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volume 5, issue 2, article 39, 2004.

Received 18 August, 2003;

accepted 11 April, 2004.

Communicated by:F. Qi

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Journal of Inequalities in Pure and Applied Mathematics

ON CERTAIN INEQUALITIES RELATED TO THE SEITZ INEQUALITY

LIANG-CHENG WANG AND JIA-GUI LUO

Department of Mathematics Daxian Teacher’s College Dazhou 635000, Sichuan Province The People’s Republic of China.

EMail:wangliangcheng@163.com Department of Mathematics Zhongshan University Guangzhou 510275 People’s Republic of China.

c

2000Victoria University ISSN (electronic): 1443-5756 112-03

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On Certain Inequalities Related to the Seitz Inequality

Liang-Cheng Wang and Jia-Gui Luo

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J. Ineq. Pure and Appl. Math. 5(2) Art. 39, 2004

http://jipam.vu.edu.au

Abstract

In this paper, we investigate the monotonicity of difference results from the G.

Seitz inequality. An application is given, with some resulting inequalities.

2000 Mathematics Subject Classification:26D15

Key words: G. Seitz inequality, Convex function, Difference, Monotonicity, Exponen- tial convex.

The first author is partially supported by the Key Research Foundation of the Daxian Teacher’s College under Grant 2003–81.

1. Introduction

For a given positive integern≥2, letX = (x₁, x₂, . . . , x_n),Y = (y₁, y₂, . . . , y_n), U = (u₁, u₂, . . . , u_n) and Z = (z₁, z₂, . . . , z_n) be known sequences of real numbers, and lett_i > 0 (i = 1,2, . . . , n),T_j = Pj

i=1t_i (j = 1,2, . . . , n) and a_ij (i, j = 1,2, . . . , n)be known real numbers. Define the functionsA, J, C, W andGby

A(n)=⁴

n

X

i=1

x_i−n

n

Y

i=1

x_i

!_n¹

(x_i >0, i= 1,2, . . . , n),

J(n)=⁴

n

X

i=1

t_if(v_i)−T_nf 1 Tn

n

X

i=1

t_iv_i

! ,

wheref is convex function on the intervalI andv_i ∈I(i= 1,2, . . . , n), C(n)=⁴

" _n X

i=1

x²_i

! _n X

i=1

y_i²

!#¹₂

−

n

X

i=1

x_iy_i,

W(n)=⁴ T_n

n

X

i=1

t_ix_iz_i−

n

X

i=1

t_ix_i

! _n X

i=1

t_iz_i

! ,

and G(n)=⁴

n

X

i,j=1

a_ijx_iz_j

! _n X

i,j=1

a_ijy_iu_j

!

−

n

X

i,j=1

a_ijx_iu_j

! _n X

i,j=1

a_ijy_iz_j

! .

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Rade investigated the monotonicity of difference forA−Gmean inequality, and obtained the following inequality [2]

(1.1) A(n)≥A(n−1).

P. M. Vasi´c and J. E. Peˇcari´c generalized inequality (1.1) to convex functions, and obtained the following inequality [4,6]

(1.2) J(n)≥J(n−1).

Recently the first author and Xu Zhang studied inequality (1.2) in depth, and obtained some inequalities. L.-C. Wang also obtained some applications, one of them is the following inequality [8]

(1.3) C(n)≥C(n−1).

Inequality (1.3) resulted from the Cauchy inequality (1.4)

n

X

i=1

x²_i

! _n X

i=1

y_i²

!

≥

n

X

i=1

x_iy_i

!2

.

In [7], L.-C. Wang proved the following inequality

(1.5) W(n)≥W(n−1),

withXandZboth increasing or both decreasing. If one ofXorZis increasing and the other decreasing, then the inequality (1.5) reverses.

