Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 295-298
295
AN INEQUALITY OF W.L. WANG AND P.F. WANG
HORST ALZER
Department of Mathemat
University of the Witwatersrand
Johannesburg, South Africa
(Received June 7, 1988 and revised form May I, 1989)
ABSTRACT.
In this note we present a proof of the inequality H /H’
G /G’
n
n
n
n
where H n and Gn (resp. H’ n and G’ n) denote the weighted harmonic and geometric means
of xl,...,xn (resp. l-Xl,...,l-xn) with x E (0, I/2], i
l,...,n.
i
KEY WORDS AND PHRASES.
Geometric and harmonic means, inequalities.
1980 AMS SUBJECT CLASSIFICATION CODE. 26D15.
I.
INTRODUCTION.
.
i--I
Let
PI’’’’’Pn
pi
and x
A’n,
(resp.
of x
,.
,xn
and
be two sequences of positive real numbers with
Xl,...,x n
(0, I/2], i=l,...,n.
i
G’ n and It’ n),
(resp.
weighted
n
An
=i
n
n
resp. A’
n
i=l
p.
n
G
PiXi
=I
i=l
geometric
and
n
and H
harmonic
n
means
n
i--I
n
xi-1
n
G’
Pi(l-xi),
arithmetic,
G
,l-x n ), i.e.
l-Xl,
. .
the
In what follows we denote by An
(l-xi)
and H
n
Pi
I/
"
i=l
(I.I)
Pi/Xl’
and
n
H’n
Setting
pl=...=
I/
Pn=I/n
i=l
pl/(l-Xl).
(I.I)
in
geometric and harmonic means of
and hn (resp.
a’n, g’n
and
and
(1.2)
(1.2)
we
obtain
the
unweighted
arithmetic,
Xl,...,x n (resp. l-x I’’’’’ l-x n ), designated by
an, gn
h’n).
In 1961 E.F. Beckenbach and R. Bellman [I] published a remarkable counterpart of
the classical arithmetic mean-geometric mean inequality which is due to Ky Fan, namely
gn / gn an/a
with equality holding in
has
been
subjected
to
(1.3)
(1.3) if and only if
considerable
Xl=...= Xn.
investigations
Since then
resulting
Fan’s inequality
in
sharpenings and refinements (see Alzer [2] and the references therein).
to
many
proofs,
It is natural
ask whether there exists a corresponding inequality for geometric and harmonic
means.
In 1984 W.L. Wang and P.F. Wang [3] have answered this question.
They
H. ALZER
296
established the inequality
(1.4)
gn/gn
hn/h
It is worth
the sign of equality is valid if and only if x.-...- x
nttontng that not only (1.3) but also (1.4) has been originally proved by using
Cauchy’s method of forward and backward induction.
where
In the last year, different authors have verified that Fan’s inequality holds
for weighted mean values,
G / G’
n
n
i.e.
A /A’
n
with equality if and only if
8].
(1.5)
n
Xl=...= Xn
[4], Levlnson [5] and Wang [6-
as in Flanders
The aim of this note is to show that inequality (1.4) can also be extended to
weighted means.
2.
AN INEQUALITY FOR WEIGHTED GEOMETRIC AND HARMONIC MEANS.
We establish the following counterpart of (1.5):
THEOREM 2.1.
If x
n
n
with equality holding in
PROOF.
(2.1)
G /G’
H /H’
n
(0, I/2], i-l,...,n, then
E
i
n
(2.1) if and only if
Xl=...=
x
n
If we set
z
xl/(l-Xl),
I
0
<
z
I,
i
1,...,n,
i
then (2.1) can be rewritten as
n
Y- Pi(t+zl)
n
i=l
n
z
]I
Pi
(2.2)
[ Pi(l+I/zi)
i=l
Since equality holds in (2.2) if z =. ..=
If the numbers
Zn,
Zl,...,z n
we have to prove that the function
f(z2)
(PlZlPl-l+z
p
<
z
is positive for 0
)z P2 +
2
<
z
Pl P2 -I
P2Zl z 2
it remains to show that
We use induction on n.
are not all equal.
P2Z2
PlZl
I.
