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Generalization of the Wolstenholme cyclic inequality and its application
Computers and Mathematics with Applications 53 (2007) 104–114 www.elsevier.com/locate/camwa
Generalization of the Wolstenholme cyclic inequality and its application Shanhe Wu a , Lokenath Debnath b,∗ a Department of Mathematics, Longyan College, Longyan Fujian 364012, China b Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA
Received 17 January 2006; received in revised form 5 June 2006; accepted 14 June 2006
Abstract This paper deals with the generalization and sharp versions of the Wolstenholme cyclic inequality and their applications. The inequalities of this paper improve and unify corresponding known results. Several interesting inequalities including the celebrated Ozeki inequality are obtained. Extensions of the Wolstenholme inequality for a complex polygon and the Wolstenholme inequality for a convex quadrilateral are derived. As example of applications, the well-known Erd¨os–Mordell inequality is improved in this paper. In addition, several extensions, unifications and refinements of Gueron–Shafrir’s inequalities and Mitrinovi´c–Pecaric’s inequality are established. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Wolstenholme’s inequality; Erd¨os–Mordell’s inequality; Weighted power means inequality; Power means inequality; Convex polygon; Improvement
1. Introduction If x1 , x2 , x3 , ϕ1 , ϕ2 , ϕ3 are real numbers and ϕ1 + ϕ2 + ϕ3 = π, the following celebrated cyclic inequality x12 + x22 + x32 ≥ 2x1 x2 cos ϕ1 + 2x2 x3 cos ϕ2 + 2x3 x1 cos ϕ3
(1)
is known as Wolstenholme’s inequality in the literature (see Mitrinovi´c et al. [1] p. 425, and Mitrinovi´c et al. [2] p. 421). In 1867, Wolstenholme first introduced inequality (1) with a historical account in his book [3]. It is important to point out that the Wolstenholme inequality provided an effective approach to combine both geometric and arithmetic inequalities. Based on the Wolstenholme inequality, a large number of new inequalities have been discovered with applications. In recent years, considerable attention has been given to study this inequality in many different directions including its various generalizations, unifications, refinements and applications (see Bottema et al. [4], Mitrinovi´c et al. [5], Murty et al. [6] and Mitrinovi´c et al. [7] and references therein). On the other hand, in 1957, Ozeki [8] proved
∗ Corresponding author. Tel.: +1 956 381 3459; fax: +1 956 384 5091.
E-mail addresses: [email protected] (S. Wu), [email protected] (L. Debnath). c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2006.06.006
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a notable generalization of (1) in his work on the extension of the Erd¨os–Mordell’s inequality for a convex polygon which reads as ! n n X π X 2 xi xi+1 cos ϕi , (2) xi ≥ sec n i=1 i=1 where x1 , x2 , . . . , xn , ϕ1 , ϕ2 , . . . , ϕn are real numbers, xn+1 = x1 , ϕ1 + ϕ2 + · · · + ϕn = π, n > 2. In spite of the above work, further study on the Wolstenholme inequality is needed. So, the purpose of this paper is to prove several new generalized and sharp versions of the Wolstenholme inequality and their applications to the investigation of geometric inequalities. Under the weak constraint condition ϕ1 +ϕ2 +· · ·+ϕn = θ , where 0 < θ ≤ π, we prove that the Ozeki type inequalities are valid and can be improved to a sharper version (see Theorem 3 below). It is shown in this paper that the improved version of (1) can be used to study geometric inequalities. In Section 5 of this paper, we demonstrate that the above inequalities can effectively be used to improve the Erd¨os–Mordell inequality and to establish some new Erd¨os–Mordell’s type inequalities. Finally, several extensions, unifications and refinements of Gueron–Shafrir’s inequalities, and Mitrinovi´c–Pe˘cari´c’s inequality are discussed. 2. Lemmas The following weighted power means inequality is well-known (see Hardy et al. [9] and Wu [10]). Lemma 1. If xi ≥ 0, λi > 0, i = 1, 2, . . . , n, 0 < p ≤ 1. Then n X
p λi xi
≤
i=1
n X i=1
!1− p λi
n X
!p λi xi
.
(3)
i=1
Inequality (3) is reversed for p ≥ 1 or p < 0. The above inequality in the special case of λ1 = λ2 = · · · = λn = 1 is called power means inequality. Lemma 2. Let x1 , x2 , x3 , ϕ1 , ϕ2 , ϕ3 be real numbers, α1 , α2 , α3 be positive numbers, and let π. Then
P3
i=1 αi
=
P3
i=1 ϕi
=
(x12 + x22 ) cot α1 + (x22 + x32 ) cot α2 + (x32 + x12 ) cot α3 ≥ 2(x1 x2 cos ϕ1 csc α1 + x2 x3 cos ϕ2 csc α2 + x3 x1 cos ϕ3 csc α3 ). P3 P3 αi = i=1 ϕ1 = π, after some simple calculations, we obtain Proof. According to i=1
(4)
(x12 + x22 ) cot α1 + (x22 + x32 ) cot α2 + (x32 + x12 ) cot α3 = x12 sin α2 csc α1 csc α3 + x22 sin α3 csc α1 csc α2 + x32 sin α1 csc α2 csc α3 .
(5)
On the other hand, applying Wolstenholme’s inequality with the substitution: x1 → x1 sin α2 , x2 → x2 sin α3 , x3 → x3 sin α1 in (1) yields x12 sin2 α2 + x22 sin2 α3 + x32 sin2 α1 ≥ 2(x1 x2 cos ϕ1 sin α2 sin α3 + x2 x3 cos ϕ2 sin α3 sin α1 + x3 x1 cos ϕ3 sin α1 sin α2 ). Or, equivalently, x12 sin α2 csc α1 csc α3 + x22 sin α3 csc α1 csc α2 + x32 sin α1 csc α2 csc α3 ≥ 2(x1 x2 cos ϕ1 csc α1 + x2 x3 cos ϕ2 csc α2 + x3 x1 cos ϕ3 csc α3 ). Combining (5) and (6) leads to inequality (4). The Lemma 2 is proved.
(6)
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3. Generalization and sharpness of Wolstenholme’s inequality Pn Theorem 1. Let xi , ϕi be real numbers, αi be positive numbers (i = 1, 2, . . . , n), n ≥ 3, and let i=1 αi = P n ϕ = θ, 0 < θ ≤ π, x = x . Then i n+1 1 i=1 ! n n n X X 1X 2 2 sin θ csc αi csc(θ − αi ) (|xi | − |xi+1 |)2 . (7) xi xi+1 cos ϕi csc αi + xi + xi+1 cot αi ≥ 2 n i=1 i=1 i=1 Proof. By using Lemma 2, we have (x1 x2 ) cos ϕ1 csc α1 + (x2 x3 ) cos ϕ2 csc α2 + (x3 x1 ) cos π −
2 X
! ϕi csc π −
i=1
2 X
! αi
i=1
!! 2 X 1 2 2 2 2 2 2 x1 + x2 cot α1 + x2 + x3 cot α2 + x3 + x1 cot π − αi , ≤ 2 i=1 ! ! ! ! 2 3 3 2 X X X X ϕi csc αi + (x3 x4 ) cos ϕ3 csc α3 + (x4 x1 ) cos π − ϕi csc π − αi (x1 x3 ) cos i=1
i=1
i=1
i=1
!! ! 3 2 X X 1 2 , αi x1 + x32 cot αi + x32 + x42 cot α3 + x42 + x12 cot π − ≤ 2 i=1 i=1 ······ ! ! n−2 n−2 X X ϕi csc αi + (xn−1 xn ) cos ϕn−1 csc αn−1 (x1 xn−1 ) cos i=1
+ (xn x1 ) cos π −
i=1 n−1 X
! ϕi csc π −
n−1 X
i=1 n−2 X 1 2 2 ≤ x1 + xn−1 cot αi 2 i=1
! αi
i=1
!