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Inequality (1.5) resulted from the following Chebyshev inequality

(1.6) T_n

n

X

i=1

t_ix_iz_i ≥

n

X

i=1

t_ix_i

! _n X

i=1

t_iz_i

! ,

withXandZboth increasing or both decreasing. If one ofXorZis increasing and the other decreasing, then the inequality (1.6) reverses.

Assume thati, j, r, s ∈Nsuch that1 ≤ i < j ≤n and1 ≤r < s ≤ n, we have

(1.7)

x_i x_j y_i y_j

z_r z_s u_r u_s

≥0

and (1.8)

a_ir a_is ajr ajs

≥0.

When both (1.7) and (1.8) are true, the following inequality by G. Seitz [1]

holds:

(1.9)

n

P

i,j=1

a_ijx_iz_j

n

P

i,j=1

a_ijx_iu_j

≥

n

P

i,j=1

a_ijy_iz_j

n

P

i,j=1

a_ijy_iu_j .

If

(1.10) X =Z, Y =U and aij =

( 1 i=j

0 i6=j (i, j = 1,2, . . . , n),

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then inequality (1.9) changes into (1.4). If (1.11) Y =U = (1,1, . . . ,1) and aij =

( t_i i=j

0 i6=j (i, j = 1,2, . . . , n), then inequality (1.9) changes into (1.6).

In this paper, we investigate inequality (1.9) in depth, obtaining the following main result.

Theorem 1.1. If both inequalities (1.7) and (1.8) are true, then we have

(1.12) G(n)≥G(n−1).

Remark 1.1. If we put (1.10) and (1.11) into (1.12), then (1.12) becomes (1.3) and (1.5), respectively. Hence, (1.12) is an extension of (1.3) and (1.5).

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2. Proof of Theorem 1.1

Using

a_ijx_iz_j

a_iny_iu_n

=

a_ijy_iz_j

a_inx_iu_n

(i, j = 1,2, . . . , n−1),

aijyiuj

ainxizn

=

aijxiuj

ainyizn

(i, j = 1,2, . . . , n−1), and (1.7) – (1.8), we have

n−1

X

i,j=1

a_ijx_iz_j

n−1

X

i=1

a_iny_iu_n−

n−1

X

i,j=1

a_ijy_iz_j

n−1

X

i=1

a_inx_iu_n (2.1)

+

n−1

X

i,j=1

a_ijy_iu_j

n−1

X

i=1

a_inx_iz_n−

n−1

X

i,j=1

a_ijx_iu_j

n−1

X

i=1

a_iny_iz_n

=

n−1

X

i=1 n−1

X

j=1

a_ijx_iz_j

n−1

X

k=1

a_kny_ku_n−

n−1

X

i=1 n−1

X

j=1

a_ijy_iz_j

n−1

X

k=1

a_knx_ku_n

+

n−1

X

i=1 n−1

X

j=1

a_ijy_iu_j

n−1

X

k=1

a_knx_kz_n−

n−1

X

i=1 n−1

X

j=1

a_ijx_iu_j

n−1

X

k=1

a_kny_kz_n

=

n−1

X

i=1 n−1

X

j=1 n−1

X

k=1,k6=i

a_ija_kn

×

x_iz_jy_ku_n+x_kz_ny_iu_j−y_iz_jx_ku_n−x_iu_jy_kz_n

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=

n−1

X

j=1 n−2

X

i=1 n−1

X

k=2,i<k

+

n−1

X

i=2 n−2

X

k=1,i>k

! a_ija_kn

×

=

n−1

X

j=1 n−2

X

i=1 n−1

X

k=2,i<k

a_ija_kn

x_iz_jy_ku_n+x_kz_ny_iu_j −y_iz_jx_ku_n−x_iu_jy_kz_n

+

n−1

X

j=1 n−1

X

k=2 n−2

X

i=1,k>i

a_kja_in

x_kz_jy_iu_n+x_iz_ny_ku_j −y_kz_jx_iu_n−x_ku_jy_iz_n

=

n−1

X

j=1

X

1≤i<k<n

aijakn−akjain

×

=

n−1

X

j=1

X

1≤i<k<n

a_ij a_in a_kj a_kn

x_i x_k y_i y_k

z_j z_n u_j u_n

≥0.

