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Hindawi Publishing CorporationJournal of Function Spaces and
ApplicationsVolume 2013, Article ID 394194, 5
pageshttp://dx.doi.org/10.1155/2013/394194
Research ArticleSome Inequalities for Bounding Toader Mean
Wen-Hui Li and Miao-Miao Zheng
Department of Mathematics, School of Science, Tianjin
Polytechnic University, Tianjin 300387, China
Correspondence should be addressed to Wen-Hui Li;
[email protected]
Received 8 May 2013; Revised 11 June 2013; Accepted 16 June
2013
Academic Editor: L. E. Persson
Copyright © 2013 W.-H. Li and M.-M. Zheng. This is an open
access article distributed under the Creative Commons
AttributionLicense, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is
properlycited.
By finding linear relations among differences between two
special means, the authors establish some inequalities for
boundingToader mean in terms of the arithmetic, harmonic,
centroidal, and contraharmonic means.
1. Introduction
It is well known that the quantities
𝐴 (𝑎, 𝑏) =𝑎 + 𝑏
2, 𝐺 (𝑎, 𝑏) = √𝑎𝑏 ,
𝐻 (𝑎, 𝑏) =2𝑎𝑏
𝑎 + 𝑏, 𝐶 (𝑎, 𝑏) =
2 (𝑎2+ 𝑎𝑏 + 𝑏
2)
3 (𝑎 + 𝑏),
𝐶 (𝑎, 𝑏) =𝑎2+ 𝑏2
𝑎 + 𝑏, 𝑆 (𝑎, 𝑏) = √
𝑎2+ 𝑏2
2,
𝑀𝑝(𝑎, 𝑏) =
{{
{{
{
(𝑎𝑝+ 𝑎𝑝
2)
1/𝑝
, 𝑝 ̸= 0
√𝑎𝑏, 𝑝 = 0
(1)
are, respectively, called in the literature the arithmetic,
geo-metric, harmonic, centroidal, contraharmonic, root-squaremeans,
and the power mean of order 𝑝 of two positivenumbers 𝑎 and 𝑏. In
[1], Toader introduced a mean
𝑇 (𝑎, 𝑏) =2
𝜋∫
𝜋/2
0
√𝑎2cos2𝜃 + 𝑏2sin2𝜃 𝑑𝜃
=
{{{{{{{{
{{{{{{{{
{
2𝑎
𝜋E(√1 − (
𝑏
𝑎)
2
) , 𝑎 > 𝑏,
2𝑏
𝜋E(√1 − (
𝑏
𝑎)
2
) , 𝑎 < 𝑏,
𝑎, 𝑎 = 𝑏,
(2)
where
E = E(𝑟) = ∫𝜋/2
0
√1 − 𝑟2sin2𝜃 𝑑𝜃,
E= E(𝑟) = E (𝑟
) , E (0) =
𝜋
2, E (1) = 1,
(3)
for 𝑟 ∈ [0, 1] and 𝑟 = √1 − 𝑟2 is Legendre’s complete
ellipticintegral of the second kind; see [2] and [3, pages
40–46].
In [4], Vuorinen conjectured that
𝑀3/2
(𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) (4)
for all 𝑎, 𝑏 > 0with 𝑎 ̸= 𝑏.This conjecturewas verified in
[5, 6],respectively. Later in [7], it was presented that
𝑇 (𝑎, 𝑏) < 𝑀(ln 2)/ ln(𝜋/2) (𝑎, 𝑏) (5)
for all𝑎, 𝑏 > 0with𝑎 ̸= 𝑏.The constants 3/2 and ln 2/ln(𝜋/2)
=1.53 . . . which appeared in (4) and (5) are the best
possible.