2 + xn−1 + xn2 cot αn−1 + xn2 + x12 cot π −
n−1 X
!! αi
.
i=1
Summing both sides in the above inequalities respectively, we obtain ! ! ! n−1 n−1 n−1 X X X xi xi+1 cos ϕi csc αi − xn x1 cos ϕi csc αi i=1
i=1
i=1
! !! n−1 n−1 X 1 X 1 2 2 2 2 ≤ x + xi+1 cot αi − xn + x1 cot αi . 2 i=1 i 2 i=1 Pn Pn Using the assumptions, i=1 αi = i=1 ϕi = θ, 0 < θ ≤ π, together with the above inequality, it follows that ! ! n n X 1 X 2 2 xi xi+1 cos ϕi csc αi − x + xi+1 cot αi 2 i=1 i i=1 1 2 ≤ xn x1 (cos ϕn csc αn + cos (θ − ϕn ) csc (θ − αn )) − xn + x12 (cot αn + cot (θ − αn )) 2 1 = xn x1 (sin (θ − αn + ϕn ) + sin (θ + αn − ϕn )) csc αn csc (θ − αn ) 2 1 − xn2 + x12 (cot αn + cot (θ − αn )) 2 1 2 = xn x1 sin θ cos (ϕn − αn ) csc αn csc (θ − αn ) − xn + x12 sin θ csc αn csc (θ − αn ) 2 1 2 ≤ |xn x1 sin θ cos (ϕn − αn ) csc αn csc (θ − αn )| − xn + x12 sin θ csc αn csc (θ − αn ) 2
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1 2 xn + x12 sin θ csc αn csc (θ − αn ) 2
≤ |xn x1 sin θ csc αn csc (θ − αn )| −
1 = − sin θ csc αn csc (θ − αn ) (|xn | − |x1 |)2 . 2 That is, ! ! n n X 1 X 1 2 2 xi xi+1 cos ϕi csc αi − xi + xi+1 cot αi ≤ − sin θ csc αn csc (θ − αn ) (|xn | − |x1 |)2 . 2 i=1 2 i=1 An argument similar to the one given above gives ! ! n n X 2 1 1 X 2 2 xi + xi+1 cot αi ≤ − sin θ csc α j csc θ − α j x j − x j+1 , xi xi+1 cos ϕi csc αi − 2 i=1 2 i=1 j = 1, 2, . . . , n − 1. Summing both sides in the above inequalities respectively from j = 1 to j = n gives !! ! n n X 1 X 2 cot αi xi2 + xi+1 n xi xi+1 cos ϕi csc αi − 2 i=1 i=1 ≤−
n 1X sin θ csc αi csc (θ − αi ) (|xi | − |xi+1 |)2 . 2 i=1
This is equivalent to inequality (7). The proof of Theorem 1 is complete.
The following corollary follows from Theorem 1 directly. Corollary 1. Let xi , ϕi be real numbers, αi be positive numbers (i = 1, 2, . . . , n), n ≥ 3, and let Pn ϕ = θ, 0 < θ ≤ π, xn+1 = x1 . Then i i=1 ! n n X X 2 2 xi xi+1 cos ϕi csc αi . xi + xi+1 cot αi ≥ 2
Pn
i=1 αi
=
(8)
i=1
i=1
Remark 1. In Corollary 1, the special case α1 = α2 = · · · = αn = πn yields the foregoing Ozeki’s inequality (2). In the following theorem, we give another generalization of the inequality (8). Theorem 2. Let xi be real numbers, 0 < αi , 0 < ϕi < 0 < θ ≤ π, xn+1 = x1 . Then, for 0 < k ≤ 1, we have n X
2 xi2 + xi+1 cot αi ≥ 2 seck−1
i=1
For k ≥ 1 and 0 < αi < n X
xi2
+
2 xi+1
π 2
X n
(i = 1, 2, . . . , n) , n ≥ 3, and let
Pn
i=1 αi
=
Pn
i=1 ϕi
= θ,
! xi xi+1 cosk ϕi csc αi .
(9)
i=1
(i = 1, 2, . . . , n), we have
cot αi ≥ 2 k
θ n
π 2
2−k
i=1
n X
! xi xi+1 cos ϕi csc αi . k
(10)
i=1
Proof. Case (I). When 0 < k ≤ 1. By the weighted power means inequality, we have n X
xi xi+1 cosk ϕi csc αi
i=1
≤
n X i=1
|xi xi+1 | cos ϕi csc αi ≤ k
n X i=1
!1−k |xi xi+1 | csc αi
n X i=1
!k |xi xi+1 | cos ϕi csc αi
.
(11)
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In addition, using Corollary 1 gives n X
|xi | |xi+1 | cos ϕi csc αi ≤
i=1
n 1X |xi |2 + |xi+1 |2 cot αi , 2 i=1
that is n X
|xi xi+1 | cos ϕi csc αi ≤
i=1
n 1X 2 xi2 + xi+1 cot αi . 2 i=1
On the other hand, we put ϕ1 = ϕ2 = · · · = ϕn = n X
|xi xi+1 | csc αi ≤
i=1
θ 1 sec 2 n
X n
θ n
(12)
in (12), so that
2 cot αi . xi2 + xi+1
(13)
i=1
Combining inequalities (11)–(13) leads to inequality (9). Case (II). When k ≥ 1. Observe that inequality (8) can be rewritten as follows: n+1 X
xi2 (cot αi−1 + cot αi ) ≥ 2
i=2
n X
xi xi+1 cos ϕi csc αi
Replacing xi by |xi | (cot αi−1 + cot αi ) n+1 X
xi2 (cot αi−1 + cot αi )k ≥ 2
n X
i=2
Since n X
(xn+1 = x1 , αn+1 = α1 ) .
i=1 k−1 2
(i = 2, 3 . . . , n + 1) in the above inequalities respectively, we obtain
|xi xi+1 | ((cot αi−1 + cot αi ) (cot αi + cot αi+1 ))
k−1 2
cos ϕi csc αi .
(14)
i=1
|xi xi+1 | ((cot αi−1 + cot αi ) (cot αi + cot αi+1 ))
k−1 2
cos ϕi csc αi
i=1
=
n X
|xi xi+1 | (sin (αi−1 + αi ) sin (αi + αi+1 ) csc αi−1 csc αi+1 )
k−1 2
cos ϕi csck αi
i=1
=
n X
|xi xi+1 | (1 + sin αi sin (αi−1 + αi + αi+1 ) csc αi−1 csc αi+1 )
k−1 2
cos ϕi csck αi ,
i=1
and note that hypotheses k ≥ 1, n X
Pn
i=1 αi
≤ π, 0 < αi <
π 2,
|xi xi+1 | ((cot αi−1 + cot αi ) (cot αi + cot αi+1 ))
0 < ϕi <
k−1 2
π 2
(i = 1, 2, . . . , n), we conclude
cos ϕi csc αi ≥
i=1
n X
xi xi+1 cos ϕi csck αi .
i=1
Applying the power means inequality together with (14) and (15), it follows that n X i=1
n+1 X 2 xi2 + xi+1 cotk αi = xi2 cotk αi−1 + cotk αi i=2
≥ 21−k
n+1 X
xi2 (cot αi−1 + cot αi )k
i=2
≥ 22−k
n X
xi xi+1 cos ϕi csck αi .
i=1
The inequality (10) is proved. This completes the proof of Theorem 2.
The following inequality is a remarkable improvement of Ozeki’s inequality:
(15)
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109
Pn Theorem 3. Let xi , ϕi be real numbers (i = 1, 2, . . . , n) , n ≥ 3, and let i=1 ϕi = θ, 0 < θ ≤ π, xn+1 = x1 . Then ! X n n n X X θ 1 2 xi ≥ sec xi xi+1 cos ϕi + (sin θ/ (1 + sin θ)) (16) (|xi | − |xi+1 |)2 . n n i=1 i=1 i=1 Proof. Taking α1 = α2 = · · · = αn = nθ in Theorem 1 yields ! X X n n n X 1 θ n−1 θ 2 xi xi+1 cos ϕi + sin θ sec csc θ xi ≥ sec (|xi | − |xi+1 |)2 . n 2n n n i=1 i=1 i=1 Since 1 n−2 θ n−1 1 sin θ sin θ + sin sin θ sec csc θ= θ . 2n n n n n
Consequently, we get the inequality (16). The proof is complete.