Using

a_ijx_iz_j

a_njy_nu_j

=

a_ijx_iu_j

a_njy_nz_j

(i, j = 1,2, . . . , n−1),

a_ijy_iu_j

a_njx_nz_j

=

a_ijy_iz_j

a_njx_nu_j

(i, j = 1,2, . . . , n−1),

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(1.7) – (1.8) and the same method as in the proof of (2.1), we obtain

n−1

X

i,j=1

a_ijx_iz_j

n−1

X

j=1

a_njy_nu_j −

n−1

X

i,j=1

a_ijx_iu_j

n−1

X

j=1

a_njy_nz_j (2.2)

+

n−1

X

i,j=1

aijyiuj n−1

X

j=1

anjxnzj −

n−1

X

i,j=1

aijyizj n−1

X

j=1

anjxnuj

=

n−1

X

i=1 n−1

X

j=1 n−1

X

k=1,k6=j

a_ija_nk

×

x_iz_jy_nu_k+y_iu_jx_nz_k−x_iu_jy_nz_k−y_iz_jx_nu_k

=

n−1

X

i=1

X

1≤j<k<n

a_ij a_ik anj ank

x_i x_n yi yn

z_j z_k uj uk

≥0.

Using

a_iny_iu_n

a_jnx_jz_n

=

a_iny_iz_n

a_jnx_ju_n

(i, j = 1,2, . . . , n) and

a_niy_nu_i

a_njx_nz_j

=

a_niy_nz_i

a_njx_nu_j

(i, j = 1,2, . . . , n−1),

we obtain (2.3)

n

X

i=1

a_iny_iu_n

n

X

j=1

a_jnx_jz_n−

n

X

i=1

a_iny_iz_n

n

X

j=1

a_jnx_ju_n= 0

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and (2.4)

n−1

X

i=1

a_niy_nu_i

n−1

X

j=1

a_njx_nz_j−

n−1

X

i=1

a_niy_nz_i

n−1

X

j=1

a_njx_nu_j = 0,

respectively.

Using

a_nny_nu_n

a_njx_nz_j

=

a_njy_nz_j

a_nnx_nu_n

(j = 1,2, . . . , n−1),

a_njy_nu_j

a_nnx_nz_n

=

a_nny_nz_n

a_njx_nu_j

(j = 1,2, . . . , n−1), and (1.7) – (1.8), we have

n

X

i=1

a_iny_iu_n

n−1

X

j=1

a_njx_nz_j−

n−1

X

j=1

a_njy_nz_j

n

X

i=1

a_inx_iu_n

! (2.5)

+

n−1

X

j=1

a_njy_nu_j

n

X

i=1

a_inx_iz_n−

n

X

i=1

a_iny_iz_n

n−1

X

j=1

a_njx_nu_j

!

+

n−1

X

i,j=1

a_ija_nnx_iz_jy_nu_n−

n−1

X

i,j=1

a_ija_nny_iz_jx_nu_n

!

+

n−1

X

i,j=1

aijannyiujxnzn−

n−1

X

i,j=1

aijannxiujynzn

!

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=

n−1

X

i=1

a_iny_iu_n

n−1

X

j=1

a_njx_nz_j −

n−1

X

j=1

a_njy_nz_j

n−1

X

i=1

a_inx_iu_n

!

+

n−1

X

j=1

anjynuj n−1

X

i=1

ainxizn−

n−1

X

i=1

ainyizn n−1

X

j=1

anjxnuj

!

+

n−1

X

i=1 n−1

X

j=1

n−1

X

i=1 n−1

X

j=1

!

