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Sep 16, 2013 - http://dx.doi.org/10.1155/2013/842542. Research Article. Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of CentroidalΒ ...
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œ† and the least value πœ‡, such that the double inequality 𝐢(πœ†π‘Ž+(1βˆ’πœ†π‘), πœ†π‘+(1βˆ’πœ†)π‘Ž) < 𝛼𝐴(π‘Ž, 𝑏)+ (1 βˆ’ 𝛼)𝑇(π‘Ž, 𝑏) < 𝐢(πœ‡π‘Ž + (1 βˆ’ πœ‡)𝑏, πœ‡π‘ + (1 βˆ’ πœ‡)π‘Ž) holds for all 𝛼 ∈ (0, 1) and π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏, where 𝐢(π‘Ž, 𝑏) = 2(π‘Ž2 + π‘Žπ‘ + 𝑏2 )/3(π‘Ž + 𝑏), πœ‹/2 𝐴(π‘Ž, 𝑏) = (π‘Ž + 𝑏)/2, and 𝑇(π‘Ž, 𝑏) = (2/πœ‹) ∫0 βˆšπ‘Ž2 cos2 πœƒ + 𝑏2 sin2 πœƒπ‘‘πœƒ denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers π‘Ž and 𝑏.

For 𝑝 ∈ R and π‘Ž, 𝑏 > 0, the centroidal mean 𝐢(π‘Ž, 𝑏) and 𝑝th power mean 𝑀𝑝 (π‘Ž, 𝑏) are, respectively, defined by

1. Introduction In [1], Toader introduced a mean

𝐢 (π‘Ž, 𝑏) =

2 πœ‹/2 𝑇 (π‘Ž, 𝑏) = ∫ βˆšπ‘Ž2 cos2 πœƒ + 𝑏2 sin2 πœƒπ‘‘πœƒ πœ‹ 0 √1 βˆ’ (𝑏/π‘Ž)2 { { { , π‘Ž > 𝑏, 2π‘ŽE { { { πœ‹ { { { { ={ √1 βˆ’ (π‘Ž/𝑏)2 { { { 2𝑏E , π‘Ž < 𝑏, { { { πœ‹ { { { π‘Ž = 𝑏, {π‘Ž,

2 (π‘Ž2 + π‘Žπ‘ + 𝑏2 ) 3 (π‘Ž + 𝑏)

,

1/𝑝

(1)

π‘Žπ‘ + π‘Žπ‘ { { {( ) 2 𝑀𝑝 (π‘Ž, 𝑏) = { { { √ { π‘Žπ‘,

, 𝑝 =ΜΈ 0, 𝑝 = 0.

In [7], Vuorinen conjectured that 𝑀3/2 (π‘Ž, 𝑏) < 𝑇 (π‘Ž, 𝑏) ,

𝑇 (π‘Ž, 𝑏) < 𝑀log 2/ log(πœ‹/2) (π‘Ž, 𝑏) , E = E (π‘Ÿ) = ∫

0

(1 βˆ’ π‘Ÿ2 sin2 πœƒ)

1/2

π‘‘πœƒ,

(4)

for all π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏. 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œ‹/2

(3)

(2)

for π‘Ÿ ∈ [0, 1] is the complete elliptic integral of the second kind. In recent years, there have been plenty of literature, such as [2–6], dedicated to the Toader mean.

(5)

for all π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏. Chu et al. [5] proved that the double inequality 𝐢 (π›Όπ‘Ž + (1 βˆ’ 𝛼) 𝑏, 𝛼𝑏 + (1 βˆ’ 𝛼) π‘Ž) < 𝑇 (π‘Ž, 𝑏) < 𝐢 (π›½π‘Ž + (1 βˆ’ 𝛽) 𝑏, 𝛽𝑏 + (1 βˆ’ 𝛽) π‘Ž)

(6)

2

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holds for all π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏 if and only if 𝛼 ≀ 3/4 and 𝛽 β‰₯ 1/2 + √4πœ‹ βˆ’ πœ‹2 /(2πœ‹). Very recently, Hua and Qi [11] proved that the double inequality 𝛼𝐢 (π‘Ž, 𝑏) + (1 βˆ’ 𝛼) 𝐴 (π‘Ž, 𝑏)

