OCTOGON MATHEMATICAL MAGAZINE
Vol. 17, No.2, October 2009, pp 714-715
ISSN 1222-5657, ISBN 978-973-88255-5-0,
www.hetfalu.ro/octogon
714
An extension of Ky Fan‘s inequality
József Sándor23
ABSTRACT. We offer a generalization of the inequality
An
A0n
≥
Gn
G0n ,
due to Ky Fan.
Let
respectivelly geometric means and of xi
¡ An , G
¢ n denote the ¡arithmetic,
¤
i = 1, n , where xi ∈ 0, 12 . Put A0n = A0n (xi ) = An (1 − xi ) ;
G0n = G0n (xi ) = Gn (1 − xi ) . The famous inequality of Ky Fan (see e.g. [1])
states that
Gn
An
≥ 0
(1)
0
An
Gn
³ k
´
¢
¡
a +...+ak 1/k
Define Ak = Ak (ai ) = 1 n n
for k 6= 0 and ai > 0 i = 1, n , while
√
A0 = Ao (ai ) = lim Ak = k a1 ...an = Gn (a) = Gn .
k→0
Then the following extension of (1) holds true:
¤
¡
Theorem. For all xi ∈ 0, 12 and all k ∈ R one has
³

1+
1−x1
x1
´k
³
+ ... +
n
1−xn
xn
´k  k1


 ≥
1
x2k−1
1
+ ... +
n
1
2k−1
xn

1
2k−1

Proof. Remark that for k = O, inequality (2) beceomes
r
1 − xn
x1 + ... + xn
n 1 − x1
...
≥
− 1,
x1
xn
n
which is relation (1), i.e. the Ky Fan inequality.
¡ 1¤
i
Put now 1−x
=
a
in
relation
(2).
Since
x
∈
0, 2 , we get ai ≥ 1.
i
i
xi
After some elementary transformations, relation (2) becomes
23
Received: 04.09.2009
2000 Mathematics Subject Classification. 26D15
Key words and phrases. Inequalities for sums.
(2)
715
An extension of Ky Fan‘s inequality
µ
1+
ak1 + ... + akn
n
¶ k1
"
(a1 + 1)2k−1 + ... + (an + 1)2k−1
≥
n
where k 6= 0.
³
1
k
#
1
2k−1
(3)
1
k
´2k−1
Let ak = yk (yk > 0) and consider the application f (y) = 1 + y
;
where y ≥ 1. It is immediate that this function
is
convex
for
k
<
0
or
£
¤
0 < k ≤ 12 , or k ≥ 1; and concave for k ∈ 21 , 1 .
Using the Jensen inequality for convex (concave) functions, relation (3)
follows at one for k 6= 12 .
For k = 12 , however we have to prove the inequality
µ√
√ ¶
p
a1 + ... + an 2
1+
≥ n (1 + a1 ) ... (1 + an )
n
¡
¢
Letting ak = yk2 and f (y) = log 1 + y 2 which is concave, by Jensen‘s
inequality follows at one again (4).
(4)
REFERENCE
[1] Sándor, J., On an inequality of Ky Fan, Babes-Bolyai University Seminar
Math. Anal., Nr. 7, 1990, pp. 29-34.
Babes-Bolyai University,
Cluj and Miercurea Ciuc,
Romania