Utilizing inequalities (4) and (5) and using the factthat the
power mean 𝑀
𝑝(𝑎, 𝑏) is continuous and strictly
increasing with respect to 𝑝 ∈ R for fixed 𝑎, 𝑏 > 0 with 𝑎 ̸=
𝑏may conclude that
𝐴 (𝑎, 𝑏) = 𝑀1(𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) < 𝑀
2(𝑎, 𝑏) = 𝑆 (𝑎, 𝑏) (6)
for all 𝑎, 𝑏 > 0 with 𝑎 ̸= 𝑏. In [8, Theorem 3.1], it
wasdemonstrated that the double inequality
𝛼𝑆 (𝑎, 𝑏) + (1 − 𝛼)𝐴 (𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) < 𝛽𝑆 (𝑎, 𝑏) + (1 − 𝛽)𝐴 (𝑎, 𝑏)
(7)
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2 Journal of Function Spaces and Applications
holds for all 𝑎, 𝑏 > 0 with 𝑎 ̸= 𝑏 if and only if
𝛼 ≤1
2, 𝛽 ≥
4 − 𝜋
(√2 − 1) 𝜋
. (8)
Recently in [9, Theorems 1.1 to 1.3], it was shown that
thedouble inequalities
𝛼1𝐶 (𝑎, 𝑏) + (1 − 𝛼
1) 𝐴 (𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) < 𝛽1𝐶 (𝑎, 𝑏) + (1 − 𝛽
1) 𝐴 (𝑎, 𝑏) ,
(9)
𝛼2
𝐴 (𝑎, 𝑏)+
1 − 𝛼2
𝐶 (𝑎, 𝑏)
<1
𝑇 (𝑎, 𝑏)<
𝛽2
𝐴 (𝑎, 𝑏)+
1 − 𝛽2
𝐶 (𝑎, 𝑏)(10)
hold for all 𝑎, 𝑏 > 0 with 𝑎 ̸= 𝑏 if and only if
𝛼1≤3
4, 𝛽1≥12
𝜋− 3, 𝛼
2≤ 𝜋 − 3, 𝛽
2≥1
4. (11)
The equation (4.4) in [10, page 1013] reads that
2 [𝐴 (𝑎, 𝑏) − 𝐻 (𝑎, 𝑏)] =3
2[𝐶 (𝑎, 𝑏) − 𝐻 (𝑎, 𝑏)]
= 𝐶 (𝑎, 𝑏) − 𝐻 (𝑎, 𝑏) =(𝑎 − 𝑏)
2
𝑎 + 𝑏.
(12)
Motivated by (12), we further find that
6 [𝐶 (𝑎, 𝑏) − 𝐴 (𝑎, 𝑏)] = 3 [𝐶 (𝑎, 𝑏) − 𝐶 (𝑎, 𝑏)]
= 2 [𝐶 (𝑎, 𝑏) − 𝐴 (𝑎, 𝑏)] =(𝑎 − 𝑏)
2
𝑎 + 𝑏.
(13)
It is not difficult to see that the double inequality (9) can
berearranged as
𝛼1<𝑇 (𝑎, 𝑏) − 𝐴 (𝑎, 𝑏)
𝐶 (𝑎, 𝑏) − 𝐴 (𝑎, 𝑏)
< 𝛽1. (14)
Therefore, replacing the denominator in (14) by one
ofdifferences in (12) and (13) yields
1
2𝛼1𝐶 (𝑎, 𝑏) −
1
2𝛼1𝐶 (𝑎, 𝑏) + 𝐴 (𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) <1
2𝛽1𝐶 (𝑎, 𝑏) −
1
2𝛽1𝐶 (𝑎, 𝑏) + 𝐴 (𝑎, 𝑏) ,
(15)
1
3𝛼1𝐶 (𝑎, 𝑏) + (1 −
1
3𝛼1)𝐴 (𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) <1
3𝛽1𝐶 (𝑎, 𝑏) + (1 −
1
3𝛽1)𝐴 (𝑎, 𝑏) ,
(16)
(1 +1
3𝛼1)𝐴 (𝑎, 𝑏) −
1
3𝛼1𝐻(𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) < (1 +1
3𝛽1)𝐴 (𝑎, 𝑏) −
1
3𝛽1𝐻(𝑎, 𝑏) ,
(17)
1
4𝛼1𝐶 (𝑎, 𝑏) −
1
4𝛼1𝐻(𝑎, 𝑏) + 𝐴 (𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) <1
4𝛽1𝐶 (𝑎, 𝑏) −
1
4𝛽1𝐻(𝑎, 𝑏) + 𝐴 (𝑎, 𝑏) ,
(18)
1
6𝛼1𝐶 (𝑎, 𝑏) −
1
6𝛼1𝐻(𝑎, 𝑏) + 𝐴 (𝑎, 𝑏)
< 𝑇 (𝑎, 𝑏) <1
6𝛽1𝐶 (𝑎, 𝑏) −
1
6𝛽1𝐻(𝑎, 𝑏) + 𝐴 (𝑎, 𝑏) ,
(19)
where𝛼1and𝛽1satisfy (11). On the other hand, the arithmetic
mean𝐴(𝑎, 𝑏) in the numerator of (14) can also be replaced bythe
harmonic, contraharmonic, or centroidal means.
