Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications
For $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ and $r ∈ \{ 1, 2, … , n\}$, a symmetric function $F_n(x, r)$ is defined by the relation $$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$ where $i_1 , i_2 , ... , i_n$ are positive integers. This pa...
Збережено в:
| Дата: | 2009 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3102 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509135632596992 |
|---|---|
| author | Wei-feng, Xia Вей, Фен-Ся |
| author_facet | Wei-feng, Xia Вей, Фен-Ся |
| author_sort | Wei-feng, Xia |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:45:28Z |
| description | For $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ and $r ∈ \{ 1, 2, … , n\}$, a symmetric function $F_n(x, r)$ is defined by the relation
$$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$
where $i_1 , i_2 , ... , i_n$ are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of $F_n(x, r)$. As applications, some inequalities are established by using the theory of majorization. |
| first_indexed | 2026-03-24T02:36:18Z |
| format | Article |
| fulltext |
UDC 517.5
Wei-feng Xia, Yu-ming Chu (Huzhou Teachers College, China)
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE
CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS
WITH APPLICATIONS*
OPUKLIST| ZA ÍUROM I MUL|TYPLIKATYVNA
OPUKLIST| ZA ÍUROM DLQ ODNOHO KLASU
SYMETRYÇNYX FUNKCIJ TA }X ZASTOSUVANNQ
For x x x xn= ( … )1 2, , , ∈ ( 0, 1 ]
n
and r ∈ { 1, 2, … , n }, the symmetric function F x rn ( ), is defined
by
F x r F x x x rn n n( ) = ( … ), , , , ;1 2 =
1
11 1 2
−
=≤ < … ≤
∏∑
x
x
i
ij
r
i i i n
j
jr
,
where i1 , i2 , … , in are positive integers.
This paper deals with the Schur convexity and Schur multiplicative convexity of F x rn ( ), . As
applications, some inequalities are established by use of the theory of majorization.
Dlq x x x xn= ( … )1 2, , , ∈ ( 0, 1 ]
n
ta r ∈ { 1, 2, … , n } symetryçna funkciq F x rn ( ), vyznaça-
[t\sq spivvidnoßennqm
F x r F x x x rn n n( ) = ( … ), , , , ;1 2 =
1
11 1 2
−
=≤ < … ≤
∏∑
x
x
i
ij
r
i i i n
j
jr
,
de i1 , i2 , … , in — dodatni cili çysla.
U statti rozhlqnuto vlastyvosti opuklosti za Íurom ta mul\typlikatyvno] opuklosti za
Íurom dlq funkci] F x rn ( ), . Qk zastosuvannq, vstanovleno deqki nerivnosti z vykorystannqm
teori] maΩoruvannq.
1. Introduction. The Schur convex function was introduced by I. Schur in 1923 [16].
It has many important applications in analytic inequalities [7 – 9, 11, 15, 19, 26, 28, 31,
32], extended mean values [4, 23, 24, 27], theory of statistical experiments [29].
graphs and matrices [6], combinational optimization [13], reliability [14], stochastic
orderings [25] and other related fields. G. H. Hardy, J. E. Littlewood and G. Pólya
were also interested in some inequalities that are related to the Schur convexity [12].
The following definition for Schur convex or concave function can be found in many
references such as [4, 9, 16, 20, 22].
Definition 1.1. Let E ⊆ Rn, n ≥ 2, be a set. A real-valued function F on E
is called a Schur convex function if
F ( x1 , x2 , … , xn ) ≤ F ( y1 , y2 , … , yn )
for each pair of n-tuples x = ( x1 , … , xn ) and y = ( y1 , … , yn ) in E such that x
is majorized by y (in symbols x ≺ y ), i.e.,
x yi
i
k
i
i
k
[ ]
=
[ ]
=
∑ ∑≤
1 1
, k = 1, 2, … , n – 1
and
*
This work was supported by the National Natural Science Foundation of China (608005) and the
National Science Foundation of Zhejiang Province (D7080080, Y7080185).
© WEI-FENG XIA, YU-MING CHU, 2009
1306 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE CONVEXITY … 1307
x yi
i
n
i
i
n
[ ]
=
[ ]
=
∑ ∑=
1 1
,
where x i[ ] denotes the i-th largest component of x. F is called Schur concave if
– F is Schur convex.
Recall that the following so-called Schur’s condition is very useful for determining
whether or not a given function is Schur convex or Schur concave.
Theorem 1.1 [8, 9, 11, 15, 16]. Let f : ( 0, 1 ]
n → R , n ≥ 2, be a continuous
symmetric function. If f is differentiable in ( 0, 1 ]
n, then f is Schur convex on
( 0, 1 ]
n if and only if
( − ) ∂ ( )
∂
− ∂ ( )
∂
x x
f x
x
f x
xi j
i j
≥ 0 (1.1)
for all i, j = 1, 2, … , n and x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 )
n. A n d f is Schur
concave if and only if inequality (1.1) is reversed. Here, f is a symmetric function
on ( 0, 1 ]
n which means that f ( P x ) = f ( x ) for all x ∈ ( 0, 1 ]
n and any n × n
permutation matrix P.
Remark 1.1. Since f is symmetric, the Schur’s condition in Theorem 1.1, i.e.,
(1.1) can be reduced as
( − ) ∂ ( )
∂
− ∂ ( )
∂
x x
f x
x
f x
x1 2
1 2
≥ 0.