Lemma 2. Let 𝑒, 𝛼 ∈ (0, 1) and

< 𝑇 (π‘Ž, 𝑏)

(7)

< 𝛽𝐢 (π‘Ž, 𝑏) + (1 βˆ’ 𝛽) 𝐴 (π‘Ž, 𝑏) is valid for all π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏 if and only if 𝛼 ≀ 3/4 and 𝛽 β‰₯ (12/πœ‹) βˆ’ 3. Where 𝐴(π‘Ž, 𝑏) = (π‘Ž + 𝑏)/2 denote the arithmetic mean. For positive numbers π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏, let 𝐽 (π‘₯) = 𝐢 (π‘₯π‘Ž + (1 βˆ’ π‘₯) 𝑏, π‘₯𝑏 + (1 βˆ’ π‘₯) π‘Ž)

(8)

be on [1/2, 1]. It is not difficult to directly verify that 𝐽(π‘₯) is continuous and strictly increasing on [1/2, 1]. The main purpose of the paper is to find the greatest value πœ† and the least value πœ‡, such that the double inequality 𝐢(πœ†π‘Ž+ (1 βˆ’ πœ†π‘), πœ†π‘ + (1 βˆ’ πœ†)π‘Ž) < 𝛼𝐴(π‘Ž, 𝑏) + (1 βˆ’ 𝛼)𝑇(π‘Ž, 𝑏) < 𝐢(πœ‡π‘Ž + (1 βˆ’ πœ‡)𝑏, πœ‡π‘ + (1 βˆ’ πœ‡)π‘Ž) holds for all 𝛼 ∈ (0, 1) and π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏. As applications, we also present new bounds for the complete elliptic integral of the second kind.

2. Preliminaries and Lemmas

πœ‹/2

0

(1 βˆ’ π‘Ÿ2 sin2 πœƒ)

βˆ’1/2

π‘‘πœƒ,

KσΈ€  = KσΈ€  (π‘Ÿ) = K (π‘ŸσΈ€  ) , πœ‹ K (0) = , K (1) = ∞, 2 πœ‹/2

E = E (π‘Ÿ) = ∫

0

2

2

(1 βˆ’ π‘Ÿ sin πœƒ)

πœ‹ , 2

1/2

(9) π‘‘πœƒ,

2

2

𝑑 (E βˆ’ π‘ŸσΈ€  K) π‘‘π‘Ÿ

= π‘ŸK,

Proof. From (11), one has 𝑓𝑒,𝛼 (0+ ) = 0,

(12)

1 4 𝑓𝑒,𝛼 (1βˆ’ ) = 𝑒 βˆ’ (1 βˆ’ 𝛼) ( βˆ’ 1) , 3 πœ‹

(13)

2 σΈ€  𝑓𝑒,𝛼 (π‘Ÿ) = π‘Ÿ [𝑒 βˆ’ 3 (1 βˆ’ 𝛼) 𝑔 (π‘Ÿ)] , 3

(14)

where 𝑔(π‘Ÿ) = (1/πœ‹)((E βˆ’ π‘ŸσΈ€  K)/π‘Ÿ2 ). We divide the proof into four cases. Case 1 (𝑒 β‰₯ 3(1βˆ’π›Ό)/πœ‹). From (14) and Lemma 1 together with the monotonicity of 𝑔(π‘Ÿ), we clearly see that 𝑓𝑒,𝛼 (π‘Ÿ) is strictly increasing on (0, 1). Therefore, 𝑓𝑒,𝛼 (π‘Ÿ) > 0, for all π‘Ÿ ∈ (0, 1).

Case 3 (3(1 βˆ’ 𝛼)/4 < 𝑒 ≀ 3(1 βˆ’ 𝛼)(4/πœ‹ βˆ’ 1)). From (13) and (14) together with the monotonicity of 𝑔(π‘Ÿ), we see that there exists πœ† ∈ (0, 1), such that 𝑓𝑒,𝛼 (π‘Ÿ) is strictly increasing in (0, πœ†] and strictly decreasing in [πœ†, 1) and (15)

E (1) = 1,

Therefore, making use of (12) and inequality (15) together with the piecewise monotonicity of 𝑓𝑒,𝛼 (π‘Ÿ) leads to the conclusion that there exists 0 < πœ† < πœ‚ < 1, such that 𝑓𝑒,𝛼 (π‘Ÿ) > 0 for π‘Ÿ ∈ (0, πœ‚) and 𝑓𝑒,𝛼 (π‘Ÿ) < 0 for π‘Ÿ ∈ (πœ‚, 1).

