Journal of Inequalities in Pure and
Applied Mathematics
ON GENERALIZATIONS OF L. YANG’S INEQUALITY
CHANG-JIAN ZHAO AND LOKENATH DEBNATH
Department of Mathematics
Shanghai University
Shanghai 200436
The People’s Republic of China;
volume 3, issue 4, article 56,
2002.
Received 03 December, 2001;
accepted 02 June, 2002.
Communicated by: Feng Qi
and
Department of Mathematics
Binzhou Teachers College
Binzhou, Shandong 256604
The People’s Republic of China.
EMail: chjzhao@163.com
Department of Mathematics
University of Texas-Pan American
Edinburg, Texas 78539-2999, U.S.A.
EMail: Debnathl@panam.edu
Abstract
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
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Abstract
A geometric inequality due to L. Yang involving cosine and sine functions is
generalized.
2000 Mathematics Subject Classification: 26D15
Key words: L. Yang’s inequality, Theory of distribution of functional values, Jensen
inequality
On Generalizations of L. Yang’s
Inequality
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Chang-jian Zhao and
Lokenath Debnath
3
4
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1.
Introduction
The well-known inequality due to L. Yang [1, pp. 116–118] can be stated as
follows:
If A > 0, B > 0, A + B ≤ π and 0 ≤ λ ≤ 1, then
(1.1)
cos2 λA + cos2 λB − 2 cos λA · cos λB · cos λπ ≥ sin2 λπ.
The equality in (1.1) holds if and only if λ = 0 or A + B = π.
L. Yang’s inequality plays an important role in the theory of distribution of
values of functions. See [1].
In this paper we will generalize inequality (1.1).
On Generalizations of L. Yang’s
Inequality
Chang-jian Zhao and
Lokenath Debnath
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2.
Main Results
In this paper, we use the following notations and abbreviations:
n
n!
;
=
r!(n − r)!
r
λ
λ
[2]
[1]
xij = (π + Ai + Aj ), xij = (π − Ai − Aj );
2
2
λ
λ
[4]
[3]
xij = (π + Ai − Aj ), xij = (π − Ai + Aj );
2
2
2
2
Hij = cos λAi + cos λAj − 2 cos λAi · cos λAj · cos λπ.
Lemma 2.1 ([2]). If Ai > 0, Aj > 0, Ai + Aj ≤ π for 1 ≤ i, j ≤ n, and
−1 ≤ λ ≤ 1, then
4
Y
[k]
2
sin xij .
Hij − sin λπ = 4
k=1
Proof. By using the following two identities
cos λ (Ai + Aj ) cos λ (Ai − Aj ) = cos2 λAi + cos2 λAj − 1,
cos λ (Ai + Aj ) + cos λ (Ai − Aj ) = 2 cos λAi cos λAj ,
it is easy to observe that
On Generalizations of L. Yang’s
Inequality
Chang-jian Zhao and
Lokenath Debnath
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2
Hij − sin λπ
= cos2 λAi + cos2 λAj − 1 − 2 cos λAi cos λAj cos λπ + cos2 λπ
= cos2 λπ + cos λ (Ai + Aj ) cos λ (Ai − Aj )
− [cos λ (Ai + Aj ) + cos λ (Ai − Aj )] cos λπ
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= [cos λπ − cos λ (Ai + Aj )] [cos λπ − cos λ (Ai − Aj )]
λ
λ
= 4 sin (π + Ai + Aj ) sin (π − Ai − Aj )
2
2
λ
λ
× sin (π + Ai − Aj ) sin (π − Ai + Aj ) .
2
2
The proof is complete.
P
Theorem 2.2. If Ai > 0, λk > 0 (i = 1, 2, . . . , n; k = 1, 2, 3, 4), ni=1 Ai ≤ π,
n ≥ 2 being a natural number, and −1 ≤ λ ≤ 1, then
n
(2.1)
sin2 λπ
2
" n
#2
n
X
X
≤ (n − 1 + cos λπ)
cos2 λAk − cos λπ
cos λAi
i=1
k=1
4
P4
n
k=1 λk
2
≤
sin λπ +
Q
2
64 4k=1 λk
X
sin4 θij ,
1≤i<j≤n
Chang-jian Zhao and
Lokenath Debnath
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where
[k]
k=1 λk xij
.
