Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 842542, 4 pages http://dx.doi.org/10.1155/2013/842542
Research Article Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean Wei-Dong Jiang Department of Information Engineering, Weihai Vocational College, Weihai, Shandong 264210, China Correspondence should be addressed to Wei-Dong Jiang;
[email protected] Received 26 August 2013; Accepted 16 September 2013 Academic Editors: F. Kittaneh and K. Sadarangani Copyright Β© 2013 Wei-Dong Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The authors find the greatest value π and the least value π, such that the double inequality πΆ(ππ+(1βππ), ππ+(1βπ)π) < πΌπ΄(π, π)+ (1 β πΌ)π(π, π) < πΆ(ππ + (1 β π)π, ππ + (1 β π)π) holds for all πΌ β (0, 1) and π, π > 0 with π =ΜΈ π, where πΆ(π, π) = 2(π2 + ππ + π2 )/3(π + π), π/2 π΄(π, π) = (π + π)/2, and π(π, π) = (2/π) β«0 βπ2 cos2 π + π2 sin2 πππ denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers π and π.
For π β R and π, π > 0, the centroidal mean πΆ(π, π) and πth power mean ππ (π, π) are, respectively, defined by
1. Introduction In [1], Toader introduced a mean
πΆ (π, π) =
2 π/2 π (π, π) = β« βπ2 cos2 π + π2 sin2 πππ π 0 β1 β (π/π)2 { { { , π > π, 2πE { { { π { { { { ={ β1 β (π/π)2 { { { 2πE , π < π, { { { π { { { π = π, {π,
2 (π2 + ππ + π2 ) 3 (π + π)
,
1/π
(1)
ππ + ππ { { {( ) 2 ππ (π, π) = { { { β { ππ,
, π =ΜΈ 0, π = 0.
In [7], Vuorinen conjectured that π3/2 (π, π) < π (π, π) ,
π (π, π) < πlog 2/ log(π/2) (π, π) , E = E (π) = β«
0
(1 β π2 sin2 π)
1/2
ππ,
(4)
for all π, π > 0 with π =ΜΈ π. This conjecture was verified by Qiu and Shen [8] and by Barnard et al. [9], respectively. In [10], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:
where π/2
(3)
(2)
for π β [0, 1] is the complete elliptic integral of the second kind. In recent years, there have been plenty of literature, such as [2β6], dedicated to the Toader mean.
(5)
for all π, π > 0 with π =ΜΈ π. Chu et al. [5] proved that the double inequality πΆ (πΌπ + (1 β πΌ) π, πΌπ + (1 β πΌ) π) < π (π, π) < πΆ (π½π + (1 β π½) π, π½π + (1 β π½) π)
(6)
2
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holds for all π, π > 0 with π =ΜΈ π if and only if πΌ β€ 3/4 and π½ β₯ 1/2 + β4π β π2 /(2π). Very recently, Hua and Qi [11] proved that the double inequality πΌπΆ (π, π) + (1 β πΌ) π΄ (π, π)
Lemma 2. Let π’, πΌ β (0, 1) and
< π (π, π)
(7)
< π½πΆ (π, π) + (1 β π½) π΄ (π, π) is valid for all π, π > 0 with π =ΜΈ π if and only if πΌ β€ 3/4 and π½ β₯ (12/π) β 3. Where π΄(π, π) = (π + π)/2 denote the arithmetic mean. For positive numbers π, π > 0 with π =ΜΈ π, let π½ (π₯) = πΆ (π₯π + (1 β π₯) π, π₯π + (1 β π₯) π)
(8)
be on [1/2, 1]. It is not difficult to directly verify that π½(π₯) is continuous and strictly increasing on [1/2, 1]. The main purpose of the paper is to find the greatest value π and the least value π, such that the double inequality πΆ(ππ+ (1 β ππ), ππ + (1 β π)π) < πΌπ΄(π, π) + (1 β πΌ)π(π, π) < πΆ(ππ + (1 β π)π, ππ + (1 β π)π) holds for all πΌ β (0, 1) and π, π > 0 with π =ΜΈ π. As applications, we also present new bounds for the complete elliptic integral of the second kind.