𝑑E E βˆ’ K = , π‘‘π‘Ÿ π‘Ÿ

Case 4 (3(1 βˆ’ 𝛼)(4/πœ‹ βˆ’ 1) ≀ 𝑒 < 3(1 βˆ’ 𝛼)/πœ‹). Equation (13) leads to

𝑑 (K βˆ’ E) π‘ŸE = 2, π‘‘π‘Ÿ π‘ŸσΈ€  2

E(

Then, 𝑓𝑒,𝛼 > 0, for all π‘Ÿ ∈ (0, 1) if and only if 𝑒 β‰₯ 3(1βˆ’π›Ό)(4/πœ‹βˆ’ 1), and 𝑓𝑒,𝛼 < 0, for all π‘Ÿ ∈ (0, 1) if and only if 𝑒 ≀ 3(1 βˆ’ 𝛼)/4.

𝑓𝑒,𝛼 (1βˆ’ ) ≀ 0.

respectively. For 0 < π‘Ÿ < 1, the formulas 𝑑K E βˆ’ π‘ŸσΈ€  K , = 2 π‘‘π‘Ÿ π‘Ÿπ‘ŸσΈ€ 

2 βˆ’ (1 βˆ’ 𝛼) ( (2E (π‘Ÿ) βˆ’ (1 βˆ’ π‘Ÿ2 ) K (π‘Ÿ)) βˆ’ 1) . πœ‹ (11)

Case 2 (𝑒 ≀ 3(1 βˆ’ 𝛼)/4). From (14) and Lemma 1 together with the monotonicity of 𝑔(π‘Ÿ), we obtain that 𝑓𝑒,𝛼 (π‘Ÿ) is strictly decreasing on (0, 1). Therefore, 𝑓𝑒,𝛼 (π‘Ÿ) < 0, for all π‘Ÿ ∈ (0, 1).

EσΈ€  = EσΈ€  (π‘Ÿ) = E (π‘ŸσΈ€  ) , E (0) =

1 𝑓𝑒,𝛼 (π‘Ÿ) = π‘’π‘Ÿ2 3

2

In order to establish our main result, we need several formulas and Lemmas below. For 0 < π‘Ÿ < 1 and π‘ŸσΈ€  = √1 βˆ’ π‘Ÿ2 , Legendre’s complete elliptic integrals of the first and second kinds are defined in [12, 13] by K = K (π‘Ÿ) = ∫

Lemma 1 (see [14, Theorem 3.21(1), 3.43 exercises 13(a)]). The 2 function (E βˆ’ π‘ŸσΈ€  K)/π‘Ÿ2 is strictly increasing from (0, 1) to 2 (πœ‹/4, 1), and the function 2E βˆ’ π‘ŸσΈ€  K is increasing from (0, 1) to (πœ‹/2, 2).

2βˆšπ‘Ÿ 2E βˆ’ π‘ŸσΈ€  K )= 1+π‘Ÿ 1+π‘Ÿ

were presented in [14, Appendix E, pages 474-475].

(10)

𝑓𝑒,𝛼 (1βˆ’ ) β‰₯ 0.

(16)

From (13) and (14) together with the monotonicity of 𝑔(π‘Ÿ), we clearly see that there exists πœ† ∈ (0, 1), such that 𝑓𝑒,𝛼 (π‘Ÿ) is strictly increasing in (0, πœ†] and strictly decreasing in [πœ†, 1). Therefore, 𝑓𝑒,𝛼 (π‘Ÿ) > 0 for π‘Ÿ ∈ (0, 1) follows from (12) and (16) together with the piecewise monotonicity of 𝑓𝑒,𝛼 (π‘Ÿ).

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3

3. Main Results

=

Now, we are in a position to state and prove our main results.