P4
k=1 λk
P4
(2.2)
On Generalizations of L. Yang’s
Inequality
θij =
The equalities in (2.1) hold if and only if λ = 0.
Proof. Since y = sin x is a continuous and convex (or concave, resp.) function
[k]
on [−π, 0] (or [0, π], resp.), and xij ∈ [−π, 0] (or [0, π], resp.) for −1 ≤ λ ≤ 0
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(or 0 ≤ λ ≤ 1, resp.), and using Jensen’s inequality (see [3]), we observe that
P4
[k]
k=1 λk sin xij
sin θij ≤ (or ≥ , resp.)
.
P4
k=1 λk
Consequently
4
X
(2.3)
!4
λk
4
(sin θij ) ≥
k=1
4
X
!4
[k]
λk sin xij
.
k=1
[k]
[k]
On the other hand, since λk sin −xij ≥ 0 or λk sin xij ≥ 0, resp. , then
4
1X
4
(2.4)
[k]
λk sin −xij
v
u 4
uY
[k]
4
≥t
λk sin xij .
v
!
u 4
4
X
uY
1
[k]
[k]
4
or
λk sin xij ≥ t
λk sin xij , resp. .
4 k=1
k=1
(2.5)
[k]
sin xij
k=1
4
k=1 λk
Q
44 4k=1 λk
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P4
≤
4
sin θij .
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By using Lemma 2.1, we have
4
λ
k
k=1
sin2 λπ ≤ Hij ≤
sin4 θij + sin2 λπ.
Q4
64 k=1 λk
P4
(2.6)
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From (2.3) and (2.4), we obtain
4
Y
Chang-jian Zhao and
Lokenath Debnath
Contents
k=1
k=1
On Generalizations of L. Yang’s
Inequality
J. Ineq. Pure and Appl. Math. 3(4) Art. 56, 2002
http://jipam.vu.edu.au
Summing both sides of (2.6) for 1 ≤ i < j ≤ n yields
X
X
(2.7)
sin2 λπ ≤
Hij
1≤i<j≤n
1≤i<j≤n
≤
X
1≤i<j≤n
!
4
λ
k=1 k
sin4 θij + sin2 λπ .
Q4
64 k=1 λk
P4
It is not difficult to see that
X
n
2
(2.8)
sin2 λπ.
sin λπ =
2
1≤i<j≤n
Direct computing yields
(2.9)
X
1≤i<j≤n
On Generalizations of L. Yang’s
Inequality
Chang-jian Zhao and
Lokenath Debnath
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!
4
λ
k
k=1
sin4 θij + sin2 λπ
Q4
64 k=1 λk
4
P4
X
n
k=1 λk
4
=
sin θij +
sin2 λπ,
Q4
2
64 k=1 λk 1≤i<j≤n
P4
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(2.10)
X
Hij
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1≤i<j≤n
=
X
(cos2 λAi + cos2 λAj − 2 cos λAi cos λAj cos λπ)
1≤i<j≤n
J. Ineq. Pure and Appl. Math. 3(4) Art. 56, 2002
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=
X
(cos2 λAi + cos2 λAj ) −
1≤i<j≤n
= (n − 1)
X
2 cos λAi cos λAj cos λπ
1≤i<j≤n
n
X

cos2 λAk − cos λπ 
!2
cos λAi
i=1
k=1
= (n − 1 + cos λπ)
n
X
n
X
k=1
cos2 λAk − cos λπ
−
n
X

cos2 λAi 
i=1
n
X
!2
cos λAi
.
i=1
Putting (2.8), (2.9) and (2.10) into (2.7), inequality (2.7) reduces to inequality (2.1). The proof is complete.
Remark 2.1. If taking λi = 14 (i = 1, 2, 3, 4) in the right-hand of (2.7), we find
that
!