Recently, C. P. Niculescu [21] introduced the multiplicatively convex function,
which reveals an entire new world of beautiful inequalities. And the Schur
multiplicative convexity was introduced and investigated by K. Z. Guan [9, 10], and
Y. M. Chu, X. M. Zhang and G. D. Wang [5].
Definition 1.2 [5, 9, 10]. Let I be a subinterval of ( 0, ∞ ). A positive real-
valued function F on I
n, n ≥ 2, is called a Schur multiplicatively convex function
if
F ( x1 , x2 , … , xn ) ≤ F ( y1 , y2 , … , yn )
for each pair of n-tuples x = ( x1 , … , xn ) and y = ( y1 , … , yn ) in I n such that
x is logarithmically majorized by y (in symbols log x ≺ log y ), i.e.,
x yi
i
k
i
i
k
[ ]
=
[ ]
=
∏ ∏≤
1 1
, k = 1, 2, … , n – 1,
and
x yi
i
n
i
i
n
[ ]
=
[ ]
=
∏ ∏=
1 1
.
F is called Schur multiplicatively concave if
1
F
is Schur multiplicatively convex.
Theorem 1.2 [5, 9, 10]. Let f : ( 0, 1 ]
n → ( 0, ∞ ), n ≥ 2, be a continuous
symmetric function. If f is differentiable in ( 0, 1 )
n, then f is Schur multi-
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1308 WEI-FENG XIA, YU-MING CHU
plicatively convex on ( 0, 1 ]
n if and only if
( − ) ∂ ( )
∂
− ∂ ( )
∂
x x x
f x
x
x
f x
x1 2 1
1
2
2
≥ 0 (1.2)
for all x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 )
n. And f is Schur multiplicatively concave if
and only if inequality (1.2) is reversed.
The main purpose of this article is to discuss the Schur convexity and Schur
multiplicative convexity of the symmetric function
Fn ( x, r ) = Fn ( x1 , x2 , … , xn ; r ) =
1 11 2
1
≤ < … ≤ =
∑ ∏
−
i i i n
i
ij
r
r
j
j
x
x
(1.3)
for x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 ]
n, n ≥ 2, and r = 1, 2, … , n, where i1 , i2 , … , in are
positive integers.
Our main results are the Theorems 1.3 and 1.4.
Theorem 1.3. 1. Fn ( x, 1 ) is Schur convex on ( 0, 1 ]
n.
2. Fn ( x, r ) is Schur convex on 0
2 1
2 2
,
n r
n
n− −
−
for 2 ≤ r ≤ n.
3. Fn ( x, r ) is Schur concave on
2 1
2 2
1
n r
n
n− −
−
, for 2 ≤ r ≤ n.
Theorem 1.4. 1. Fn ( x, 1 ) is Schur multiplicatively convex on ( 0, 1 ]
n.
2. Fn ( x, n ) is Schur multiplicatively concave on ( 0, 1 ]
n.
3. Fn ( x, r ) is Schur multiplicatively convex on 0
1
,
n r
n
n−
−
for n ≥ 3 and
2 ≤ r ≤ n – 1.
4. Fn ( x, r ) is Schur multiplicatively concave on
n r
n
n−
−
1
1, for n ≥ 3 and
2 ≤ r ≤ n – 1.
As applications of Theorems 1.3 and 1.4, some inequalities are established by use
of the theory of majorization in Section 4.
2. Lemmas. In this section, we establish and introduce several lemmas, which are
used in the next sections.
For t = ( t1 , t2 , … , tn ) ∈ ( 0, ∞ )
n and r ∈ { 0, 1, 2, … , n }, n ≥ 2, the r-th
elementary symmetric function (see [3]) is defined as
En ( t, r ) = En ( t1 , t2 , … , tn , r ) = 1 11 2
1 2
1 0
≤ < <…< ≤ =∑ ∏ = …
=
i i i n ij
r
r j
t r n
r
, , , , ,
, ,
where i1 , i2 , … , in are positive integers.
Lemma 2.1. If 1 ≤ r ≤ n – 1, then
E t rn
2( ), ≥ En ( t, r – 1 ) En ( t, r + 1 )
for t = ( t1 , t2 , … , tn ) ∈ ( 0, ∞ )
n.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE CONVEXITY … 1309
Proof. We use mathematical induction to prove Lemma 2.1.
(i) By simple computation, it is not difficult to verify that Lemma 2.1 is true for
n = 2 and 3, and n ≥ 4 and r = 2.
(ii) Assume that Lemma 2.1 is true for 3 ≤ n ≤ m – 1 and 2 ≤ r ≤ n. Then the
definition of En ( t, r ) yields that
Em ( t, r ) = Em−1 ( t1 , t2 , … , tm−1 ; r – 1 ) tm + Em−1 ( t1 , t2 , … , tm−1; r ),
Em ( t, r – 1 ) = Em−1 ( t1 , t2 , … , tm−1 ; r – 2 ) tm + Em−1 ( t1 , t2 , … , tm−1 ; r – 1 ), (2.1)
Em ( t, r + 1 ) = Em−1 ( t1 , t2 , … , tm−1 ; r ) tm + Em−1 ( t1 , t2 , … , tm−1 ; r + 1 ).