1 π‘Ž 2 [ (1 βˆ’ 2𝑝) π‘Ÿ2 + 1 βˆ’ 𝛼 1+π‘Ÿ 3 βˆ’ (1 βˆ’ 𝛼)

Theorem 3. If 𝛼 ∈ (0, 1) and πœ†, πœ‡ ∈ (1/2, 1), then the double inequality 𝐢 (πœ†π‘Ž + (1 βˆ’ πœ†) 𝑏, πœ†π‘ + (1 βˆ’ πœ†) π‘Ž) < 𝛼𝐴 (π‘Ž, 𝑏) + (1 βˆ’ 𝛼) 𝑇 (π‘Ž, 𝑏)

(17) Let 𝛼 = 1/4, πœ† = 7/8, πœ‡ = (1/2)(1 + (3√4/πœ‹ βˆ’ 1/2)). Then, from Theorem 3, we get new bounds for the complete elliptic integral E(π‘Ÿ) of the second kind in terms of elementary functions as follows.

holds for all π‘Ž, 𝑏 > 0 with π‘Ž =ΜΈ 𝑏 if and only if

πœ‡β‰₯

1 √3 (1 βˆ’ 𝛼) + , 2 4

4 1 (1 + √ 3 (1 βˆ’ 𝛼) ( βˆ’ 1)) . 2 πœ‹

Corollary 4. For π‘Ÿ ∈ (0, 1) and π‘ŸσΈ€  = √1 βˆ’ π‘Ÿ2 , one has (18)

Proof. Since 𝐴(π‘Ž, 𝑏), 𝑇(π‘Ž, 𝑏), and 𝐢(π‘Ž, 𝑏) are symmetric and homogeneous of degree one, without loss of generality, we assume that π‘Ž > 𝑏. Let 𝑝 ∈ (1/2, 1), 𝑑 = 𝑏/π‘Ž ∈ (0, 1), and π‘Ÿ = (1 βˆ’ 𝑑)/(1 + 𝑑). Then, 𝐢 (π‘π‘Ž + (1 βˆ’ 𝑝) 𝑏, 𝑝𝑏 + (1 βˆ’ 𝑝) π‘Ž) βˆ’ 𝛼𝐴 (π‘Ž, 𝑏) βˆ’ (1 βˆ’ 𝛼) 𝑇 (π‘Ž, 𝑏) 2 𝑏 2 = π‘Ž ((𝑝 + (1 βˆ’ 𝑝) ) 3 π‘Ž 𝑏 𝑏 + (𝑝 + (1 βˆ’ 𝑝) ) (𝑝 + 1 βˆ’ 𝑝) π‘Ž π‘Ž 2 𝑏 βˆ’1 𝑏 +(𝑝 + 1 βˆ’ 𝑝) ) (1 + ) π‘Ž π‘Ž

βˆ’ π›Όπ‘Ž

1 + (𝑏/π‘Ž) 2

𝑏 2 2π‘Ž βˆ’ (1 βˆ’ 𝛼) E (√ 1 βˆ’ ( ) ) πœ‹ π‘Ž 2 2 = π‘Ž { ((𝑝 + (1 βˆ’ 𝑝) 𝑑) 3 + (𝑝 + (1 βˆ’ 𝑝) 𝑑) (𝑝𝑑 + 1 βˆ’ 𝑝) 2

+ (𝑝𝑑 + 1 βˆ’ 𝑝) ) (1 + 𝑑)βˆ’1 βˆ’π›Ό

1+𝑑 2 βˆ’ (1 βˆ’ 𝛼) E (√1 βˆ’ 𝑑2 )} 2 πœ‹ 2

= π‘Ž{

(1 βˆ’ 2𝑝) π‘Ÿ2 + 3 1 βˆ’π›Ό 3 (1 + π‘Ÿ) 1+π‘Ÿ 2

2 2E βˆ’ π‘ŸσΈ€  K βˆ’ (1 βˆ’ 𝛼) } πœ‹ 1+π‘Ÿ

(19) Therefore, Theorem 3 follows easily from Lemma 2 and (19).