+

n−1

X

i=1 n−1

X

j=1

a_ija_nny_iu_jx_nz_n−

n−1

X

i=1 n−1

X

j=1

a_ija_nnx_iu_jy_nz_n

!

=

n−1

X

i=1 n−1

X

j=1

a_ina_nj

y_iu_nx_nz_j+x_iz_ny_nu_j−x_iu_ny_nz_j−y_iz_nx_nu_j

+

n−1

X

i=1 n−1

X

j=1

a_ija_nn

x_iz_jy_nu_n+x_nz_ny_iu_j−y_iz_jx_nu_n−x_iu_jy_nz_n

=

n−1

X

i=1 n−1

X

j=1

a_ija_nn−a_ina_nj

×

x_iz_jy_nu_n+x_nz_ny_iu_j −y_iz_jx_nu_n−x_iu_jy_nz_n

=

n−1

X

i=1 n−1

X

j=1

a_ij a_in a_nj a_nn

x_i x_n y_i y_n

z_j z_n u_j u_n

≥0.

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By (2.1)-(2.5) and definition ofG(n), we have G(n)−G(n−1) =

n−1

X

i,j=1

a_ijx_iz_j +

n

X

i=1

a_inx_iz_n+

n−1

X

j=1

a_njx_nz_j

!

×

n−1

X

i,j=1

a_ijy_iu_j +

n

X

i=1

a_iny_iu_n+

n−1

X

j=1

a_njy_nu_j

!

−

n−1

X

i,j=1

a_ijx_iu_j +

n

X

i=1

a_inx_iu_n+

n−1

X

j=1

a_njx_nu_j

!

×

n−1

X

i,j=1

a_ijy_iz_j+

n

X

i=1

a_iny_iz_n+

n−1

X

j=1

a_njy_nz_j

!

−

n−1

X

i,j=1

a_ijx_iz_j

n−1

X

i,j=1

a_ijy_iu_j −

n−1

X

i,j=1

a_ijx_iu_j

n−1

X

i,j=1

a_ijy_iz_j

!

=

n−1

X

i,j=1

aijxizj n−1

X

i=1

ainyiun−

n−1

X

i,j=1

aijyizj n−1

X

i=1

ainxiun

!

+

n−1

X

i,j=1

n−1

X

i,j=1

!

+

n−1

X

i,j=1

a_ijy_iu_j

n−1

X

i=1

a_inx_iz_n−

n−1

X

i,j=1

a_ijx_iu_j

n−1

X

i=1

a_iny_iz_n

!

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+

n−1

X

i,j=1

a_ija_nny_iu_jx_nz_n−

n−1

X

i,j=1

a_ija_nnx_iu_jy_nz_n

!

+

n−1

X

i,j=1

aijxizj n−1

X

j=1

anjynuj−

n−1

X

i,j=1

aijxiuj n−1

X

j=1

anjynzj

!

+

n−1

X

i,j=1

a_ijy_iu_j

n−1

X

j=1

a_njx_nz_j−

n−1

X

i,j=1

a_ijy_iz_j

n−1

X

j=1

a_njx_nu_j

!

+

n

X

i=1

ainyiun n

X

j=1

ajnxjzn−

n

X

i=1

ainyizn n

X

j=1

ajnxjun

!

+

n−1

X

i=1

a_niy_nu_i

n−1

X

j=1

a_njx_nz_j−

n−1

X

i=1

a_niy_nz_i

n−1

X

j=1

a_njx_nu_j

!

+

n

X

i=1

a_iny_iu_n

n−1

X

j=1

a_njx_nz_j−

n−1

X

j=1

a_njy_nz_j

n

X

i=1

a_inx_iu_n

!

+

n−1

X

j=1

a_njy_nu_j

n

X

i=1

a_inx_iz_n−

n

X

i=1

a_iny_iz_n

n−1

X

j=1

a_njx_nu_j

!