Remark 2. It is obvious that inequality (16) represents the sharpened version of the Wolstenholme’s inequality and Ozeki’s inequality. Especially, if ϕ1 , ϕ2 , . . . , ϕn are the acute angles, we have the following exponential version of inequality (16). Pn Theorem 4. Let xi be real numbers, 0 < ϕi < π2 (i = 1, 2, . . . , n) , n ≥ 3, and let i=1 ϕi = θ, 0 < θ ≤ π , 0 < k ≤ 1, xn+1 = x1 . Then ! X n n n X X 1 k 2 k θ xi xi+1 cos ϕi + (sin θ/ (1 + sin θ)) (17) xi ≥ sec (|xi | − |xi+1 |)2 . n n i=1 i=1 i=1 Proof. When k = 1, inequality (17) is equivalent to inequality (16). When 0 < k < 1, by using the weighted power means inequality, we obtain !k !1−k n n n n X X X X k k |xi xi+1 | cos ϕi ≤ |xi xi+1 | |xi xi+1 | cos ϕi . (18) xi xi+1 cos ϕi ≤ i=1
i=1
i=1
i=1
In addition, Theorem 3 gives ! # " X n n n X X 1 θ 2 2 |xi xi+1 | cos ϕi ≤ cos xi − (sin θ/ (1 + sin θ)) (|xi | − |xi+1 |) . n n i=1 i=1 i=1 On the other hand, we note that ! n n X X 1 2 xi − (sin θ/ (1 + sin θ)) (|xi | − |xi+1 |)2 ≥ n i=1 i=1
!
n 1X (|xi | − |xi+1 |)2 n i=1 i=1 ! ! X n n X 2 2 |xi xi+1 | = 1− xi2 + n n i=1 i=1
≥
n X
(19)
n X
xi2
−
|xi xi+1 | ,
i=1
that is, n X i=1
|xi xi+1 | ≤
n X i=1
! xi2
−
n X 1 (sin θ/ (1 + sin θ)) (|xi | − |xi+1 |)2 . n i=1
(20)
Combining inequalities (18)–(20) leads to inequality (17) immediately. This completes the proof of Theorem 4.
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4. Another extension of Wolstenholme’s inequality In this section we discuss a new type of Wolstenholme’s inequality, and a class of Wolstenholme’s inequality for convex polygon. The Wolstenholme’s inequality for a triangle is well-known, which states as x 2 + y 2 + z 2 > 2x y cos A + 2yz cos B + 2zx cos C,
(21)
where x, y, z are real numbers, A, B, C are the angles of a triangle. Here we extend the inequality (21) to a new class of inequalities involving two convex polygons. Proposition 1. Let x1 , x2 , . . . , xn be real numbers, and let A1 , A2 , . . . , An and A01 , A02 , . . . , A0n be the angles of two convex polygons, n ≥ 3, 0 < λ ≤ 1, xn+1 = x1 . Then n n X X λAi0 λAi0 λAi0 λAi 2 sin ≥2 xi xi+1 cos sin xi2 + xi+1 cos n−2 n−2 n−2 n−2 i=1 i=1 X n λAi0 λAi0 sin λπ + csc λπ − csc (22) (|xi | − |xi+1 |)2 . n n − 2 n − 2 i=1 Proof. From the known relation for the convex polygon 0 < λ ≤ 1, it follows that n X i=1
λAi / (n − 2) =
n X
Pn
i=1
Ai =
Pn
i=1
Ai0 = (n − 2) π and the hypothesis
λAi0 / (n − 2) = λπ ≤ π.
i=1
Now utilizing Theorem 1 with the substitution ϕi = λAi / (n − 2), αi = λAi0 / (n − 2) (i = 1, 2, . . . , n) in (7) yields inequality (22) immediately. The Proposition 1 is proved. Putting λ = 1 in (22) gives the following result: Proposition 2. Let x1 , x2 , . . . , xn be real numbers, and let A1 , A2 , . . . , An and A01 , A02 , . . . , A0n be the angles of two convex polygons, n ≥ 3, xn+1 = x1 . Then n n X X Ai0 Ai0 Ai0 Ai 2 2 xi + xi+1 cos sin ≥2 sin . (23) xi xi+1 cos n−2 n−2 n−2 n−2 i=1 i=1 Remark 3. In the special case x1 = x2 = · · · = xn = 1, inequality (23) becomes X n n X Ai0 Ai0 Ai0 Ai cos sin ≥ cos sin , n−2 n−2 n−2 n−2 i=1 i=1
(24)
where A1 , A2 , . . . , An and A01 , A02 , . . . , A0n are the angles of two convex polygons. A0n
Clearly, inequality (24) is an interesting inequality related to two convex polygons. Putting A01 = A02 = · · · = = (n − 2) π/n in (22), we get
Proposition 3. Let x1 , x2 , . . . , xn be real numbers, and let A1 , A2 , . . . , An be the angles of a convex polygon, n ≥ 3, 0 < λ ≤ 1, xn+1 = x1 . Then ! X n n n X λA 1 sin λπ λπ λπ(n − 1) X λπ i 2 xi xi+1 cos + sec csc xi ≥ sec (|xi | − |xi+1 |)2 . (25) n n−2 2 n n n i=1 i=1 i=1 In particular, taking λ = 1 in (25), we obtain the following Wolstenholme type inequality for a convex polygon.
S. Wu, L. Debnath / Computers and Mathematics with Applications 53 (2007) 104–114
111
Proposition 4. For any real numbers x1 , x2 , . . . , xn and angles A1 , A2 , . . . , An of a convex polygon, n ≥ 3, we have the following inequality π A1 A2 An 2 2 2 . (26) x1 + x2 + · · · + xn ≥ sec x1 x2 cos + x2 x3 cos + · · · + xn x1 cos n n−2 n−2 n−2 Remark 4. When n = 3, inequality (26) is the Wolstenholme’s inequality for a triangle. In particular, taking n = 4 in (26), the following Wolstenholme’s inequality for convex quadrilateral is derived. √ A B C D 2 2 2 2 x + y + z + u ≥ 2 x y cos + yz cos + zu cos + ux cos , (27) 2 2 2 2 where x, y, z, u are real numbers, A, B, C, D are the angles of a convex quadrilateral. 5. Application to improvement of Erd¨os–Mordell’s inequality As examples of applications, we improve the well-known Erd¨os–Mordell inequality by using the above results and some interesting results are obtained in this section. In what follows, we state that A1 , A2 , . . . , An denote the vertices of the convex polygon Pn (n ≥ 3), Q is an interior point of Pn . Let Ri be the distance from Q to the vertex Ai , ri the distance from Q to the side Ai Ai+1 , wi the segment of the bisector of the angle Ai Q Ai+1 from Q to its intersection with the side Ai Ai+1 (i = 1, 2, . . . , n, An+1 = A1 ). The well-known Erd¨os–Mordell’s inequality states that if Q is a point in a triangle A1 A2 A3 whose distances are R1 , R2 , R3 from the vertices and r1 , r2 , r3 from the sides. Then R1 + R2 + R3 ≥ 2(r1 + r2 + r3 ).
(28)
This inequality was proposed by Erd¨os [11] as a conjecture and proved by Mordell and Barrow [12]. Some related results with historical comments on this problem can be found in [13,14]. Lenhard [15] established a remarkable inequality concerning the convex polygon as an extension of Erd¨os–Mordell’s inequality as follows n X i=1
n n π X π X Ri ≥ sec wi ≥ sec ri . n i=1 n i=1
(29)
Dar and Gueron [16] proved a weighted Erd¨os–Mordell’s inequality: p p p λ1 R 1 + λ2 R 2 + λ3 R 3 ≥ 2 λ1 λ2 r 1 + λ2 λ3 r 2 + λ3 λ1 r 3 ,
(30)
where λ1 , λ2 , λ3 are positive numbers. In a recent paper, Gueron and Shafrir [17] generalized (30) as follows: n X
n p π X λi Ri ≥ sec λi λi+1ri , n i=1 i=1
(31)
where λ1 , λ2 , . . . , λn are positive numbers and λn+1 = λ1 . We give here several unified and further extensions of the above inequalities. Proposition 5. Let λi , µi , αi be positive numbers (i = 1, 2, . . . , n), and let the inequality n X k λi Rik + λi+1 Ri+1 cot αi
Pn
i=1 αi
= π. Then for k ≥ 1, we have
i=1
≥2
n p X i=1
λi λi+1 µi µi+1 (Ri + Ri+1 )
(µi Ri + µi+1 Ri+1 ) wik csc αi .