2
A simple calculation yields
f’’(z 2)
-P2 z P2 -3
plP2Zl
2
(pl+Zl)(zO-z 2)
z
(2-P2)Zl/(Pl+Zl)
0
so
f’’(z
and
2) >
0
for
z
<
0
for
z
f’’(z 2)
0
with
<
z
<
z
2
<
z
2
<
I.
e (z
1,1),
(2.2) is strict
2; then
Let n
INEQUALITY OF W.L. WANG AND P.F. WANG
Since f(z
I)
f(z
Next we
2) >
<
for z
0
0 we obtain
z
I.
2
(2.2) is true for n
that
assume
>
0 and f’(1)
f’(zl
...
297
2.
Let us put z
Zn+
and p
Pn+l"
Without loss of generality we set
<
0
z
Since
z
1, z
z
n
<
(2.3)
z.
n
"
l-p
Pi
i=l
we get from the induction hypothesis
l-p
n
z
p
n
H
z
i=l
Pl
z
i
[ Pi( l+z i
p
i=l
n
[I Pi(l+I/zi
i=
and it remains to prove
l-p
z
p
i
.
n
n
[__
[! Pl (l+I/z)
Pl (+zl)
.
We set
.
i
i=l
(2.4)
i=l
Pi(l+I/zi)
+ p(l+I/z)
n
n
a
+ p(l+z)
n
n
i=
pi(l+zi)
and
Pi(l+zi)
b
i=l
pi(l+I/zl).
Then (2.4) can be written as
z
p
(a/b)l-P >
a
b
+ p(l+z)
+ p(l+i/z)
and this is equivalent to
pin(z) + (l-p)In(a)
g(a,b,z)
(l-p)In(b)
In(a+p(l+z)) + In(b+p(l+I/z))
Partial differentiation reveals
---g(a
a
b z)
p
a
[(l-p)(l+z) -a] / [a+p(l+z)]
and
g(a b z)
[(p-1)(l+l/z)+b] / [b+p(l+I/z)]
From (2.3) we conclude
a
<
(l-p)(l+z)
hence we obtain
and
b
>
(l-p)(l+I/z)
>
O.
298
H. ALZER
’<
Since
g(a,b,z)
p
b
a
g(a,b,z)
>
>
0
and
g(a,b,z)
>
0
we get
g(l-p,l-p,z).
We define
h(p)
g(l-p,l-p,z)
then we get
(z/(l+pz)) 2
h’’(p)
(I/(p+z)) 2
0
and because of
h(O)
h(1)
0
we have
0 for 0 < p < I,
h(p)
which completes the proof of inequality (2.4).
REMARK 2.1. We notice that the method used to establish inequality (2.2) for
n
2, can also be used to prove (2.4).
And the technique applied to establish
inequality (2.4) can be used to prove (2.2) for n
2 as well.
ACKNOWLEDGEMENT.
I want to thank the referee for helpful remarks.
REFERENCES
1.
BECKENBACK, E.F. and BELLMAN, R., Inequalities, Springer Verlag, Berlin 1961.
2.
ALZER, H., Verschrfung elner Unglelchung von Ky Fan, Aequat. Math. 36 (1988),
246-250.
3.
WANG, W.L. and WANG, P.F., A Class of Inequalities for the Symmetric Functions,
(Chinese), Acta. Math. Sinlca 27 (1984), 485-497.
4.
FLANDERS, H., Less Than or Equal to an Exercise, Amer. Math. Monthly 86 (1979),
583-584.
5.
LEVINSON, N., Generalization of an Inequality of Ky Fan, J. Math. Anal. Appl. 8
(1964), 133-134.
6.
WANG, C.L., On a Ky Fan Inequality of the Complementary A-G Type and
Variants. J. Math. Anal. Appl. 73 (1980), 501-505.
7.
WANG, C.L., Functional Equation Approach to Inequalities II, J. Math. Anal.
Appl. 78 (1980), 522-530.
8.
WANG,
C.L., Inequalities of the Rado-Popoviciu Type for
Applications, J. Math. Anal. Appl. I00 (1984), 436-446.
Functions
its
and
Their