≥0,

i.e., inequality (1.12) is true. This completes the proof of theorem .

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3. Applications

Let E be a convex subset of an arbitrary real linear spaceK, and letf : E 7→

(0,+∞). f is an exponential convex function onE, if and only if

(3.1) f

tu+ (1−t)v

≤f^t(u)f^1−t(v)

for anyu, v ∈E and anyt ∈[0,1]. f is an exponential concave function onE, if and only if the inequality (3.1) reverses (see [4]).

For anyu, v ∈E(u6=v)andα_ki, β_ki ∈[0,1], we letx_ki =α_kiu+ (1−α_ki)v and yki = βkiu+ (1−βki)v (k = 1,2;i = 1,2, . . . , n;n > 2). Define a functionLby

L(n) =

n

X

i,j=1

a_ijf(x_1i)f(x_2j)

n

X

i,j=1

a_ijf(y_1i)f(y_2j)

−

n

X

i,j=1

a_ijf(x_1i)f(y_2j)

n

X

i,j=1

a_ijf(y_1i)f(x_2j).

Proposition 3.1. Let f be an exponential convex (or concave) function on E and inequality (1.8) be true. Fork = 1,2and every pair of positive integersi andj such that1≤i < j ≤n, if

(3.2) αki ≤βki ≤αkj and αkj −αki =βkj−βki, then we have

(3.3) L(n)≥L(n−1).

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Proof. 1. Supposef is an exponential convex function onE. Fork = 1,2 and1≤i < j ≤n, from (3.2), we haveβkj =αkj +βki−αki ≥αkj. Case 1. Whenαki < βki ≤αkj < βkj, we taket = ^β_β^ki^−α^ki

kj−α_ki, then1−t =

βkj−β_ki

βkj−α_ki. Hence, we have

(3.4) tykj + (1−t)xki =βkiu+ (1−βki)v =yki. From (3.1) and (3.4), we have

(3.5) f(yki)≤f^t(ykj)f^1−t(xki).

From (3.2), we get the other form oftand1−t: t = ^β_β^kj^−α^kj

kj−α_ki and1−t =

αkj−α_ki

βkj−αki. Then we have

(3.6) (1−t)y_kj +tx_ki =α_kju+ (1−α_kj)v =x_kj. From (3.1) and (3.6), we have

(3.7) f(x_kj)≤f^1−t(y_kj)f^t(x_ki).

From (3.5) and (3.7), we obtain (3.8)

f(x_ki) f(x_kj) f(yki) f(ykj)

≥0.

Case 2. Whenα_ki =β_ki (orα_kj =β_kj), by (3.2), then we haveα_kj =β_kj (orαki =βki). Hence, the equality of (3.8) holds.

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For any1≤i < j ≤nand any1≤r < s≤n, by (3.8), we obtain (3.9)

f(x_1i) f(x_1j) f(y1i) f(y1j)

f(x_2r) f(x_2s) f(y2r) f(y2s)

≥0.

2. Letfbe an exponential concave function onE. Then (3.5), (3.7) and (3.8) reverse. Hence, (3.9) is still valid.

Replacingxi, yi,ziandui in Theorem1.1withf(x1i), f(y1i),f(x2i)and f(y_2i)(i = 1,2, . . . , n), respectively, we obtain (3.3). This completes the proof of Proposition3.1.

Remark 3.1. In Proposition3.1, whenEis a real intervalI, we only need

xki ≤yki ≤xkj and xkj−xki =ykj−yki,

wherek = 1,2,i, j are every pair of positive integers such that1≤i < j ≤n, x_ki, x_kj, y_ki, y_kj ∈I.

In order to verify Proposition3.3, the following lemma is necessary.

Lemma 3.2. Let c, d : [a, b] 7→ R(b > a) be the monotonic functions, both increasing or both decreasing. Furthermore, let q : [a, b] 7→ (0,+∞)be an integrable function. Then

(3.10) Z b

a

q(x)c(x)dx Z b

a

q(x)d(x)dx≤ Z b

a

q(x)dx Z b

a

q(x)c(x)d(x)dx.