Equation (2.1) leads to
E t rm
2 ( ), – Em ( t, r – 1 ) Em ( t, r + 1 ) =
= [ −Em 1
2 ( t1 , t2 , … , tm – 1 ; r – 1 ) – Em – 1 ( t1 , t2 , … , tm – 1 ; r – 2 ) ×
× Em – 1 ( t1 , t2 , … , tm – 1 ; r ) ]tm
2 + [ Em – 1 ( t1 , t2 , … , tm – 1 ; r – 1 ) ×
× Em – 1 ( t1 , t2 , … , tm – 1 ; r ) – Em – 1 ( t1 , t2 , … , tm – 1 ; r – 2 ) ×
× Em – 1 ( t1 , t2 , … , tm – 1 ; r + 1 ) ] tm + Em−1
2 ( t1 , t2 , … , tm – 1 ; r ) –
– Em – 1 ( t1 , t2 , … , tm – 1 ; r – 1 ) Em – 1 ( t1 , t2 , … , tm – 1 ; r + 1 ). (2.2)
By induction hypothesis we have
E t t t r
E t t t r
m m
m m
− −
− −
( … )
( … + )
≥1 1 2 1
1 1 2 1 1
, , , ;
, , , ;
EE t t t r
E t t t r
m m
m m
− −
− −
( … − )
( … )
1 1 2 1
1 1 2 1
1, , , ;
, , , ;
≥
≥
E t t t r
E t t t r
m m
m m
− −
− −
( … − )
( … −
1 1 2 1
1 1 2 1
2
1
, , , ;
, , , ; ))
. (2.3)
Now, equations (2.2) and (2.3) imply that
E t rm
2 ( ), ≥ Em ( t, r – 1 ) Em ( t, r + 1 ).
Therefore, Lemma 2.1 follows from (i) and (ii) together with the mathematical
induction.
Lemma 2.2. If n ≥ 3 and 1 ≤ r ≤ n – 1, then the function
ϕn ( x1 , x2 , … , xn ; r ) =
F x x x r
F x x x r
n n
n n
( … + )
( … )
1 2
1 2
1, , , ;
, , , ;
is decreasing with respect to each xi in ( 0, 1 ), i = 1, 2, … , n.
Proof. Let ψn ( t1 , t2 , … , tn ; r ) =
E t t t r
E t t t r
n n
n n
( … + )
( … )
1 2
1 2
1, , , ;
, , , ;
and ti =
1− x
x
i
i
, then
from the symmetry of ϕn and ψn , and the monotonicity of
1− x
x
, we need only to
prove that ψn ( t1 , t2 , … , tn ; r ) is increasing with respect to t1 in ( 0, ∞ ). The proof is
divided into three cases.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1310 WEI-FENG XIA, YU-MING CHU
Case 1. If r = 1, then
ψn ( t1 , t2 , … , tn ; 1 ) =
t t t t
t
ii
n
i ji j n
ii
n
1 2 2
1
= ≤ < ≤
=
∑ ∑
∑
+
and
∂ ( … )
∂
=
+
= ≤ < ≤∑ ∑ψn n ii
n
i ji j nt t t
t
t t t
t
1 2
1
2
2 21, , , ;
iii
n
=∑( )1
2
> 0.
Case 2. If r = n – 1, then
ψn ( t1 , t2 , … , tn ; n – 1 ) =
1
1
1 ti
i
n
=∑
,
we clearly see that ψn ( t1 , t2 , … , tn ; n – 1 ) is increasing with respect to t1 in ( 0, ∞ ).
Case 3. If n ≥ 4 and 2 ≤ r ≤ n – 2, then
ψn ( t1 , t2 , … , tn ; r ) =
t E t t t r E t t t r
t
n n n n1 1 2 3 1 2 3
1
1− −( … ) + ( … + ), , , ; , , , ;
EE t t t r E t t t rn n n n− −( … − ) + ( … )1 2 3 1 2 31, , , ; , , , ;
and
∂ ( … )
∂
ψn nt t t r
t
1 2
1
, , , ;
≥ 0.
From above Cases 1 – 3 we know that ψn ( t1 , t2 , … , tn ; r ) is increasing with
respect to t1 in ( 0, ∞ ) for n ≥ 3 and 1 ≤ r ≤ n – 1, and the proof of Lemma 2.2 is
completed.
Lemma 2.3. Let x = ( x1 , x2 , … , xn ) ∈ ( 0, ∞ )
n and xii
n
=∑ 1
= s . If λ ≤ 1,
then
s x
n
s x
n
s x
n
s x
n
n−
−
=
−
−
−
−
…
−
−
λ
λ
λ
λ
λ
λ
λ
λ
1 2, , , ≺ (( … )x x xn1 2, , , = x.
Proof. For any x = ( x1 , x2 , … , xn ) ∈ ( 0, ∞ )
n, we clearly see that
1
1
1
1
1
11 2n
x
n
x
n
x xi
i
i
i
i
i n− −
…
−
(
≠ ≠ ≠
∑ ∑ ∑, , , ≺ 11 2, , ,x xn… ) = x,
multiply by n – 1, add ( 1 – λ ) x to both sides and divided by n – λ, we get
s x
n
s x
n
s x
n
s x
n
n−
−
=
−
−
−
−
…
−
−
λ
λ
λ
λ
λ
λ
λ
λ
1 2, , , ≺ (( … )x x xn1 2, , , = x.