(32)
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For 0 < k ≤ 1, we have the inequality n X k λi Rik + λi+1 Ri+1 cot αi i=1
k−1
≥ 2 sec
n p π X k k k k µi Ri + µi+1 Ri+1 wik csc αi , λi λi+1 µi µi+1 Ri + Ri+1 n i=1
(33)
where λn+1 = λ1 , µn+1 = µ1 , Rn+1 = R1 . Proof. We use ϕi to signify n X
ϕi = π,
16 2
0 < ϕi <
i=1
Ai Q Ai+1 (i = 1, 2, . . . , n) and infer that π , i = 1, 2, . . . , n. 2
q λi Rik (i = 1, 2, . . . , n) in (8) and (9), Applying Corollary 1 and Theorem 2 with the substitution xi = respectively, we obtain the inequalities: n p n p k X X k λi Rik + λi+1 Ri+1 cot αi ≥ 2 λi λi+1 Ri Ri+1 cos ϕi csc αi i=1
i=1
≥2
n p X
λi λi+1
p
Ri Ri+1 cos ϕi
k
csc αi
(k ≥ 1) ,
(34)
i=1
and n X i=1
n p k p π X k λi Rik + λi+1 Ri+1 cot αi ≥ 2 seck−1 λi λi+1 Ri Ri+1 cos ϕi csc αi n i=1
(0 < k ≤ 1) .
(35)
In view of the formula for the bisector length of the triangle Ai Q Ai+1 : wi = (2Ri Ri+1 cos ϕi ) / (Ri + Ri+1 ) ,
i = 1, 2, . . . , n,
we have n p X
λi λi+1
i=1
p
Ri Ri+1 cos ϕi
k
csc αi =
n p X
λi λi+1
k p wik csc αi . 2 Ri Ri+1 (Ri + Ri+1 )
(36)
i=1
Case (I). When k ≥ 1. Applying the arithmetic–geometric means inequality and the weighted power means inequality to (36), respectively, it follows that n p k X p λi λi+1 (Ri + Ri+1 ) 2 Ri Ri+1 wik csc αi i=1
=
n p X
q k 2 1/k 1/k λi λi+1 µi µi+1 (Ri + Ri+1 ) µi µi+1 Ri Ri+1 wik csc αi
i=1
n p k X 1/k 1/k µi Ri + µi+1 Ri+1 ≥ λi λi+1 µi µi+1 (Ri + Ri+1 ) wik csc αi i=1
≥
n p X
=
n p X
k λi λi+1 µi µi+1 (Ri + Ri+1 ) wik csc αi (µi Ri + µi+1 Ri+1 )1/k (Ri + Ri+1 )1−(1/k)
i=1
λi λi+1 µi µi+1 (Ri + Ri+1 ) (µi Ri + µi+1 Ri+1 ) wik csc αi .
i=1
Combining (34) and (36) and the above inequality yield inequality (32).
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Case (II). When 0 < k ≤ 1. Utilizing the power means inequality and the arithmetic–geometric means inequality to (36), respectively, it follows that n p k X p 2 Ri Ri+1 wik csc αi λi λi+1 (Ri + Ri+1 ) i=1
=
n p X
≥
i=1 n p X
=
n p X
≥
n p X
λi λi+1
λi λi+1
k q ! k k Ri Ri+1 wik csc αi (Ri + Ri+1 ) 2
k Rik + Ri+1
q k 2 Rik Ri+1 wik csc αi
i=1
λi λi+1 µi µi+1
k Rik + Ri+1
q k 2 µi µi+1 Rik Ri+1 wik csc αi
i=1
k k λi λi+1 µi µi+1 Rik + Ri+1 µi Rik + µi+1 Ri+1 wik csc αi .
i=1
Combining (35) and (36) and the above inequality lead to inequality (33). The proof of Proposition 5 is complete. We give here some direct consequences of Proposition 5. Choosing µi = λi and αi =
π n
(i = 1, 2, . . . , n) in Proposition 5, we obtain
Proposition 6. Let λ1 , λ2 , . . . , λn be positive numbers. Then for k ≥ 1, we have n n X π X k λi λi+1 (Ri + Ri+1 ) (λi Ri + λi+1 Ri+1 ) wik . λi Ri ≥ sec n i=1 i=1
(37)
For 0 < k ≤ 1, we have n X
n π X k k λi Rik ≥ seck λi λi+1 Rik + Ri+1 λi Rik + λi+1 Ri+1 wik , n i=1 i=1
(38)
where λn+1 = λ1 , Rn+1 = R1 . Remark 5. Substituting λ1 = λ2 = · · · = λn = k = 1 in Proposition 6 yields the foregoing Lenhard’s inequality (29). Moreover, taking λi = 1/Rik , i = 1, 2, . . . , n in (38), an extension and refinement of Mitrinovi´c–Pe˘cari´c’s [18] inequality is derived as follows: !n !k !k n n n n n Y Y Y 1X k k k k k π Ri wi + wi+1 Ri+1 Ri wik + wi+1 , (39) 2 cos ≥ ≥ n n i=1 i=1 i=1 i=1 where 0 < k ≤ 1, wn+1 = w1 , Rn+1 = R1 . Putting µ1 = µ2 = · · · µn = 1 in Proposition 5 with the assumption γ = min{k, 1}, we get the following inequality Proposition 7. Let λ1 , λ2 , . . . , λn , α1 , α2 , . . . , αn , k be positive numbers, and let λn+1 = λ1 , Rn+1 = R1 . Then n n p X π X k λi Rik + λi+1 Ri+1 cot αi ≥ 2 secγ −1 λi λi+1 wik csc αi . n i=1 i=1
Pn
i=1 αi
= π, γ = min{k, 1},
(40)
In particular, choosing α1 = α2 = · · · = αn = πn in inequality (40) together with the well-known inequality wi ≥ ri (i = 1, 2, . . . , n), we get the following generalization of Gueron–Shafrir’s inequality.
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Proposition 8. Let λ1 , λ2 , . . . , λn , k be positive numbers, and let γ = min{k, 1}, λn+1 = λ1 . Then n X
n p n p π X π X λi Rik ≥ secγ λi λi+1 wik ≥ secγ λi λi+1rik . n n i=1 i=1 i=1
(41)
Remark 6. In Proposition 8, taking k = 1 gives Gueron–Shafrir’s inequality (31), specially, when n = 3 and λ1 = λ2 = λ3 = k = 1 it yields the Erd¨os–Mordell’s inequality. Acknowledgments The authors would like to express hearty thanks to the anonymous referees for their valuable comments on this article. The first author expresses thanks to Natural Science Foundation of Fujian province of China (No. S0650003) for support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
D.S. Mitrinovi´c, J.E. Pe˘cari´c, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993. D.S. Mitrinovi´c, J.E. Pe˘cari´c, V. Volenec, Recent Advances in Geometric Inequalities, Kluwer Academic Publishers, Dordrecht, 1989. J. Wolstenholme, A Book of Mathematical Problems, Cambridge, London, 1867. O. Bottema, R.Z. Djordjevi´c, R.R. Jani´c, D.S. Mitrinovi´c, P.M. Vasi´c, Geometric Inequalities, Wolters-Noordhoff, Groningen, 1969. D.S. Mitrinovi´c, J.E. Pe˘cari´c, V. Volenec, J. Chen, Addenda to the monograph: Recent advances in geometric inequalities (I), J. Ningbo Univ. 4 (2) (1991) 108–120. V.N. Murty, K. Satyanarayana, M.S. Klamkin, Problem 715, Crux Math. 9 (1983) 58–62. ˇ Milovanovi´c, Some discrete inequality of Opial’s type, Acta Sci. Math. 47 (1984) 413–417. G.V. Milovanovi´c, I.Z. N. Ozeki, On P. Erd¨os’s inequality for the triangle, J. College Arts Sci., Chiba Univ. 2 (1957) 247–250. G. Hardy, J.E. Littlewood, G. P´olya, Inequalities, second ed., Cambridge Univ. Press, Cambridge, UK, 1952. S. Wu, Generalization and sharpness of power means inequality and their applications, J. Math. Anal. Appl. 312 (2005) 637–652. P. Erd¨os, Problem 3740, Amer. Math. Monthly 42 (1935) 396. L.J. Mordell, D.F. Barrow, Solution of problem 3740, Amer. Math. Monthly 44 (1937) 252–254. V. Komornik, A short proof of the Erd¨os–Mordell Theorem, Amer. Math. Monthly 104 (1997) 57–60. R.A. Satnoianu, Erd¨os–Mordell type inequality in a triangle, Amer. Math. Monthly 110 (2003) 727–729. H.C. Lenhard, Verallgemeinerung und Versch¨arfung der Erd¨os–Mordellschen Ungleichung f¨ur polygone, Arch. Math. 12 (1961) 311–314. S. Dar, S. Gueron, A weighted Erd¨os–Mordell inequality, Amer. Math. Monthly 108 (2001) 165–168. S. Gueron, I. Shafrir, A weighted Erd¨os–Mordell inequality for polygons, Amer. Math. Monthly 112 (2005) 257–263. D.S. Mitrinovi´c, J.E. Pe˘cari´c, On the Erd¨os–Mordell’s inequality for a polygon, J. College Arts Sci., Chiba Univ. B-19 (1986) 3–6.