If one of the functions ofcordis increasing and the other decreasing, then the inequality (3.10) reverses. (see [2,3]).

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Letp : [a, b]7→ (0,+∞)be continuous,g : [a, b]7→(1,+∞)be monotonic continuous. Define a functionM by

M(n) =

n

X

i,j

a_ijh^(k+i)(x)h^(m+j)(x)

n

X

i,j

a_ijh^(l+i)(x)h^(w+j)(x)

−

n

X

i,j

a_ijh^(k+i)(x)h^(w+j)(x)

n

X

i,j

a_ijh^(l+i)(x)h^(m+j)(x),

wherek, l, m, w∈N,i, j = 1,2, . . . , nand (3.11) h:R7→R⁺, h(x) =

Z b a

p(t) g(t)x

dt (see [5]).

Proposition 3.3. Let the inequalities in (1.8) hold. Ifk < l,m < w ork > l, m > w, then we have

(3.12) M(n)≥M(n−1).

Proof. For (3.11), by continuity ofpandg,we may change the order of deriva- tion and integration. By direct computation, we get

(3.13) h⁽ⁿ⁾(x) =

Z b a

p(t) (g(t))^x(lng(t))ⁿdt.

For every pair of integersi, j such that1≤ i < j ≤ n, when k < l, replaceq, canddin Lemma3.2byp(t) (g(t))^x(lng(t))^k+i,(lng(t))^j−i and(lng(t))^l−k, respectively. Using (3.13), we get

(3.14) h^(k+i)(x)h^(l+j)(x)≥h^(k+j)(x)h^(l+i)(x).

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By (3.14), we have (3.15)

h^(k+i)(x) h^(k+j)(x) h^(l+i)(x) h^(l+j)(x)

≥0.

Similarly we obtain (3.16)

h^(m+r)(x) h^(m+s)(x) h^(w+r)(x) h^(w+s)(x)

≥0,

wherer,sare pair of integer such that1≤r < s ≤nandm < w.

Replacingx_i,y_i,z_iandu_iin Theorem1.1byh^(k+i)(x),h^(l+i)(x),h^(m+i)(x) andh^(w+i)(x)(i= 1,2, . . . , n), respectively, we obtain (3.12).

By Lemma 3.2, when k > l and m > w, both (3.15) and (3.16) reverse.

Hence, the product on the left for both (3.15) and (3.16) is still nonnegative, hence, by Theorem1.1, (3.12) is also satisfied.

This completes the proof of Proposition3.3.

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References

[1] G. SEITZ, Une remarque aux inégalités, Aktuarské Vˇedy, 6 (1936), 167–

171.

[2] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, Berlin/New York, 1970.

[3] BAI-NI GUO AND FENG QI, Inequalities for generalized weighted mean values of convex function, Math. Ineq. Appl., 4(2) (2001), 195–202.

[4] CHUNG-LIE WANG, Inequalities of the Rado-Popoviciu type for functions and their applications, J. Math. Anal. Appl., 100 (1984), 436–446.

[5] FENG QI, Generalized weighted mean values with two parameters , Proc.

R. Soc. Lond. A., 454 (1998), 2723–2732.

[6] P.M. VASI ´CANDJ.E. PE ˇCARI ´C, On the Jensen inequality, Univ. Beograd.

Publ. Elektrotehn. Fak. Ser. Math. Fiz., 634–677 (1979), 50–54.

[7] L.-C. WANG, On the monotonicity of difference generated by the inequal- ity of Chebyshev type, J. Sichuan University (Natural Science Edition), 39 (2002), 338–403.

[8] L.-C. WANG, Convex Functions and Their Inequalities, Sichuan University Press, Chengdu, China, 2001. (In Chinese).