Remark 2.1. Lemma 2.3 was prove by S. H. Wu [30] in the case of 0 ≤ λ ≤ 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE CONVEXITY … 1311
3. Proof of Theorems 1.3 and 1.4. Proof of Theorem 1.3. 1. If r = 1 and x =
= ( x1 , x2 , … , xn ) ∈ ( 0, 1 )
n, then (1.3) leads to
Fn ( x, 1 ) = Fn ( x1 , x2 , … , xn , 1 ) =
1
1
−
=
∑ x
x
i
ii
n
(3.1)
and
( − )
∂ ( )
∂
−
∂ ( )
∂
=
( −
x x
F x
x
F x
x
x xn n
1 2
1 2
1 21 1, , )) ( + )2
1 2
1
2
2
2
x x
x x
≥ 0. (3.2)
Therefore, Theorem 1.3(1) follows from (3.2) and Theorem 1.1 together with
Remark 1.1.
2. If 2 ≤ r ≤ n and x = ( x1 , x2 , … , xn ) ∈ 0
2 1
2 2
,
n r
n
n− −
−
, then the proof is
divided into six cases.
Case 2.1. If n = 2, r = 2 and x = ( x1 , x2 ) ∈ 0
1
2
,
n
, then
F2 ( x, 2 ) = F2 ( x1 , x2 ; 2 ) =
( − )( − )1 11 2
1 2
x x
x x
(3.3)
and
( − )
∂ ( )
∂
−
∂ ( )
∂
=
( −
x x
F x
x
F x
x
x x
1 2
2
1
2
2
1 22 2, , )) ( − − )2
1 2
1
2
2
2
1 x x
x x
≥ 0.
Case 2.2. If n ≥ 3, r = n and x = ( x1 , x2 , … , xn ) ∈ 0
1
2
,
n
, then
Fn ( x, n ) = Fn ( x1 , x2 , … , xn ; n ) =
1
1
−
=
∏ x
x
i
ii
n
(3.4)
and
( − )
∂ ( )
∂
−
∂ ( )
∂
=
( −
x x
F x n
x
F x n
x
x xn n
1 2
1 2
1 2, , )) ( )
( − )( − )
( − − )
2
1 2 1 2
1 21 1
1
F x n
x x x x
x xn ,
≥ 0.
Case 2.3. If n = 3, r = 2 and x = ( x1 , x2 , x3 ) ∈ 0
3
4
,
n
, then
F3 ( x, 2 ) = F3 ( x1 , x2 , x3 ; 2 ) =
=
( − )( − )
+
( − )( − )
+
( − )( −1 1 1 1 1 11 2
1 2
1 3
1 3
2x x
x x
x x
x x
x xx
x x
3
2 3
)
(3.5)
and
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1312 WEI-FENG XIA, YU-MING CHU
( − )
∂ ( )
∂
−
∂ ( )
∂
x x
F x
x
F x
x1 2
3
1
3
2
2 2, ,
=
=
( − )
( − − ) + ( + ) −
x x
x x
x x x x
x
1 2
2
1
2
2
2 1 2 1 2
3
1
1
1
≥
≥
( − )
− ( + )
x x
x x
x x1 2
2
1
2
2
2 1 21
2
3
≥ 0.
Case 2.4. If n ≥ 4, r = 2 and x = ( x1 , x2 , … , xn ) ∈ 0
2 3
2 2
,
n
n
n−
−
, then
Fn ( x, 2 ) = Fn ( x1 , x2 , … , xn ; 2 ) =
=
( − )( − )
+
−
+
−
−1 1 1 1 11 2
1 2
1
1
2
2
x x
x x
x
x
x
x
x
x
i
ii==
∑
3
n
+
( − )( − )
≤ < ≤
∑
1 1
3
x x
x x
i j
i ji j n
(3.6)
and
( − )
∂ ( )
∂
−
∂ ( )
∂
x x
F x
x
F x
x
n n
1 2
1 2
2 2, ,
=
=
( − )
( − − ) + ( + )
−
=
∑x x
x x
x x x x
x
x
i
ii
n
1 2
2
1
2
2
2 1 2 1 2
3
1
1
≥
≥
( − )
− −
−
( + )
x x
x x
n
n
x x1 2
2
1
2
2
2 1 21
1
2 3
≥ 0.
Case 2.5. If n ≥ 4, r = n – 1 and x = ( x1 , x2 , … , xn ) ∈ 0
2 2
,
n
n
n
−
, then
Fn ( x, n – 1 ) = Fn ( x1 , x2 , … , xn ; n – 1 ) =
=
( − )( − )
−
+
−
+
−
=
∑1 1
1
1 11 2
1 2 3
1
2
2
2
x x
x x
x
x
x
x
x
x
i
ii
n
−
=
∏ 1
3
x
x
i
ii
n
(3.7)
and
( − )
∂ ( − )
∂
−
∂ ( − )
∂
x x
F x n
x
F x n
x
n n
1 2
1 2
1 1, ,
=
=
( − )
− − +
+
−
=∑
x x
x x
x x
x x
x
x
i
i
i
n
1 2
2
1
2
2
2 1 2
1 2
3
1
1
−
−
= =
∑ ∏x
x
x
x
i
ii
n
i
ii
n
1
1
3 3
≥
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE CONVEXITY … 1313
≥
( − )
− − ( + )
−=
x x
x x
n
n
x x
x
x
i
ii
n
1 2
2
1
2
2
2 1 2
3
1
1
1
∑∑ ∏
−
=
1
3
x
x
i
ii
n
≥ 0.