2. Preliminaries and Lemmas
π/2
0
(1 β π2 sin2 π)
β1/2
ππ,
KσΈ = KσΈ (π) = K (πσΈ ) , π K (0) = , K (1) = β, 2 π/2
E = E (π) = β«
0
2
2
(1 β π sin π)
π , 2
1/2
(9) ππ,
2
2
π (E β πσΈ K) ππ
= πK,
Proof. From (11), one has ππ’,πΌ (0+ ) = 0,
(12)
1 4 ππ’,πΌ (1β ) = π’ β (1 β πΌ) ( β 1) , 3 π
(13)
2 σΈ ππ’,πΌ (π) = π [π’ β 3 (1 β πΌ) π (π)] , 3
(14)
where π(π) = (1/π)((E β πσΈ K)/π2 ). We divide the proof into four cases. Case 1 (π’ β₯ 3(1βπΌ)/π). From (14) and Lemma 1 together with the monotonicity of π(π), we clearly see that ππ’,πΌ (π) is strictly increasing on (0, 1). Therefore, ππ’,πΌ (π) > 0, for all π β (0, 1).
Case 3 (3(1 β πΌ)/4 < π’ β€ 3(1 β πΌ)(4/π β 1)). From (13) and (14) together with the monotonicity of π(π), we see that there exists π β (0, 1), such that ππ’,πΌ (π) is strictly increasing in (0, π] and strictly decreasing in [π, 1) and (15)
E (1) = 1,
Therefore, making use of (12) and inequality (15) together with the piecewise monotonicity of ππ’,πΌ (π) leads to the conclusion that there exists 0 < π < π < 1, such that ππ’,πΌ (π) > 0 for π β (0, π) and ππ’,πΌ (π) < 0 for π β (π, 1).
πE E β K = , ππ π
Case 4 (3(1 β πΌ)(4/π β 1) β€ π’ < 3(1 β πΌ)/π). Equation (13) leads to
π (K β E) πE = 2, ππ πσΈ 2
E(
Then, ππ’,πΌ > 0, for all π β (0, 1) if and only if π’ β₯ 3(1βπΌ)(4/πβ 1), and ππ’,πΌ < 0, for all π β (0, 1) if and only if π’ β€ 3(1 β πΌ)/4.
ππ’,πΌ (1β ) β€ 0.
respectively. For 0 < π < 1, the formulas πK E β πσΈ K , = 2 ππ ππσΈ
2 β (1 β πΌ) ( (2E (π) β (1 β π2 ) K (π)) β 1) . π (11)
Case 2 (π’ β€ 3(1 β πΌ)/4). From (14) and Lemma 1 together with the monotonicity of π(π), we obtain that ππ’,πΌ (π) is strictly decreasing on (0, 1). Therefore, ππ’,πΌ (π) < 0, for all π β (0, 1).
EσΈ = EσΈ (π) = E (πσΈ ) , E (0) =
1 ππ’,πΌ (π) = π’π2 3
2
In order to establish our main result, we need several formulas and Lemmas below. For 0 < π < 1 and πσΈ = β1 β π2 , Legendreβs complete elliptic integrals of the first and second kinds are defined in [12, 13] by K = K (π) = β«
Lemma 1 (see [14, Theorem 3.21(1), 3.43 exercises 13(a)]). The 2 function (E β πσΈ K)/π2 is strictly increasing from (0, 1) to 2 (π/4, 1), and the function 2E β πσΈ K is increasing from (0, 1) to (π/2, 2).
2βπ 2E β πσΈ K )= 1+π 1+π
were presented in [14, Appendix E, pages 474-475].
(10)
ππ’,πΌ (1β ) β₯ 0.
