwamoto, R.J. Tomkins and C.L. Wang [6].
Rado and Popoviciu type inequalities for pseudo arithmetic and geometric
means were given in [2], [9], [10].
We note that inequality (1.5) is an example of a so called reverse inequality.
One of the first reverse inequalities was published by J. Aczél [1] who proved
the following intriguing variant of the Cauchy-Schwarz inequality:
Pn 2
If P
xi and yi (i = 1, 2, . . . , n) are real numbers with x21 >
i=2 xi and
n
2
2
y1 > i=2 yi , then
(1.6)
x1 y1 −
n
X
i=2
!2
xi yi
≥
x21
−
n
X
i=2
!
x2i
y12
−
n
X
!
yi2
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
.
i=2
Further interesting reverse inequalities were given in [3], [5], [6], [7], [8], [11],
[12].
The aim of this paper is to prove a comparison theorem and Rado type inequalities for the weighted power pseudo means.
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2.
Comparison Theorem
Our first result is a comparison theorem for the weighted power pseudo means.
Theorem 2.1. If 0 ≤ r ≤ s, x ∈ Rs then x ∈ Rr and
mn[s] (x,p) ≤ m[r]
n (x,p).
(2.1)
If r ≤ s ≤ 0, x ∈ Rr then x ∈ Rs and
Inequalities for Weighted Power
Pseudo Means
[r]
m[s]
n (x;p) ≤ mn (x;p).
(2.2)
[r]
[s]
If r < 0 < s then Rr ∩ Rs = ∅, hence mn (x,p), mn (x,p) cannot both be
defined, they are not comparable.
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[s]
Proof. To prove (2.1) let a = mn (x, p) > 0, then we obtain by (1.4) and (1.2)
x1 =
p 1 as +
Pn
i=2
1
s s
pi xi
≥
Pn
p 1 ar +
Pn
i=2
1
r r
pi xi
,
Pn
hence
Pn
r
Pn r
i=2 pi xi
x1 −
≥ ar > 0
p1
p1
which shows that x ∈ Rr . Taking the r th root, we obtain (2.1).
[r]
To prove (2.2) let b = mn (x, p) > 0, then we obtain by (1.4) and (1.2),
x1 =
p 1 br +
Pn
i=2
Pn
Vasile Mihesan
pi xri
r1
≤
p 1 bs +
Pn
i=2
Pn
pi xbi
1s
.
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Hence
xs1
and
≥
Pn s
x −
p1 1
p 1 bs +
Pn
i=2
pi xsi
Pn
Pn
i=2
pi xsi
p1
≥ bs > 0,
which shows that x ∈ Rs . Taking the (−s) th root, we obtain (2.2).
If r < 0 < s we infer for n = 2, p1 = p2 that x1 , x2 > 0, xr1 > xr2 , xs1 > xs2
hence x1 < x2 and x1 > x2 , which is impossible.
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
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3.
Rado Type Inequalities for Weighted Power
Pseudo Means
The well-known extension of the arithmetic mean-geometric mean inequality
(1.3) is the following inequality of Rado [11]:
(3.1)
Qn (An (y; q) − Gn (y; q)) ≥ Qn−1 (An−1 (y; q) − Gn−1 (y; q)).
The next proposition provides an analog of the Rado inequality (3.1) for pseudo
arithmetic and geometric means [2].
Proposition 3.1. For all positive real numbers xi (i = 1, 2, . . . , n; n ≥ 2) we
have
(3.2)
gn (x,p) − an (x;p) ≥ gn−1 (x;p) − an−1 (x;p).
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
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The most obvious extension is to allow the means in the Rado inequality to
have different weights [4]
Qn An (y; q) −
qn
qn
Pn Gn (y; p) ≥ Qn−1 An−1 (y; q) − Pn−1 Gn−1 (y; p).
pn
pn
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Using this inequality we obtain the following generalization of the inequality
(3.2) [10].
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Proposition 3.2. For all positive real numbers xi (i = 1, 2, . . . , n; n ≥ 2) we
have
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J. Ineq. Pure and Appl. Math. 6(3) Art. 75, 2005
(3.3)
gn (x;p) − an (x;q) ≥ gn−1 (x;p) − an−1 (x;q).
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An extension of the Rado inequality for weighted power means is the following inequality [4]: If r, s, t ∈ R such that r/t ≤ 1 and s/t ≥ 1 then
(3.4) Pn
t
t Mn[s] (y; p) − Mn[r] (y; p)
t t [s]
[r]
Mn−1 (y; p) − Mn−1 (y, p)
.
≥ Pn−1
Using inequality (3.4) we obtain generalizations of the inequality of Rado type
(3.2) for the weighted power pseudo means.
Theorem 3.3. If r ≤ 1, x ∈ Rr and xr1 ≤ xrn then
(3.5)
Vasile Mihesan
[r]
mn[r] (x,p) − an (x,p) ≥ mn−1 (x,p) − an−1 (x,p).
If s ≥ 1, x ∈ Rs and x1 ≤ xn then
(3.6)
Inequalities for Weighted Power
Pseudo Means
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[s]
an (x,p) − mn[s] (x,p) ≥ an−1 (x,p) − mn−1 (x,p).