Case 2.6. If n ≥ 5, 3 ≤ r ≤ n – 2 and x = ( x1 , x2 , … , xn ) ∈ 0
2 1
2 2
,
n r
n
n− −
−
,
then (1.3) and Lemma 2.2 yield that
Fn ( x, r ) = Fn ( x1 , x2 , … , xn ; r ) =
1 1
21
1
2
2
2 3 4
− −
( … − )−
x
x
x
x
F x x x rn n, , , ; +
+
1 1
11
1
2
2
2 3 4
−
+
−
( … − ) +− −
x
x
x
x
F x x x r Fn n n, , , ; 22 3 4( … )x x x rn, , , ; (3.8)
and
( − )
∂ ( )
∂
−
∂ ( )
∂
x x
F x r
x
F x r
x
n n
1 2
1 2
, ,
=
( − )
( … − )−
x x
x x
F x x x rn n
1 2
2
1
2
2
2 2 3 4 2, , , ; ×
× ( − − ) +
( … − )
( …
−
−
1
1
1 2
2 3 4
2 3 4
x x
F x x x r
F x x
n n
n
, , , ;
, , , xx r
x x
n; − )
( + )
2 1 2 ≥
≥
( − )
( … − )−
x x
x x
F x x x rn n
1 2
2
1
2
2
2 2 3 4 2, , , ; ×
× ( − − ) +
( − )
( − ) ( − − )
( − )
( − ) (
1
2
1 1
2
2
1 2x x
n
r n r
n
r
!
! !
!
! nn r
r
n r
x x
− )
−
− −
( + )
!
1
2 1 1 2 =
=
( − )
( … − ) − −
−−
x x
x x
F x x x r
n
n rn n
1 2
2
1
2
2
2 2 3 4 2 1
1
2
, , , ;
−−
( + )
1 1 2x x ≥ 0.
Therefore, Theorem 1.3(2) follows from Cases 2.1 – 2. 6 and Theorem 1.1 together
with Remark 1.1.
3. If 2 ≤ r ≤ n and x = ( x1 , x2 , … , xn ) ∈
2 1
2 2
1
n r
n
n− −
−
, , then the similar
proofs as in Theorem 1.3(2) show that Fn ( x, r ) is Schur concave on
2 1
2 2
1
n r
n
n− −
−
, .
Proof of Theorem 1.4. 1. If r = 1 and x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 )
n, then (3.1)
yields that
( − )
∂ ( )
∂
−
∂ ( )
∂
=
(
x x x
F x
x
x
F x
x
xn n
1 2 1
1
2
2
1 1, , 11 2
2
1 2
− )x
x x
≥ 0. (3.9)
Therefore, Theorem 1.4(1) follows from (3.9) and Theorem 1.2.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1314 WEI-FENG XIA, YU-MING CHU
2. If r = n and x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 )
n, then (3.3) and (3.4) lead to that
( − )
∂ ( )
∂
−
∂ ( )
∂
= −
(
x x x
F x n
x
x
F x n
x
n n
1 2 1
1
2
2
, , xx x
x x
F x nn
1 2
2
1 21 1
− )
( − )( − )
( ), ≤ 0. (3.10)
Therefore, Theorem 1.4(2) follows from (3.10) and Theorem 1.2.
3. If n ≥ 3, 2 ≤ r ≤ n – 1 and x = ( x1 , x2 , … , xn ) ∈ 0
1
,
n r
n
n−
−
, then the proof
is divided into three cases.
Case 3.1. If n ≥ 3, r = 2 and x = ( x1 , x2 , … , xn ) ∈ 0
2
1
,
n
n
n−
−
, then (3.5) and
(3.6) yield that
( − )
∂ ( )
∂
−
∂ ( )
∂
=
(
x x x
F x
x
x
F x
x
xn n
1 2 1
1
2
2
2 2, , 11 2
2
1 2 3
1
1− )
− +
−
=
∑x
x x
x
x
i
ii
n
≥ 0.
Case 3.2. If n ≥ 4, r = n – 1 and x = ( x1 , x2 , … , xn ) ∈ 0
1
1
,
n
n
−
, then (3.7)
implies that
( − )
∂ ( − )
∂
−
∂ ( − )
∂
x x x
F x n
x
x
F x n
x
n n
1 2 1
1
2
2
1 1, ,
=
=
( − )
−
−
−
= =
∑ ∏x x
x x
x
x
x
x
i
ii
n
i
ii
n
1 2
2
1 2 3 3
1
1
1
≥ 0.