For our own convenience, we denote the difference ofmeans in
(12) and (13) by
𝑀𝐶𝐻
(𝑎, 𝑏) =(𝑎 − 𝑏)
2
𝑎 + 𝑏. (20)
The quantity 𝑀𝐶𝐻(𝑎, 𝑏) is nonnegative and convex on
(0,∞) × (0,∞). See [11, Theorem 2.1].Now we naturally pose the
following problem.
Problem 1. What are the best constants 𝛼 and 𝛽 such that
thedouble inequality
𝛼 <𝑇 (𝑎, 𝑏) − 𝐻 (𝑎, 𝑏)
𝑀𝐶𝐻
(𝑎, 𝑏)< 𝛽 (21)
holds for all positive numbers 𝑎 and 𝑏 with 𝑎 ̸= 𝑏?
The main purposes of this paper are to answer the previ-ous
problem, to provide an alternative proof for inequalities(14) to
(19), and, finally, to remark the connection betweenToader mean and
the complete elliptic integral of the secondkind.
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Journal of Function Spaces and Applications 3
2. Lemmas
To attain our main purposes, we need the following lemmas.For 0
< 𝑟 < 1, denote 𝑟 = √1 − 𝑟2. Legendre’s complete
elliptic integrals of the first kind may be defined in [12, 13]
by
K = K(𝑟) = ∫𝜋/2
0
1
√1 − 𝑟2sin2𝜃𝑑𝜃,
K= K(𝑟) = K(𝑟
) , K(0) =
𝜋
2, K(1) = ∞.
(22)
Lemma 2 (see [14, Appendix E, pages 474-475]). For 0 < 𝑟
<1 and 𝑟 = √1 − 𝑟2 , one has
𝑑K
𝑑𝑟=
E − (𝑟)2
K
𝑟(𝑟)2
,𝑑E
𝑑𝑟=E −K
𝑟,
𝑑 (E − (𝑟)2
K)
𝑑𝑟= 𝑟K, E(
2√𝑟
1 + 𝑟) =
2E − (𝑟)2
K
1 + 𝑟.
(23)
Lemma 3 (see [14, Theorem 1.25]). For −∞ < 𝑎 < 𝑏 <
∞,let 𝑓, 𝑔 : [𝑎, 𝑏] → R be continuous on [𝑎, 𝑏], differentiableon
(𝑎, 𝑏), and 𝑔(𝑥) ̸= 0 on (𝑎, 𝑏). If 𝑓(𝑥)/𝑔(𝑥) is
(strictly)increasing (or (strictly) decreasing, resp.) on (𝑎, 𝑏),
so are thefunctions
𝑓 (𝑥) − 𝑓 (𝑎)
𝑔 (𝑥) − 𝑔 (𝑎),
𝑓 (𝑥) − 𝑓 (𝑏)
𝑔 (𝑥) − 𝑔 (𝑏). (24)
Lemma 4 (see [14, Theorem 3.21]). The function
ℎ (𝑟) =
E − (𝑟)2
K
𝑟2(25)
is strictly increasing and convex from (0, 1) onto (𝜋/4, 1).
Lemma 5. The function
𝑓 (𝑟) =
2E − (𝑟)2
K
𝑟2(26)
is strictly decreasing on (0, 1) and satisfies
lim𝑟→0
+
𝑓 (𝑟) = ∞, lim𝑟→1
−
𝑓 (𝑟) = 2. (27)
Proof. By the first three formulas in Lemma 2, simple
com-putations lead to
𝑓(𝑟) =
(𝑟)2
K − 3E
𝑟3≜𝑓1(𝑟)
𝑟3,
𝑓
1(𝑟) =
(𝑟)2
K +K − 2E
𝑟≜𝑓2(𝑟)
𝑟,
𝑓
2(𝑟) =
𝑟
(𝑟)2[E − (𝑟
)2
K] ≜𝑟3
(𝑟)2ℎ (𝑟) ,
(28)
where the function ℎ(𝑟) is defined by (25) in Lemma 4.