Generalization of the Wolstenholme cyclic inequality and its application Shanhe Wu a , Lokenath Debnath b,∗ a Department of Mathematics, Longyan College, Longyan Fujian 364012, China b Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA
Received 17 January 2006; received in revised form 5 June 2006; accepted 14 June 2006
Abstract This paper deals with the generalization and sharp versions of the Wolstenholme cyclic inequality and their applications. The inequalities of this paper improve and unify corresponding known results. Several interesting inequalities including the celebrated Ozeki inequality are obtained. Extensions of the Wolstenholme inequality for a complex polygon and the Wolstenholme inequality for a convex quadrilateral are derived. As example of applications, the well-known Erd¨os–Mordell inequality is improved in this paper. In addition, several extensions, unifications and refinements of Gueron–Shafrir’s inequalities and Mitrinovi´c–Pecaric’s inequality are established. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Wolstenholme’s inequality; Erd¨os–Mordell’s inequality; Weighted power means inequality; Power means inequality; Convex polygon; Improvement
1. Introduction If x1 , x2 , x3 , ϕ1 , ϕ2 , ϕ3 are real numbers and ϕ1 + ϕ2 + ϕ3 = π, the following celebrated cyclic inequality x12 + x22 + x32 ≥ 2x1 x2 cos ϕ1 + 2x2 x3 cos ϕ2 + 2x3 x1 cos ϕ3
(1)
is known as Wolstenholme’s inequality in the literature (see Mitrinovi´c et al. [1] p. 425, and Mitrinovi´c et al. [2] p. 421). In 1867, Wolstenholme first introduced inequality (1) with a historical account in his book [3]. It is important to point out that the Wolstenholme inequality provided an effective approach to combine both geometric and arithmetic inequalities. Based on the Wolstenholme inequality, a large number of new inequalities have been discovered with applications. In recent years, considerable attention has been given to study this inequality in many different directions including its various generalizations, unifications, refinements and applications (see Bottema et al. [4], Mitrinovi´c et al. [5], Murty et al. [6] and Mitrinovi´c et al. [7] and references therein). On the other hand, in 1957, Ozeki [8] proved
∗ Corresponding author. Tel.: +1 956 381 3459; fax: +1 956 384 5091.
E-mail addresses: [email protected] (S. Wu), [email protected] (L. Debnath). c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2006.06.006
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a notable generalization of (1) in his work on the extension of the Erd¨os–Mordell’s inequality for a convex polygon which reads as ! n n X π X 2 xi xi+1 cos ϕi , (2) xi ≥ sec n i=1 i=1 where x1 , x2 , . . . , xn , ϕ1 , ϕ2 , . . . , ϕn are real numbers, xn+1 = x1 , ϕ1 + ϕ2 + · · · + ϕn = π, n > 2. In spite of the above work, further study on the Wolstenholme inequality is needed. So, the purpose of this paper is to prove several new generalized and sharp versions of the Wolstenholme inequality and their applications to the investigation of geometric inequalities. Under the weak constraint condition ϕ1 +ϕ2 +· · ·+ϕn = θ , where 0 < θ ≤ π, we prove that the Ozeki type inequalities are valid and can be improved to a sharper version (see Theorem 3 below). It is shown in this paper that the improved version of (1) can be used to study geometric inequalities. In Section 5 of this paper, we demonstrate that the above inequalities can effectively be used to improve the Erd¨os–Mordell inequality and to establish some new Erd¨os–Mordell’s type inequalities. Finally, several extensions, unifications and refinements of Gueron–Shafrir’s inequalities, and Mitrinovi´c–Pe˘cari´c’s inequality are discussed. 2. Lemmas The following weighted power means inequality is well-known (see Hardy et al. [9] and Wu [10]). Lemma 1. If xi ≥ 0, λi > 0, i = 1, 2, . . . , n, 0 < p ≤ 1. Then n X
p λi xi
≤
i=1
n X i=1
!1− p λi
n X
!p λi xi
.
(3)
i=1
Inequality (3) is reversed for p ≥ 1 or p < 0. The above inequality in the special case of λ1 = λ2 = · · · = λn = 1 is called power means inequality. Lemma 2. Let x1 , x2 , x3 , ϕ1 , ϕ2 , ϕ3 be real numbers, α1 , α2 , α3 be positive numbers, and let π. Then
P3
i=1 αi
=
P3
i=1 ϕi
=
(x12 + x22 ) cot α1 + (x22 + x32 ) cot α2 + (x32 + x12 ) cot α3 ≥ 2(x1 x2 cos ϕ1 csc α1 + x2 x3 cos ϕ2 csc α2 + x3 x1 cos ϕ3 csc α3 ). P3 P3 αi = i=1 ϕ1 = π, after some simple calculations, we obtain Proof. According to i=1
(4)
(x12 + x22 ) cot α1 + (x22 + x32 ) cot α2 + (x32 + x12 ) cot α3 = x12 sin α2 csc α1 csc α3 + x22 sin α3 csc α1 csc α2 + x32 sin α1 csc α2 csc α3 .
(5)
On the other hand, applying Wolstenholme’s inequality with the substitution: x1 → x1 sin α2 , x2 → x2 sin α3 , x3 → x3 sin α1 in (1) yields x12 sin2 α2 + x22 sin2 α3 + x32 sin2 α1 ≥ 2(x1 x2 cos ϕ1 sin α2 sin α3 + x2 x3 cos ϕ2 sin α3 sin α1 + x3 x1 cos ϕ3 sin α1 sin α2 ). Or, equivalently, x12 sin α2 csc α1 csc α3 + x22 sin α3 csc α1 csc α2 + x32 sin α1 csc α2 csc α3 ≥ 2(x1 x2 cos ϕ1 csc α1 + x2 x3 cos ϕ2 csc α2 + x3 x1 cos ϕ3 csc α3 ). Combining (5) and (6) leads to inequality (4). The Lemma 2 is proved.
(6)
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3. Generalization and sharpness of Wolstenholme’s inequality Pn Theorem 1. Let xi , ϕi be real numbers, αi be positive numbers (i = 1, 2, . . . , n), n ≥ 3, and let i=1 αi = P n ϕ = θ, 0 < θ ≤ π, x = x . Then i n+1 1 i=1 ! n n n X X 1X 2 2 sin θ csc αi csc(θ − αi ) (|xi | − |xi+1 |)2 . (7) xi xi+1 cos ϕi csc αi + xi + xi+1 cot αi ≥ 2 n i=1 i=1 i=1 Proof. By using Lemma 2, we have (x1 x2 ) cos ϕ1 csc α1 + (x2 x3 ) cos ϕ2 csc α2 + (x3 x1 ) cos π −
2 X
! ϕi csc π −
i=1
2 X
! αi
i=1
!! 2 X 1 2 2 2 2 2 2 x1 + x2 cot α1 + x2 + x3 cot α2 + x3 + x1 cot π − αi , ≤ 2 i=1 ! ! ! ! 2 3 3 2 X X X X ϕi csc αi + (x3 x4 ) cos ϕ3 csc α3 + (x4 x1 ) cos π − ϕi csc π − αi (x1 x3 ) cos i=1
i=1
i=1
i=1
!! ! 3 2 X X 1 2 , αi x1 + x32 cot αi + x32 + x42 cot α3 + x42 + x12 cot π − ≤ 2 i=1 i=1 ······ ! ! n−2 n−2 X X ϕi csc αi + (xn−1 xn ) cos ϕn−1 csc αn−1 (x1 xn−1 ) cos i=1
+ (xn x1 ) cos π −
i=1 n−1 X
! ϕi csc π −
n−1 X
i=1 n−2 X 1 2 2 ≤ x1 + xn−1 cot αi 2 i=1
! αi
i=1
!