Case 3.3. If n ≥ 5, 3 ≤ r ≤ n – 2 and x = ( x1 , x2 , … , xn ) ∈ 0
1
,
n r
n
n−
−
, then
from (3.8) and Lemma 2.2 together with (1.3) we see that
( − )
∂ ( )
∂
−
∂ ( )
∂
x x x
F x r
x
x
F x r
x
n n
1 2 1
1
2
2
, ,
=
=
( − )
( … − )
(
−
−x x
x x
F x x x r
F x x
n n
n1 2
2
1 2
2 3 4
2 3 42, , , ;
, ,…… − )
( … − )
−
−
, ;
, , , ;
x r
F x x x r
n
n n
1
2
1
2 3 4
≥
≥
( − )
( … − )
( − )
( − )
−
x x
x x
F x x x r
n
r
n n
1 2
2
1 2
2 3 4 2
2
1
, , , ;
!
!! !
!
! !
( − − )
( − )
( − ) ( − )
−
−
−
n r
n
r n r
r
n r
1
2
2
1
1
= 0.
Therefore, Theorem 1.4(3) follows from Cases 3.1 – 3.3 and Theorem 1.2.
4. The proofs is completely parallel to that in Theorem 1.4(3).
4. Applications. In this section, we establish some inequalities by use of
Theorems 1.3, 1.4 and the theory of majorization.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE CONVEXITY … 1315
Theorem 4.1. If n ≥ 2, x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 ]
n and s = xii
n
=∑ 1
, then
(1)
1 1
1 3
−
≥
( − ) − ( − )
−= =
∑ ∑x
x
n s x
s x
i
ii
n
i
ii
n λ
λ
for λ ≤ 1;
(2) Fn ( x, r ) ≥ F
s x
n
rn
−
−
λ
λ
; for 2 ≤ r ≤ n, x ∈ 0
2 1
2 2
,
n r
n
n− −
−
and λ ≤ 1;
(3) Fn ( x, r ) ≤ F
s x
n
rn
−
−
λ
λ
; for 2 ≤ r ≤ n, x ∈
2 1
2 2
1
n r
n
n− −
−
, and λ ≤ 1.
Proof. Theorem 4.1(1) follows from Theorem 1.3(1), Lemma 2.3 and (1.3);
Theorem 4.1(2) follows from Theorem 1.3(2) and Lemma 2.3; and Theorem 4.1(3)
follows from Theorem 1.3(3) and Lemma 2.3.
If we take s = 1 in Theorem 4.1(1), then we get the following corollary.
Corollary 4.1. If n ≥ 2, x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 )
n with xii
n
=∑ 1
= 1, and
λ ≤ 1, then
1 1
11 1x
n
xii
n
ii
n
= =
∑ ∑≥ ( − )
−
λ
λ
.
If we take r = n in Theorem 4.1(2) and (3), respectively, then we have the
following corollary.
Corollary 4.2. If n ≥ 2, s = xii
n
=∑ 1
and λ ≤ 1, then
(1)
1
1 1
1 1x
n
s xii
n
ii
n
−
≥ −
−
−
= =
∏ ∏ λ
λ
for ( x1 , x2 , … , xn ) ∈ 0
1
2
,
n
;
(2)
1
1 1
1 1x
n
s xii
n
ii
n
−
≤ −
−
−
= =
∏ ∏ λ
λ
for ( x1 , x2 , … , xn ) ∈
1
2
1,
n
.
Theorem 4.2. If n ≥ 2, x = ( x1 , x2 , … , xn ) ∈ ( 0, 1 ]
n, An ( x ) =
x
n
ii
n
=∑ 1 , G n ( x ) =
= xii
n n
=∏( )1
1/
and Hn ( x ) =
n
xi
i
n 1
1=∑
, then
(1) An ( x ) ≥ Hn ( x ) ;
(2)
1 11 2
1
1
≤ < <…< ≤ =
∑ ∏ −
≥
( − )i i i n ij
r
r j
x
n
r n r
!
! !
AA x
A x
n
n
r
( − )
( )
1
for 2 ≤ r ≤ n and
x ∈ 0
2 1
2 2
,
n r
n
n− −
−
;
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1316 WEI-FENG XIA, YU-MING CHU
(3)
1 11 2
1
1
≤ < <…< ≤ =
∑ ∏ −
≤
( − )i i i n ij
r
r j
x
n
r n r
!
! !
AA x
A x
n
n
r
( − )
( )
1
for 2 ≤ r ≤ n and
x ∈
2 1
2 2
1
n r
n
n− −
−
, ;
(4) Gn ( x ) ≥ Hn ( x );
(5) Gn ( x ) + Gn ( 1 – x ) ≤ 1;
(6)
1 11 2
1
1
≤ < <…< ≤ =
∑ ∏ −
≥
( − )i i i n ij
r
r j
x
n
r n r
!
! !
GG x
G x
n
n
r
( ) −
( )
1
for n ≥ 3, 2 ≤ r ≤ n –
– 1 and x ∈ 0
1
,
n r
n
n−
−
;
(7)
1 11 2
1
1
≤ < <…< ≤ =
∑ ∏ −
≤
( − )i i i n ij
r
r j
x
n
r n r
!
! !
GG x
G x
n
n
r
( ) −
( )
1
for n ≥ 3, 2 ≤ r ≤ n –
– 1 and x ∈
n r
n
n−
−
1
1, .