< 𝐢 (πœ‡π‘Ž + (1 βˆ’ πœ‡) 𝑏, πœ‡π‘ + (1 βˆ’ πœ‡) π‘Ž)

πœ†β‰€

2 2 (2E βˆ’ π‘ŸσΈ€  K)] . πœ‹

2

2 π‘ŸσΈ€  + (2/πœ‹) (1 βˆ’ π‘ŸσΈ€  ) πœ‹ 5 + 6π‘ŸσΈ€  + 5π‘ŸσΈ€  ]. [ ] < E < πœ‹ [ (π‘Ÿ) 2 1 + π‘ŸσΈ€  8 (1 + π‘ŸσΈ€  ) ] [ (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œ‹ [ 1 √ 1 + π‘ŸσΈ€  1 + π‘ŸσΈ€  ] + 2 2 2 4 [ ] < E (π‘Ÿ)
0, for all π‘₯ ∈ (0, 1). Remark 7. The following equivalence relations for π‘₯ ∈ (0, 1) show that the lower bound in (20) for E(π‘Ÿ) is better than the lower bound in (23): 2 5 + 6π‘₯ + 5π‘₯2 √6 + 2π‘₯ βˆ’ 3 (1 βˆ’ π‘₯ ) > 8 (1 + π‘₯) 2√2

⇐⇒ (5π‘₯2 + 6π‘₯ + 5)

2

(25)

> 8(π‘₯ + 1)2 (3π‘₯2 + 2π‘₯ + 3) ⇐⇒ (π‘₯ βˆ’ 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.

References [1] G. Toader, β€œSome mean values related to the arithmetic-geometric mean,” Journal of Mathematical Analysis and Applications, vol. 218, no. 2, pp. 358–368, 1998. [2] Y.-M. Chu and M.-K. Wang, β€œInequalities between arithmeticgeometric, Gini, and Toader means,” Abstract and Applied Analysis, vol. 2012, Article ID 830585, 11 pages, 2012. [3] Y.-M. Chu and M.-K. Wang, β€œOptimal lehmer mean bounds for the Toader mean,” Results in Mathematics, vol. 61, no. 3-4, pp. 223–229, 2012. [4] Y.-M. Chu, M.-K. Wang, and S.-L. Qiu, β€œOptimal combinations bounds of root-square and arithmetic means for Toader mean,” Proceedings of the Indian Academy of Sciences, vol. 122, no. 1, pp. 41–51, 2012. [5] Y.-M. Chu, M.-K. Wang, and X.-Y. Ma, β€œSharp bounds for Toader mean in terms of contraharmonic mean with applications,” Journal of Mathematical Inequalities, vol. 7, no. 2, pp. 161– 166, 2013. [6] Y.-M. Chu, M.-K. Wang, S.-L. Qiu, and Y.-F. Qiu, β€œSharp generalized seiffert mean bounds for toader mean,” Abstract and Applied Analysis, vol. 2011, Article ID 605259, 8 pages, 2011.

[7] M. Vuorinen, β€œHypergeometric functions in geometric function theory,” in Proceedings of the Workshop on Special Functions and Differential Equations, pp. 119–126, The Institute of Mathematical Sciences, Madras, India, January 1998. [8] S.-L. Qiu and J.-M. Shen, β€œOn two problems concerning means,” Journal of Hangzhou Insitute of Electronic Engineering, vol. 17, no. 3, pp. 1–7, 1997 (Chinese). [9] R. W. Barnard, K. Pearce, and K. C. Richards, β€œAn inequality involving the generalized hypergeometric function and the arc length of an ellipse,” SIAM Journal on Mathematical Analysis, vol. 31, no. 3, pp. 693–699, 2000. [10] H. Alzer and S.-L. Qiu, β€œMonotonicity theorems and inequalities for the complete elliptic integrals,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 289–312, 2004. [11] Y. Hua and F. Qi, β€œThe best bounds for toader mean in terms of the centroidal and arithmetic mean,” http://arxiv.org/abs/ 1303.2451. [12] F. Bowman, Introduction to Elliptic Functions with Applications, Dover, New York, NY, USA, 1961. [13] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, New York, NY, USA, 1971. [14] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, NY, USA, 1997. [15] B.-N. Guo and F. Qi, β€œSome bounds for the complete elliptic integrals of the first and second kinds,” Mathematical Inequalities and Applications, vol. 14, no. 2, pp. 323–334, 2011. [16] L. Yin and F. Qi, β€œSome inequalities for complete elliptic integrals,” http://arxiv.org/abs/1301.4385.