(16)
From (13) and (14) together with the monotonicity of π(π), we clearly see that there exists π β (0, 1), such that ππ’,πΌ (π) is strictly increasing in (0, π] and strictly decreasing in [π, 1). Therefore, ππ’,πΌ (π) > 0 for π β (0, 1) follows from (12) and (16) together with the piecewise monotonicity of ππ’,πΌ (π).
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3. Main Results
=
Now, we are in a position to state and prove our main results.
1 π 2 [ (1 β 2π) π2 + 1 β πΌ 1+π 3 β (1 β πΌ)
Theorem 3. If πΌ β (0, 1) and π, π β (1/2, 1), then the double inequality πΆ (ππ + (1 β π) π, ππ + (1 β π) π) < πΌπ΄ (π, π) + (1 β πΌ) π (π, π)
(17) Let πΌ = 1/4, π = 7/8, π = (1/2)(1 + (3β4/π β 1/2)). Then, from Theorem 3, we get new bounds for the complete elliptic integral E(π) of the second kind in terms of elementary functions as follows.
holds for all π, π > 0 with π =ΜΈ π if and only if
πβ₯
1 β3 (1 β πΌ) + , 2 4
4 1 (1 + β 3 (1 β πΌ) ( β 1)) . 2 π
Corollary 4. For π β (0, 1) and πσΈ = β1 β π2 , one has (18)
Proof. Since π΄(π, π), π(π, π), and πΆ(π, π) are symmetric and homogeneous of degree one, without loss of generality, we assume that π > π. Let π β (1/2, 1), π‘ = π/π β (0, 1), and π = (1 β π‘)/(1 + π‘). Then, πΆ (ππ + (1 β π) π, ππ + (1 β π) π) β πΌπ΄ (π, π) β (1 β πΌ) π (π, π) 2 π 2 = π ((π + (1 β π) ) 3 π π π + (π + (1 β π) ) (π + 1 β π) π π 2 π β1 π +(π + 1 β π) ) (1 + ) π π
β πΌπ
1 + (π/π) 2
π 2 2π β (1 β πΌ) E (β 1 β ( ) ) π π 2 2 = π { ((π + (1 β π) π‘) 3 + (π + (1 β π) π‘) (ππ‘ + 1 β π) 2
+ (ππ‘ + 1 β π) ) (1 + π‘)β1 βπΌ
1+π‘ 2 β (1 β πΌ) E (β1 β π‘2 )} 2 π 2
= π{
(1 β 2π) π2 + 3 1 βπΌ 3 (1 + π) 1+π 2
2 2E β πσΈ K β (1 β πΌ) } π 1+π
(19) Therefore, Theorem 3 follows easily from Lemma 2 and (19).
< πΆ (ππ + (1 β π) π, ππ + (1 β π) π)
πβ€
2 2 (2E β πσΈ K)] . π
2
2 πσΈ + (2/π) (1 β πσΈ ) π 5 + 6πσΈ + 5πσΈ ]. [ ] < E < π [ (π) 2 1 + πσΈ 8 (1 + πσΈ ) ] [ (20)
4. Remarks Remark 5. In the recent past, the complete elliptic integrals have attracted the attention of numerous mathematicians. In [4], it was established that 2
π [ 1 β 1 + πσΈ 1 + πσΈ ] + 2 2 2 4 [ ] < E (π)
0, for all π₯ β (0, 1). Remark 7. The following equivalence relations for π₯ β (0, 1) show that the lower bound in (20) for E(π) is better than the lower bound in (23): 2 5 + 6π₯ + 5π₯2 β6 + 2π₯ β 3 (1 β π₯ ) > 8 (1 + π₯) 2β2
ββ (5π₯2 + 6π₯ + 5)
2
(25)
> 8(π₯ + 1)2 (3π₯2 + 2π₯ + 3) ββ (π₯ β 1)4 > 0.
Acknowledgments The author is thankful to the anonymous referees for their valuable and profound comments on and suggestions to the original version of this paper. This work was supported by the project of Shandong Higher Education Science and Technology Program under Grant no. J11LA57.
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