2 + xn−1 + xn2 cot αn−1 + xn2 + x12 cot π −
n−1 X
!! αi
.
i=1
Summing both sides in the above inequalities respectively, we obtain ! ! ! n−1 n−1 n−1 X X X xi xi+1 cos ϕi csc αi − xn x1 cos ϕi csc αi i=1
i=1
i=1
! !! n−1 n−1 X 1 X 1 2 2 2 2 ≤ x + xi+1 cot αi − xn + x1 cot αi . 2 i=1 i 2 i=1 Pn Pn Using the assumptions, i=1 αi = i=1 ϕi = θ, 0 < θ ≤ π, together with the above inequality, it follows that ! ! n n X 1 X 2 2 xi xi+1 cos ϕi csc αi − x + xi+1 cot αi 2 i=1 i i=1 1 2 ≤ xn x1 (cos ϕn csc αn + cos (θ − ϕn ) csc (θ − αn )) − xn + x12 (cot αn + cot (θ − αn )) 2 1 = xn x1 (sin (θ − αn + ϕn ) + sin (θ + αn − ϕn )) csc αn csc (θ − αn ) 2 1 − xn2 + x12 (cot αn + cot (θ − αn )) 2 1 2 = xn x1 sin θ cos (ϕn − αn ) csc αn csc (θ − αn ) − xn + x12 sin θ csc αn csc (θ − αn ) 2 1 2 ≤ |xn x1 sin θ cos (ϕn − αn ) csc αn csc (θ − αn )| − xn + x12 sin θ csc αn csc (θ − αn ) 2
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1 2 xn + x12 sin θ csc αn csc (θ − αn ) 2
≤ |xn x1 sin θ csc αn csc (θ − αn )| −
1 = − sin θ csc αn csc (θ − αn ) (|xn | − |x1 |)2 . 2 That is, ! ! n n X 1 X 1 2 2 xi xi+1 cos ϕi csc αi − xi + xi+1 cot αi ≤ − sin θ csc αn csc (θ − αn ) (|xn | − |x1 |)2 . 2 i=1 2 i=1 An argument similar to the one given above gives ! ! n n X 2 1 1 X 2 2 xi + xi+1 cot αi ≤ − sin θ csc α j csc θ − α j x j − x j+1 , xi xi+1 cos ϕi csc αi − 2 i=1 2 i=1 j = 1, 2, . . . , n − 1. Summing both sides in the above inequalities respectively from j = 1 to j = n gives !! ! n n X 1 X 2 cot αi xi2 + xi+1 n xi xi+1 cos ϕi csc αi − 2 i=1 i=1 ≤−
n 1X sin θ csc αi csc (θ − αi ) (|xi | − |xi+1 |)2 . 2 i=1
This is equivalent to inequality (7). The proof of Theorem 1 is complete.
The following corollary follows from Theorem 1 directly. Corollary 1. Let xi , ϕi be real numbers, αi be positive numbers (i = 1, 2, . . . , n), n ≥ 3, and let Pn ϕ = θ, 0 < θ ≤ π, xn+1 = x1 . Then i i=1 ! n n X X 2 2 xi xi+1 cos ϕi csc αi . xi + xi+1 cot αi ≥ 2
Pn
i=1 αi
=
(8)
i=1
i=1
Remark 1. In Corollary 1, the special case α1 = α2 = · · · = αn = πn yields the foregoing Ozeki’s inequality (2). In the following theorem, we give another generalization of the inequality (8). Theorem 2. Let xi be real numbers, 0 < αi , 0 < ϕi < 0 < θ ≤ π, xn+1 = x1 . Then, for 0 < k ≤ 1, we have n X
2 xi2 + xi+1 cot αi ≥ 2 seck−1
i=1
For k ≥ 1 and 0 < αi < n X
xi2
+
2 xi+1
π 2
X n
(i = 1, 2, . . . , n) , n ≥ 3, and let
Pn
i=1 αi
=
Pn
i=1 ϕi
= θ,
! xi xi+1 cosk ϕi csc αi .
(9)
i=1
(i = 1, 2, . . . , n), we have
cot αi ≥ 2 k
θ n
π 2
2−k
i=1
n X
! xi xi+1 cos ϕi csc αi . k
(10)
i=1
Proof. Case (I). When 0 < k ≤ 1. By the weighted power means inequality, we have n X
xi xi+1 cosk ϕi csc αi
i=1
≤
n X i=1
|xi xi+1 | cos ϕi csc αi ≤ k
n X i=1
!1−k |xi xi+1 | csc αi
n X i=1
!k |xi xi+1 | cos ϕi csc αi
.
(11)
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In addition, using Corollary 1 gives n X
|xi | |xi+1 | cos ϕi csc αi ≤
i=1
n 1X |xi |2 + |xi+1 |2 cot αi , 2 i=1
that is n X
|xi xi+1 | cos ϕi csc αi ≤
i=1
n 1X 2 xi2 + xi+1 cot αi . 2 i=1
On the other hand, we put ϕ1 = ϕ2 = · · · = ϕn = n X
|xi xi+1 | csc αi ≤
i=1
θ 1 sec 2 n
X n
θ n
(12)
in (12), so that
2 cot αi . xi2 + xi+1
(13)
i=1
Combining inequalities (11)–(13) leads to inequality (9). Case (II). When k ≥ 1. Observe that inequality (8) can be rewritten as follows: n+1 X
xi2 (cot αi−1 + cot αi ) ≥ 2
i=2
n X
xi xi+1 cos ϕi csc αi
Replacing xi by |xi | (cot αi−1 + cot αi ) n+1 X
xi2 (cot αi−1 + cot αi )k ≥ 2
n X
i=2
Since n X
(xn+1 = x1 , αn+1 = α1 ) .
i=1 k−1 2
(i = 2, 3 . . . , n + 1) in the above inequalities respectively, we obtain
|xi xi+1 | ((cot αi−1 + cot αi ) (cot αi + cot αi+1 ))
k−1 2
cos ϕi csc αi .
(14)
i=1
|xi xi+1 | ((cot αi−1 + cot αi ) (cot αi + cot αi+1 ))
k−1 2
cos ϕi csc αi
i=1
=
n X
|xi xi+1 | (sin (αi−1 + αi ) sin (αi + αi+1 ) csc αi−1 csc αi+1 )
k−1 2
cos ϕi csck αi
i=1
=
n X
|xi xi+1 | (1 + sin αi sin (αi−1 + αi + αi+1 ) csc αi−1 csc αi+1 )
k−1 2
cos ϕi csck αi ,
i=1
and note that hypotheses k ≥ 1, n X
Pn
i=1 αi
≤ π, 0 < αi <
π 2,
|xi xi+1 | ((cot αi−1 + cot αi ) (cot αi + cot αi+1 ))
0 < ϕi <
k−1 2
π 2
(i = 1, 2, . . . , n), we conclude
cos ϕi csc αi ≥
i=1
n X
xi xi+1 cos ϕi csck αi .
i=1
Applying the power means inequality together with (14) and (15), it follows that n X i=1
n+1 X 2 xi2 + xi+1 cotk αi = xi2 cotk αi−1 + cotk αi i=2
≥ 21−k
n+1 X
xi2 (cot αi−1 + cot αi )k
i=2
≥ 22−k
n X
xi xi+1 cos ϕi csck αi .
i=1
The inequality (10) is proved. This completes the proof of Theorem 2.