Proof. To prove (3.5) we put in (3.4) s = t = 1 and we obtain for r ≤ 1 the
inequality:
[r]
(3.7) Pn An (y; p) − Mn[r] (y; p) ≥ Pn−1 An−1 (y; p) − Mn−1 (y; p) .
[r]
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If we set in (3.7) y1 = mn (x, p), yi = xi (i = 2, 3, . . . , n) then we have:
Pn An (y; p) − Mn[r] (y; p) = p1 m[r]
n (x; p) − an (x; p) ,
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which leads to inequality (3.5). We observe that for r ≤ 1, x ∈ Rr and xr1 ≤ xrn
we have
n
n−1
X
X
0 < Pn xr1 −
pi xri ≤ Pn−1 xr1 −
pi xri
i=2
i=2
[r]
mn−1 (x, p)
and
exist.
To prove (3.6) we set in (3.4) r = t = 1 and we obtain for s ≥ 1 the
inequality
[s]
(s)
(3.8) Pn Mn (y; p) − An (y; p) ≥ Pn−1 Mn−1 (y; p) − An−1 (y; p) .
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
If we put in (3.8) y1 =
Pn
[s]
mn (x, p),
yi = xi (i = 2, 3, . . . , n) then we have
Mn[s] (y; p) − An (y; p) = p1 an (x; p) − m[s]
n (x; p) ,
which leads to inequality (3.6). For s ≥ 1, x ∈ Rs and x1 ≤ xn ,
exist.
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[s]
mn−1 (x; p)
Theorem 3.4. If 0 < r ≤ s, x ∈ Rs and x1 ≤ xn then
s s
s
s [r]
[s]
[s]
(3.9)
m[r]
(x;p)
−
m
(x;p)
≥
m
(x;p)
−
m
(x;p)
n−1
n−1
n
n
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and
(3.10)
r
m[r]
n (x;p)
−
r
mn[s] (x;p)
≥
r
[s]
mn−1 (x;p)
−
r
[s]
mn−1 (x;p)
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Proof. To prove (3.9) we put in (3.4) t = s and we obtain for 0 < r ≤ s the
inequality
(3.11) Pn
s
Mn[s] (y; p)
−
s
Mn[r] (y; p)
≥ Pn−1
s s [s]
[r]
Mn−1 (y, p) − Mn−1 (y, p)
.
[r]
If we set in (3.11) y1 = mn (x; p), yi = xi (i = 2, 3, . . . , n) then we have
s
s s
s [s]
Pn Mn[s] (y; p) − Mn[r] (y; p)
= p1 m[r]
(x;
p)
−
m
(x;
p)
,
n
n
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
which leads to inequality (3.9) If 0 < r ≤ s, x ∈ Rs then x ∈ Rr and if x1 ≤ xn
[r]
then mn−1 (x; p) exists.
To prove (3.10) we set in (3.4) t = r and we obtain for 0 < r ≤ s the
inequality
(3.12) Pn
r
Mn[s] (y; p)
−
r
Mn[r] (y; p)
≥ Pn−1
r r [s]
[r]
Mn−1 (y; p) − Mn−1 (y; p)
.
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[s]
If we put in (3.12) y1 = mn (x, p)yi = xi (i = 2, 3, . . . , n) then we have
r
r
r [s]
Pn Mn[s] (y; p) − Mn[r] (y; p)r = p1 m[r]
(x;
p)
−
m
(x;
p)
,
n
n
which leads to inequality (3.10).
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References
[1] J. ACZÉL, Some general methods in the theory of functional equations in
one variable.New applications of functional equations (Russian), Uspehi
Mat. Nauk (N.S.), 11(3) (69) (1956), 3–58.
[2] H. ALZER, Inequalities for pseudo arithmetic and geometric means, International Series of Numerical Mathematics, Vol. 103, Birkhauser-Verlag
Basel, 1992, 5–16.
[3] R. BELLMAN, On an inequality concerning an indefinite form, Amer.
Math. Monthly, 63 (1956), 108–109.
[4] P.S. BULLEN, D.S. MITRINOVIĆ AND P.M. VASIĆ, Means and Their
Inequalities, Reidel Publ. Co., Dordrecht, 1988.
[5] Y.J. CHO, M. MATIĆ AND J. PEČARIĆ, Improvements of some inequalities of Aczél’s type, J. Math. Anal. Appl., 256 (2001), 226–240.
[6] S. IWAMOTO, R.J. TOMKINS AND C.L. WANG, Some theorems on reverse inequalities, J. Math. Anal. Appl., 119 (1986), 282–299.
Inequalities for Weighted Power
Pseudo Means
Vasile Mihesan
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[7] L. LOSONCZI, Inequalities for indefinite forms, J. Math. Anal. Appl., 285
(1997),148–156.
[8] V. MIHEŞAN, Applications of continuous dynamic programing to inverse
inequalities, General Mathematics, 2(1994), 53–60.
[9] V. MIHEŞAN, Popoviciu type inequalities for pseudo arithmetic and geometric means, (in press)
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[10] V. MIHEŞAN, Rado and Popoviciu type inequalities for pseudo arithmetic
and geometric means, (in press)
[11] D.S. MITRINOVIĆ, Analytic Inequalities, Springer Verlag, New York,
1970.
[12] X.H. SUN, Aczél-Chebyshev type inequality for positive linear functional,
J. Math. Anal. Appl., 245 (2000), 393–403.
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Pseudo Means
Vasile Mihesan
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