From
𝑓
1(0) = 𝑓
2(0) = 𝑓
2(0) = 0, 𝑓
1(1) = −3, (29)
it follows that
𝑓
2(𝑟) > 0, 𝑓
2(𝑟) > 0, 𝑓
1(𝑟) > 0,
𝑓1(𝑟) < 0, 𝑓
(𝑟) < 0.
(30)
Hence, the function 𝑓(𝑟) is strictly decreasing on (0,
1).Further, by easily obtained limits in (27), the proof ofLemma 5
is complete.
3. Some Inequalities for BoundingToader Mean
Now, we are in a position to give an affirmative solution
toProblem 1 and to provide an alternative proof for
inequalities(14) to (19).
Theorem 6. The double inequality
𝛼𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐻 (𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) < 𝛽𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐻 (𝑎, 𝑏)
(31)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ̸= 𝑏 if and only if
𝛼 ≤5
8= 0.625, 𝛽 ≥
2
𝜋= 0.636 . . . . (32)
Proof. Without loss of generality, assume that 𝑎 > 𝑏 > 0.
Let𝑡 = 𝑏/𝑎. Then, 𝑡 ∈ (0, 1) and
𝑇 (𝑎, 𝑏) − 𝐻 (𝑎, 𝑏)
𝑀𝐶𝐻
(𝑎, 𝑏)=
(2/𝜋)E (√1 − 𝑡2 ) − (2𝑡/ (1 + 𝑡))
(1 − 𝑡)2/ (1 + 𝑡)
.
(33)
Let 𝑟 = (1− 𝑡)/(1+ 𝑡). Then, 𝑟 ∈ (0, 1), and by the last
formulain Lemma 2,
𝑇 (𝑎, 𝑏) − 𝐻 (𝑎, 𝑏)
𝑀𝐶𝐻
(𝑎, 𝑏)=(2/𝜋)E (2√𝑟 / (1 + 𝑟)) + 𝑟 − 1
2𝑟2/ (1 + 𝑟)
=𝑓1(𝑟)
𝑓2(𝑟)
+1
2,
(34)
where
𝑓1(𝑟) =
2
𝜋[2E − (𝑟
)2
K] − 1, 𝑓2(𝑟) = 2𝑟
2. (35)
By the middle two formulas in Lemma 2, a
straightforwardcalculation leads to
𝑓1(0) = 𝑓
2(0) = 0, 𝑓
1(𝑟) =
2
𝜋
E − (𝑟)2
K
𝑟, 𝑓
2(𝑟) = 4𝑟.
(36)
So, we have
𝑓
1(𝑟)
𝑓
2(𝑟)
=1
2𝜋
E − (𝑟)2
K
𝑟2=ℎ (𝑟)
2𝜋. (37)
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4 Journal of Function Spaces and Applications
Combining this with Lemmas 3 and 4 reveals that thefunction𝑓
1(𝑟)/𝑓2(𝑟) is strictly increasing on (0, 1). Moreover,
using L’Hôpital’s rule, we obtain
lim𝑟→0
+
𝑓1(𝑟)
𝑓2(𝑟)
+1
2= lim𝑟→0
+
1
2𝜋
E − (𝑟)2
K
𝑟2+1
2=5
8,
lim𝑟→1
−
𝑓1(𝑟)
𝑓2(𝑟)
+1
2= lim𝑟→1
−
(2/𝜋) [2E − (𝑟)2
K] − 1
2𝑟2+1
2
= lim𝑟→1
−
E
𝜋𝑟2+ lim𝑟→1
−
ℎ (𝑟)
𝜋=2
𝜋.
(38)
The proof of Theorem 6 is thus complete.
Theorem 7. For all 𝑎, 𝑏 > 0 with 𝑎 ̸= 𝑏,(1) the double
inequality
𝛼1𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐴 (𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) < 𝛽1𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐴 (𝑎, 𝑏)
(39)
holds if and only if
𝛼1≤1
8= 0.125, 𝛽
1≥2
𝜋−1
2= 0.136 . . . , (40)
(2) the double inequality
𝛼2𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐶 (𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) < 𝛽2𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐶 (𝑎, 𝑏)
(41)
holds if and only if
𝛼2≤ −
3
8= −0.375, 𝛽
2≥2
𝜋− 1 = −0.363 . . . , (42)
(3) the double inequality
𝛼3𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐶 (𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) < 𝛽3𝑀𝐶𝐻
(𝑎, 𝑏) + 𝐶 (𝑎, 𝑏)
(43)
holds if and only if
𝛼3≤ −
1
24= −0.041 . . . , 𝛽
3≥2
𝜋−2
3= −0.030 . . . .