The following inequality is a remarkable improvement of Ozeki’s inequality:
(15)
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109
Pn Theorem 3. Let xi , ϕi be real numbers (i = 1, 2, . . . , n) , n ≥ 3, and let i=1 ϕi = θ, 0 < θ ≤ π, xn+1 = x1 . Then ! X n n n X X θ 1 2 xi ≥ sec xi xi+1 cos ϕi + (sin θ/ (1 + sin θ)) (16) (|xi | − |xi+1 |)2 . n n i=1 i=1 i=1 Proof. Taking α1 = α2 = · · · = αn = nθ in Theorem 1 yields ! X X n n n X 1 θ n−1 θ 2 xi xi+1 cos ϕi + sin θ sec csc θ xi ≥ sec (|xi | − |xi+1 |)2 . n 2n n n i=1 i=1 i=1 Since 1 n−2 θ n−1 1 sin θ sin θ + sin sin θ sec csc θ= θ . 2n n n n n
Consequently, we get the inequality (16). The proof is complete.
Remark 2. It is obvious that inequality (16) represents the sharpened version of the Wolstenholme’s inequality and Ozeki’s inequality. Especially, if ϕ1 , ϕ2 , . . . , ϕn are the acute angles, we have the following exponential version of inequality (16). Pn Theorem 4. Let xi be real numbers, 0 < ϕi < π2 (i = 1, 2, . . . , n) , n ≥ 3, and let i=1 ϕi = θ, 0 < θ ≤ π , 0 < k ≤ 1, xn+1 = x1 . Then ! X n n n X X 1 k 2 k θ xi xi+1 cos ϕi + (sin θ/ (1 + sin θ)) (17) xi ≥ sec (|xi | − |xi+1 |)2 . n n i=1 i=1 i=1 Proof. When k = 1, inequality (17) is equivalent to inequality (16). When 0 < k < 1, by using the weighted power means inequality, we obtain !k !1−k n n n n X X X X k k |xi xi+1 | cos ϕi ≤ |xi xi+1 | |xi xi+1 | cos ϕi . (18) xi xi+1 cos ϕi ≤ i=1
i=1
i=1
i=1
In addition, Theorem 3 gives ! # " X n n n X X 1 θ 2 2 |xi xi+1 | cos ϕi ≤ cos xi − (sin θ/ (1 + sin θ)) (|xi | − |xi+1 |) . n n i=1 i=1 i=1 On the other hand, we note that ! n n X X 1 2 xi − (sin θ/ (1 + sin θ)) (|xi | − |xi+1 |)2 ≥ n i=1 i=1
!
n 1X (|xi | − |xi+1 |)2 n i=1 i=1 ! ! X n n X 2 2 |xi xi+1 | = 1− xi2 + n n i=1 i=1
≥
n X
(19)
n X
xi2
−
|xi xi+1 | ,
i=1
that is, n X i=1
|xi xi+1 | ≤
n X i=1
! xi2
−
n X 1 (sin θ/ (1 + sin θ)) (|xi | − |xi+1 |)2 . n i=1
(20)
Combining inequalities (18)–(20) leads to inequality (17) immediately. This completes the proof of Theorem 4.
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4. Another extension of Wolstenholme’s inequality In this section we discuss a new type of Wolstenholme’s inequality, and a class of Wolstenholme’s inequality for convex polygon. The Wolstenholme’s inequality for a triangle is well-known, which states as x 2 + y 2 + z 2 > 2x y cos A + 2yz cos B + 2zx cos C,
(21)
where x, y, z are real numbers, A, B, C are the angles of a triangle. Here we extend the inequality (21) to a new class of inequalities involving two convex polygons. Proposition 1. Let x1 , x2 , . . . , xn be real numbers, and let A1 , A2 , . . . , An and A01 , A02 , . . . , A0n be the angles of two convex polygons, n ≥ 3, 0 < λ ≤ 1, xn+1 = x1 . Then n n X X λAi0 λAi0 λAi0 λAi 2 sin ≥2 xi xi+1 cos sin xi2 + xi+1 cos n−2 n−2 n−2 n−2 i=1 i=1 X n λAi0 λAi0 sin λπ + csc λπ − csc (22) (|xi | − |xi+1 |)2 . n n − 2 n − 2 i=1 Proof. From the known relation for the convex polygon 0 < λ ≤ 1, it follows that n X i=1
λAi / (n − 2) =
n X
Pn
i=1
Ai =
Pn
i=1
Ai0 = (n − 2) π and the hypothesis
λAi0 / (n − 2) = λπ ≤ π.
i=1
Now utilizing Theorem 1 with the substitution ϕi = λAi / (n − 2), αi = λAi0 / (n − 2) (i = 1, 2, . . . , n) in (7) yields inequality (22) immediately. The Proposition 1 is proved. Putting λ = 1 in (22) gives the following result: Proposition 2. Let x1 , x2 , . . . , xn be real numbers, and let A1 , A2 , . . . , An and A01 , A02 , . . . , A0n be the angles of two convex polygons, n ≥ 3, xn+1 = x1 . Then n n X X Ai0 Ai0 Ai0 Ai 2 2 xi + xi+1 cos sin ≥2 sin . (23) xi xi+1 cos n−2 n−2 n−2 n−2 i=1 i=1 Remark 3. In the special case x1 = x2 = · · · = xn = 1, inequality (23) becomes X n n X Ai0 Ai0 Ai0 Ai cos sin ≥ cos sin , n−2 n−2 n−2 n−2 i=1 i=1
(24)
where A1 , A2 , . . . , An and A01 , A02 , . . . , A0n are the angles of two convex polygons. A0n
Clearly, inequality (24) is an interesting inequality related to two convex polygons. Putting A01 = A02 = · · · = = (n − 2) π/n in (22), we get
Proposition 3. Let x1 , x2 , . . . , xn be real numbers, and let A1 , A2 , . . . , An be the angles of a convex polygon, n ≥ 3, 0 < λ ≤ 1, xn+1 = x1 . Then ! X n n n X λA 1 sin λπ λπ λπ(n − 1) X λπ i 2 xi xi+1 cos + sec csc xi ≥ sec (|xi | − |xi+1 |)2 . (25) n n−2 2 n n n i=1 i=1 i=1 In particular, taking λ = 1 in (25), we obtain the following Wolstenholme type inequality for a convex polygon.
S. Wu, L. Debnath / Computers and Mathematics with Applications 53 (2007) 104–114
111
Proposition 4. For any real numbers x1 , x2 , . . . , xn and angles A1 , A2 , . . . , An of a convex polygon, n ≥ 3, we have the following inequality π A1 A2 An 2 2 2 . (26) x1 + x2 + · · · + xn ≥ sec x1 x2 cos + x2 x3 cos + · · · + xn x1 cos n n−2 n−2 n−2 Remark 4. When n = 3, inequality (26) is the Wolstenholme’s inequality for a triangle. In particular, taking n = 4 in (26), the following Wolstenholme’s inequality for convex quadrilateral is derived. √ A B C D 2 2 2 2 x + y + z + u ≥ 2 x y cos + yz cos + zu cos + ux cos , (27) 2 2 2 2 where x, y, z, u are real numbers, A, B, C, D are the angles of a convex quadrilateral. 5. Application to improvement of Erd¨os–Mordell’s inequality As examples of applications, we improve the well-known Erd¨os–Mordell inequality by using the above results and some interesting results are obtained in this section. In what follows, we state that A1 , A2 , . . . , An denote the vertices of the convex polygon Pn (n ≥ 3), Q is an interior point of Pn . Let Ri be the distance from Q to the vertex Ai , ri the distance from Q to the side Ai Ai+1 , wi the segment of the bisector of the angle Ai Q Ai+1 from Q to its intersection with the side Ai Ai+1 (i = 1, 2, . . . , n, An+1 = A1 ). The well-known Erd¨os–Mordell’s inequality states that if Q is a point in a triangle A1 A2 A3 whose distances are R1 , R2 , R3 from the vertices and r1 , r2 , r3 from the sides. Then R1 + R2 + R3 ≥ 2(r1 + r2 + r3 ).