Proof. We clearly see that
(An ( x ), An ( x ), … , An ( x ) ) ≺ ( x1 , x2 , … , xn ) (4.1)
and
log (Gn ( x ), Gn ( x ), … , Gn ( x ) ) ≺ log ( x1 , x2 , … , xn ). (4.2)
Therefore, Theorem 4.2(1) follows from Theorem 1.3(1), (4.1) and (1.3). Theorem
4.2(2) and (3) follow from (4.1), (1.3) and Theorem 1.3(2) and (3), respectively.
Theorem 4.2(4) follows from Theorem 1.4(1), (4.2) and (1.3). Theorem 4.2(5) follows
from Theorem 1.4(2), (4.2) and (1.3). Theorem 4.2(6) and (7) follow from (4.2), (1.3)
and Theorem 1.4(3)and (4), respectively.
If we take r = n in Theorem 4.2(2), then we get the following corollary.
Corollary 4.3. If n ≥ 2 and x = ( x1 , x2 , … , xn ) ∈ 0
1
2
,
n
, then
G x
G x
A x
A x
n
n
n
n
( − )
( )
≥
( − )
( )
1 1
.
Remark 4.1. The inequality in Corollary 4.3 is known as Ky Fan’s inequality [20,
p. 363; 2, p. 5]. There are already at least ten proofs of this result, see, for example, [1,
17, 18] and references cited therein.
Theorem 4.3. Let A = A1 A2 … An + 1 be a n-dimensional simplex in R n, n ≥
2, and P be an arbitrary point in the interior of A . If B i is the intersection
point of straight line Ai P and the hyperplane
i∑ = A1 A2 … Ai – 1 Ai + 1 … An + 1 , i
= 1, 2, … … , n + 1, then
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
THE SCHUR CONVEXITY AND SCHUR MULTIPLICATIVE CONVEXITY … 1317
(1)
PA
PB
n ni
ii
n
=
+
∑ ≥ ( + )
1
1
1 ;
(2)
PB
PA
n
n
i
ii
n
=
+
∑ ≥ +
1
1 1
.
Proof. One can easily see that
PB
A B
i
i i
i
n
=
+∑ 1
1
= 1 and
PA
A B
i
i i
i
n
=
+∑ 1
1
= n. Therefore,
Theorem 4.3 follows from Theorem 1.3(1) and (1.3) together with the fact that
1
1
1
1
1
1
1
1 1
2
2 2n n n
PB
A B
PB
A B
PB
+ +
…
+
…, , , , , ,≺ nn
n nA B
+
+ +
1
1 1
and
n
n
n
n
n
n
PA
A B
PA
A B
PA
+ +
…
+
…
1 1 1
1
1 1
2
2 2
, , , , , ,≺ nn
n nA B
+
+ +
1
1 1
.
Acknowledgements. The authors cordially thank the referee's valuable suggestions
which lead to improvement of this paper.
1. Alzer H. A short proof of Ky Fan’s inequality // Arch. Math. (Brno). – 1991. – 27B. – P. 199 –
200.
2. Beckenbach E. F., Bellman R. Inequalities. – Berlin: Springer-Verlag, 1961.
3. Bullen P. S. Handbook of means and their inequalities. – Dordrecht: Kluwer Acad. Publ., 2003.
4. Chu Y. M., Zhang X. M. Necessary and sufficient conditions such that extended mean values are
Schur-convex or Schur-concave // J. Math. Kyoto Univ. – 2008. – 48, # 1. – P. 229 – 238.
5. Chu Y. M., Zhang X. M., Wang G. D. The Schur geometrical convexity of the extended mean
values // J. Convex Anal. – 2008. – 15, # 4. – P. 707 – 718.
6. Constantine G. M. Schur-convex functions on the spectra of graphs // Discrete Math. – 1985. – 45,
# 2 – 3. – P. 181 – 188.
7. Guan K. Z. The Hamy symmetric function and its generalization // Math. Inequal. and Appl. –
2006. – 9, # 4. – P. 797 – 805.
8. Guan K. Z. Schur-convexity of the complete symmetric function // Ibid. – P. 567 – 576.
9. Guan K. Z. Some properties of a class of symmetric functions // J. Math. Anal. and Appl. – 2007. –
336, # 1. – P. 70 – 80.
10. Guan K. Z. A class of symmetric functions for multiplicatively convex function // Math. Inequal.
and Appl. – 2007. – 10, # 4. – P. 745 – 753.
11. Guan K. Z., Shen J. H. Schur-convexity for a class of symmetric function and its applications //
Ibid. – 2006. – 9, # 2. – P. 199 – 210.
12. Hardy G. H., Littlewood J. E., Pólya G. Some simple inequalities satisfied by convex functions //
Messenger Math. – 1929. – 58. – P. 145 – 152.
13. Hwang F. K., Rothblum U. G. Partition-optimization with Schur convex sum objective functions //
SIAM J. Discrete Math. – 2004/2005. – 18, # 3. – P. 512 – 524.
14. Hwang F. K., Rothblum U. G., Shepp L. Monotone optimal multipartitions using Schur convexity
with respect to partial orders // Ibid. – 1993. – 6, # 4. – P. 533 – 574.
15. Jiang W. D. Some properties of dual form of the Hamy's symmetric function // J. Math. Inequal. –
2007. – 1, # 1. – P. 117 – 125.
16. Marshall A. W., Olkin I. Inequalities: theory of majorization and its applications. – New York:
Acad. Press, 1979.