4
P4
X
λ
k
k=1
(2.11)
sin4 θij + sin2 λπ
Q4
64
λ
k=1 k
1≤i<j≤n


!
4
P4
P4
[k] 4
X
k=1 λk xij
k=1 λk
 Q
+ sin2 λπ 
=
sin P
4
4
64 k=1 λk
k=1 λk
1≤i<j≤n


!!4
4
X
X
[k]
4 sin 1
=
xij
+ sin2 λπ 
4
1≤i<j≤n
k=1
X n
λπ
4 λ
2
=
4 sin π + sin λπ = 4
sin2
.
2
2
2
1≤i<j≤n
On Generalizations of L. Yang’s
Inequality
Chang-jian Zhao and
Lokenath Debnath
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Therefore, inequality (2.1) reduces to the following inequality
n
(2.12)
sin2 λπ
2
≤ (n − 1 + cos λπ)
n
X
cos2 λAk − cos λπ
k=1
n
X
!2
cos λAi
i=1
n
λ
≤4
sin2 π.
2
2
On Generalizations of L. Yang’s
Inequality
The equalities in (2.12) hold if and only if λ = 0.
Letting n = 2 in (2.12), we have the following result: If A > 0, B > 0,
A + B ≤ π, and −1 ≤ λ ≤ 1, then
λ
(2.13) sin2 λπ ≤ cos2 λA + cos2 λB − 2 cos λA cos λB cos λπ ≤ 4 sin2 π.
2
The equalities in (2.13) holds if and only if λ = 0.
It is obvious that inequality (2.13) is the same as L. Yang’s inequality (1.1).
Theorem
2.3. If Ai > 0 and λk > 0 for i = 1, 2, . . . , n and k = 1, 2, 3, 4,
Pn
i=1 Ai ≤ π, n ≥ 2, −1 ≤ λ ≤ 1, then
(2.14)
n
X
n
0 ≤ (n − 1)
cos λAk −
sin2 λπ
2
k=1
X
− cos λπ
cos λAi cos λAj
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1≤i,j≤n
i6=j
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P4
≤
128
k=1 λk
Q4
4
X
k=1 λk
where
sin4 θij ,
1≤i,j≤n
i6=j
[k]
k=1 λk xij
.
P4
k=1 λk
P4
θij =
On Generalizations of L. Yang’s
Inequality
The equalities in (2.14) hold if and only if λ = 0.
Proof. It follows from inequality (2.6) that
4
P4
λ
k
k=1
sin4 θij .
(2.15)
0 ≤ Hij − sin2 λπ ≤
Q4
64 k=1 λk
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Summing both sides of (2.15) for i 6= j, first over j from 1 to n and then over i
from 1 to n of the resulting inequality, then
4
P4
X
X
n
k=1 λk
2
(2.16)
0≤
Hij − 2
sin λπ ≤
sin4 θij .
Q4
2
64 k=1 λk 1≤i,j≤n
1≤i,j≤n
i6=j
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On the other hand
(2.17)
X
1≤i,j≤n
i6=j
Hij = 2(n − 1)
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n
X
cos2 λAk − 2 cos λπ
k=1
X
1≤i,j≤n
i6=j
cos λAi cos λAj .
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Putting (2.17) into (2.16), we get inequality (2.14).
Remark 2.2. In (2.14), if λk =
1
4
for k = 1, 2, 3, 4, then
n
X
n
cos λAk −
0 ≤ (n − 1)
sin2 λπ
2
k=1
X
− cos λπ
cos λAi cos λAj
2
1≤i,j≤n
i6=j
n
λ
≤
sin4 π.
2
2
The equalities in (2.18) hold if and only if λ = 0.
Letting n = 2, then (2.18) reduces to inequality (2.13).
On Generalizations of L. Yang’s
Inequality
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Lokenath Debnath
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References
[1] L. YANG, Distribution of Values and New Research, Science Press, Beijing,
1982. (Chinese).
[2] Ch.-J. ZHAO, Further research of L. Yang’s inequality, J. Binzhou Teachers
College, 12(4) (1996), 32–34. (Chinese).
[3] D. S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, 1970.
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