(28)
This inequality was proposed by Erd¨os [11] as a conjecture and proved by Mordell and Barrow [12]. Some related results with historical comments on this problem can be found in [13,14]. Lenhard [15] established a remarkable inequality concerning the convex polygon as an extension of Erd¨os–Mordell’s inequality as follows n X i=1
n n π X π X Ri ≥ sec wi ≥ sec ri . n i=1 n i=1
(29)
Dar and Gueron [16] proved a weighted Erd¨os–Mordell’s inequality: p p p λ1 R 1 + λ2 R 2 + λ3 R 3 ≥ 2 λ1 λ2 r 1 + λ2 λ3 r 2 + λ3 λ1 r 3 ,
(30)
where λ1 , λ2 , λ3 are positive numbers. In a recent paper, Gueron and Shafrir [17] generalized (30) as follows: n X
n p π X λi Ri ≥ sec λi λi+1ri , n i=1 i=1
(31)
where λ1 , λ2 , . . . , λn are positive numbers and λn+1 = λ1 . We give here several unified and further extensions of the above inequalities. Proposition 5. Let λi , µi , αi be positive numbers (i = 1, 2, . . . , n), and let the inequality n X k λi Rik + λi+1 Ri+1 cot αi
Pn
i=1 αi
= π. Then for k ≥ 1, we have
i=1
≥2
n p X i=1
λi λi+1 µi µi+1 (Ri + Ri+1 )
(µi Ri + µi+1 Ri+1 ) wik csc αi .
(32)
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S. Wu, L. Debnath / Computers and Mathematics with Applications 53 (2007) 104–114
For 0 < k ≤ 1, we have the inequality n X k λi Rik + λi+1 Ri+1 cot αi i=1
k−1
≥ 2 sec
n p π X k k k k µi Ri + µi+1 Ri+1 wik csc αi , λi λi+1 µi µi+1 Ri + Ri+1 n i=1
(33)
where λn+1 = λ1 , µn+1 = µ1 , Rn+1 = R1 . Proof. We use ϕi to signify n X
ϕi = π,
16 2
0 < ϕi <
i=1
Ai Q Ai+1 (i = 1, 2, . . . , n) and infer that π , i = 1, 2, . . . , n. 2
q λi Rik (i = 1, 2, . . . , n) in (8) and (9), Applying Corollary 1 and Theorem 2 with the substitution xi = respectively, we obtain the inequalities: n p n p k X X k λi Rik + λi+1 Ri+1 cot αi ≥ 2 λi λi+1 Ri Ri+1 cos ϕi csc αi i=1
i=1
≥2
n p X
λi λi+1
p
Ri Ri+1 cos ϕi
k
csc αi
(k ≥ 1) ,
(34)
i=1
and n X i=1
n p k p π X k λi Rik + λi+1 Ri+1 cot αi ≥ 2 seck−1 λi λi+1 Ri Ri+1 cos ϕi csc αi n i=1
(0 < k ≤ 1) .
(35)
In view of the formula for the bisector length of the triangle Ai Q Ai+1 : wi = (2Ri Ri+1 cos ϕi ) / (Ri + Ri+1 ) ,
i = 1, 2, . . . , n,
we have n p X
λi λi+1
i=1
p
Ri Ri+1 cos ϕi
k
csc αi =
n p X
λi λi+1
k p wik csc αi . 2 Ri Ri+1 (Ri + Ri+1 )
(36)
i=1
Case (I). When k ≥ 1. Applying the arithmetic–geometric means inequality and the weighted power means inequality to (36), respectively, it follows that n p k X p λi λi+1 (Ri + Ri+1 ) 2 Ri Ri+1 wik csc αi i=1
=
n p X
q k 2 1/k 1/k λi λi+1 µi µi+1 (Ri + Ri+1 ) µi µi+1 Ri Ri+1 wik csc αi
i=1
n p k X 1/k 1/k µi Ri + µi+1 Ri+1 ≥ λi λi+1 µi µi+1 (Ri + Ri+1 ) wik csc αi i=1
≥
n p X
=
n p X
k λi λi+1 µi µi+1 (Ri + Ri+1 ) wik csc αi (µi Ri + µi+1 Ri+1 )1/k (Ri + Ri+1 )1−(1/k)
i=1
λi λi+1 µi µi+1 (Ri + Ri+1 ) (µi Ri + µi+1 Ri+1 ) wik csc αi .
i=1
Combining (34) and (36) and the above inequality yield inequality (32).
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S. Wu, L. Debnath / Computers and Mathematics with Applications 53 (2007) 104–114
Case (II). When 0 < k ≤ 1. Utilizing the power means inequality and the arithmetic–geometric means inequality to (36), respectively, it follows that n p k X p 2 Ri Ri+1 wik csc αi λi λi+1 (Ri + Ri+1 ) i=1
=
n p X
≥
i=1 n p X
=
n p X
≥
n p X
λi λi+1
λi λi+1
k q ! k k Ri Ri+1 wik csc αi (Ri + Ri+1 ) 2
k Rik + Ri+1
q k 2 Rik Ri+1 wik csc αi
i=1
λi λi+1 µi µi+1
k Rik + Ri+1
q k 2 µi µi+1 Rik Ri+1 wik csc αi
i=1
k k λi λi+1 µi µi+1 Rik + Ri+1 µi Rik + µi+1 Ri+1 wik csc αi .
i=1
Combining (35) and (36) and the above inequality lead to inequality (33). The proof of Proposition 5 is complete. We give here some direct consequences of Proposition 5. Choosing µi = λi and αi =
π n
(i = 1, 2, . . . , n) in Proposition 5, we obtain
Proposition 6. Let λ1 , λ2 , . . . , λn be positive numbers. Then for k ≥ 1, we have n n X π X k λi λi+1 (Ri + Ri+1 ) (λi Ri + λi+1 Ri+1 ) wik . λi Ri ≥ sec n i=1 i=1
(37)
For 0 < k ≤ 1, we have n X
n π X k k λi Rik ≥ seck λi λi+1 Rik + Ri+1 λi Rik + λi+1 Ri+1 wik , n i=1 i=1
(38)
where λn+1 = λ1 , Rn+1 = R1 . Remark 5. Substituting λ1 = λ2 = · · · = λn = k = 1 in Proposition 6 yields the foregoing Lenhard’s inequality (29). Moreover, taking λi = 1/Rik , i = 1, 2, . . . , n in (38), an extension and refinement of Mitrinovi´c–Pe˘cari´c’s [18] inequality is derived as follows: !n !k !k n n n n n Y Y Y 1X k k k k k π Ri wi + wi+1 Ri+1 Ri wik + wi+1 , (39) 2 cos ≥ ≥ n n i=1 i=1 i=1 i=1 where 0 < k ≤ 1, wn+1 = w1 , Rn+1 = R1 . Putting µ1 = µ2 = · · · µn = 1 in Proposition 5 with the assumption γ = min{k, 1}, we get the following inequality Proposition 7. Let λ1 , λ2 , . . . , λn , α1 , α2 , . . . , αn , k be positive numbers, and let λn+1 = λ1 , Rn+1 = R1 . Then n n p X π X k λi Rik + λi+1 Ri+1 cot αi ≥ 2 secγ −1 λi λi+1 wik csc αi . n i=1 i=1
Pn
i=1 αi
= π, γ = min{k, 1},
(40)
In particular, choosing α1 = α2 = · · · = αn = πn in inequality (40) together with the well-known inequality wi ≥ ri (i = 1, 2, . . . , n), we get the following generalization of Gueron–Shafrir’s inequality.
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Proposition 8. Let λ1 , λ2 , . . . , λn , k be positive numbers, and let γ = min{k, 1}, λn+1 = λ1 . Then n X
n p n p π X π X λi Rik ≥ secγ λi λi+1 wik ≥ secγ λi λi+1rik . n n i=1 i=1 i=1
(41)
Remark 6. In Proposition 8, taking k = 1 gives Gueron–Shafrir’s inequality (31), specially, when n = 3 and λ1 = λ2 = λ3 = k = 1 it yields the Erd¨os–Mordell’s inequality. Acknowledgments The authors would like to express hearty thanks to the anonymous referees for their valuable comments on this article. The first author expresses thanks to Natural Science Foundation of Fujian province of China (No. S0650003) for support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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