17. McGregor M. T. On some inequalities of Ky Fan and Wang-Wang // J. Math. Anal. and Appl. –
1993. – 180, # 1. – P. 182 – 188.
18. Mercer A. McD. A short proof of Ky Fan’s arithmetic-geometric inequality // Ibid. – 1996. – 204,
# 3. – P. 940 – 942.
19. Merkle M. Convexity, Schur-convexity and bounds for the gamma function involving the digamma
function // Rocky Mountain J. Math. – 1998. – 28, # 3. – P. 1053 – 1066.
20. Mitrinovic D. S. Analytic inequalities. – New York: Springer-Verlag, 1970.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1318 WEI-FENG XIA, YU-MING CHU
21. Niculescu C. P. Convexity according to the geometric mean // Math. Inequal. and Appl. – 2000. –
3, # 2. – P. 155 – 167.
22. Pečarić J., Proschan F., Tong Y. L. Conves functions, partial orderings, and statistical applications.
– New York: Acad. Press, 1992.
23. Qi F. A note on Schur-convexity of extended mean values // Rocky Mountain J. Math. – 2005. –
35, # 5. – P. 1787 – 1793.
24. Qi F., Sándor J., Dragomir S. S., Sofo A. Note on the Schur-convexity of the extended mean values
// Taiwan. J. Math. – 2005. – 9, # 3. – P. 411 – 420.
25. Shaked M., Shanthikumar J. G., Tong Y. L. Parametric Schur convexity and arrangement
monotonicity properties of partial sums // J. Multivar. Anal. – 1995. – 53, # 2. – P. 293 – 310.
26. Shi H. N. Schur-convex functions related to Hadamard-type inequalities // J. Math. Inequal. –
2007. – 1, # 1. – P. 127 – 136.
27. Shi H. N., Wu S. H., Qi F. An alternative note on the Schur-convexity of the extended mean values
// Math. Inequal. and Appl. – 2006. – 9, # 2. – P. 219 – 224.
28. Stepniak C. An effective characterization of Schur-convex function with applications // J. Convex
Anal. – 2007. – 14, # 1. – P. 103 – 108.
29. Stepniak C. Stochastic ordering and Schur-convex functions in comparison of linear experiments //
Metrika. – 1989. – 36, # 5. – P. 291 – 298.
30. Wu S. H. Generalization and sharpness of the power means inequality and their applications // J.
Math. Anal. and Appl. – 2005. – 312, # 2. – P. 637 – 652.
31. Zhang X. M. Optimization of Schur-convex functions // Math. Inequal. and Appl. – 1998. – 1, # 3.
– P. 319 – 330.
32. Zhang X. M. Schur-convex functions and isperimetric inequalities // Proc. Amer. Math. Soc. –
1998. – 126, # 2. – P. 461 – 470.
Received 24.12.08,
after revision — 16.06.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
|
| id | umjimathkievua-article-3102 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:18Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bf/b05f300d7cb8f35f9ff79fae110665bf.pdf |
| spelling | umjimathkievua-article-31022020-03-18T19:45:28Z Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications Опуклість за Шуром i мультиплікативна опуклість за Шуром для одного класу симетричних функцій та їх застосування Wei-feng, Xia Вей, Фен-Ся For $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ and $r ∈ \{ 1, 2, … , n\}$, a symmetric function $F_n(x, r)$ is defined by the relation $$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$ where $i_1 , i_2 , ... , i_n$ are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of $F_n(x, r)$. As applications, some inequalities are established by using the theory of majorization. Для $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ та $r ∈ \{ 1, 2, … , n\}$ симетрична функція $F_n(x, r)$ визначається співвідношенням $$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$ де $i_1 , i_2 , ... , i_n$ — додатні цілі числа. У статті розглянуто властивості опуклості за Шуром та мультиплікативної опуклості за Шуром для функції $F_n(x, r)$. Як застосування, встановлено деякі нерівності з використанням теорії мажорування Institute of Mathematics, NAS of Ukraine 2009-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3102 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 10 (2009); 1306-1318 Український математичний журнал; Том 61 № 10 (2009); 1306-1318 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3102/2952 https://umj.imath.kiev.ua/index.php/umj/article/view/3102/2953 Copyright (c) 2009 Wei-feng Xia |
| spellingShingle | Wei-feng, Xia Вей, Фен-Ся Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications |
| title | Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications |
| title_alt | Опуклість за Шуром i мультиплікативна опуклість за Шуром для одного класу симетричних функцій та їх застосування |
| title_full | Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications |
| title_fullStr | Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications |
| title_full_unstemmed | Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications |
| title_short | Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications |
| title_sort | schur convexity and schur multiplicative convexity for a class of symmetric functions with applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3102 |
| work_keys_str_mv | AT weifengxia schurconvexityandschurmultiplicativeconvexityforaclassofsymmetricfunctionswithapplications AT vejfensâ schurconvexityandschurmultiplicativeconvexityforaclassofsymmetricfunctionswithapplications AT weifengxia opuklístʹzašuromimulʹtiplíkativnaopuklístʹzašuromdlâodnogoklasusimetričnihfunkcíjtaíhzastosuvannâ AT vejfensâ opuklístʹzašuromimulʹtiplíkativnaopuklístʹzašuromdlâodnogoklasusimetričnihfunkcíjtaíhzastosuvannâ |