(44)
Proof. From the identities in (12), it follows that
𝐻(𝑎, 𝑏) = 𝐴 (𝑎, 𝑏) −1
2𝑀𝐶𝐻
(𝑎, 𝑏)
= 𝐶 (𝑎, 𝑏) − 𝑀𝐶𝐻
(𝑎, 𝑏) = 𝐶 (𝑎, 𝑏) −2
3𝑀𝐶𝐻
(𝑎, 𝑏) .
(45)
Substituting these into the inequality (31) in Theorem 6acquires
inequalities (39) to (43) inTheorem 7.
Theorem 8. The inequality
𝑇 (𝑎, 𝑏) > 𝜆𝑀𝐶𝐻
(𝑎, 𝑏) (46)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ̸= 𝑏 if and only if 𝜆 ≤
2/𝜋.
Proof. Without loss of generality, assume that 𝑎 > 𝑏 > 0.
Let𝑡 = 𝑏/𝑎. Then, 𝑡 ∈ (0, 1) and
𝑇 (𝑎, 𝑏)
𝑀𝐶𝐻
(𝑎, 𝑏)=
(2/𝜋)E (√1 − 𝑡2 )
(1 − 𝑡)2/ (1 + 𝑡)
. (47)
Let 𝑟 = (1− 𝑡)/(1+ 𝑡). Then, 𝑟 ∈ (0, 1), and by the last
formulain Lemma 2,
𝑇 (𝑎, 𝑏)
𝑀𝐶𝐻
(𝑎, 𝑏)=(2/𝜋)E (2√𝑟 / (1 + 𝑟))
2𝑟2/ (1 + 𝑟)=1
𝜋
2E − (𝑟)2
K
𝑟2.
(48)
Therefore, from Lemma 5, it follows that function 𝑇(𝑎, 𝑏)/𝑀𝐶𝐻(𝑎,
𝑏) is strictly decreasing and
lim𝑟→1
−
𝑇 (𝑎, 𝑏)
𝑀𝐶𝐻
(𝑎, 𝑏)= lim𝑟→1
−
1
𝜋
2E − (𝑟)2
K
𝑟2=2
𝜋. (49)
Theorem 8 is thus proved.
4. Remarks
Finally, we would like to remark several things, including
theconnection between Toader mean and the complete ellipticintegral
of the second kind.
Remark 9. The double inequality (39) is equivalent to
(9).Consequently, inequalities (14) to (19) are recovered
onceagain.
Remark 10. The coefficient 3/2 in (12) corrects an error
whichappeared at the corresponding position in (4.4) in [10,
page1013]. Luckily, this error does not influence the correctness
ofany other conclusions in [10].
Remark 11. We point out that Toader mean 𝑇(𝑎, 𝑏) satisfies
𝑇 (𝑎, 𝑏) = 𝑅𝐸(𝑎2, 𝑏2) , (50)
where
𝑅𝐸(𝑥, 𝑦) =
1
𝜋∫
∞
0
(𝑥
𝑡 + 𝑥+
𝑦
𝑡 + 𝑦)
𝑡
√(𝑡 + 𝑥) (𝑡 + 𝑦)
𝑑𝑡
(51)
is the complete symmetric elliptic integral of the secondkind
and is a symmetric and homogeneous function; see[15, equation
(9.2-3)] and [16, page 250, equation (1.6)].Numerous inequalities
involving 𝑅
𝐸,E, andK are known in
the mathematical literature; see [2, 16–19] and [3, pages 40–46]
and closely related references therein. In the past years,the fact
that Toader mean 𝑇 and the elliptic integral 𝑅
𝐸are
the same has been overlooked by several researchers.
Acknowledgments
The authors would like to thank the anonymous referees fortheir
valuable comments on the original version of this paperand
Professor Feng Qi for his kind help in the whole processof
composing this paper.
-
Journal of Function Spaces and Applications 5
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