Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 146945, 11 pages doi:10.1155/2010/146945 Research Article An Optimal Double Inequality between Power-Type Heron and Seiffert Means Yu-Ming Chu,1 Miao-Kun Wang,2 and Ye-Fang Qiu2 Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn Received 29 August 2010; Accepted 16 November 2010 Academic Editor: Alexander I Domoshnitsky Copyright q 2010 Yu-Ming Chu et al 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 For k ∈ 0, ∞ , the power-type Heron mean Hk a, b and the Seiffert mean T a, b of two positive √ ak ab k/2 bk /3 1/k , k / 0; Hk a, b ab, real numbers a and b are defined by Hk a, b k and T a, b a − b /2 arctan a − b / a b , a / b; T a, b a, a b, respectively In this paper, we find the greatest value p and the least value q such that the double inequality Hp a, b < T a, b < Hq a, b holds for all a, b > with a / b Introduction For k ∈ 0, ∞ , the power-type Heron mean Hk a, b and the Seiffert mean T a, b of two positive real numbers a and b are defined by Hk a, b T a, b respectively ⎧ ⎪ ⎪ ⎨ ⎪√ ⎪ ⎩ ⎧ ⎪ ⎨ ak ab k/2 bk 1/k , ab, a−b arctan a − b / a ⎪ ⎩a, k / 0, k b , 0, a / b, a 1.1 b, 1.2 Journal of Inequalities and Applications Recently, the means of two variables have been the subject of intensive research 1– 15 In particular, many remarkable inequalities for Hk a, b and T a, b can be found in the literature 16–20 It is well known that Hk a, b is continuous and strictly increasing with respect to 1/ b−a , k ∈ 0, ∞ for fixed a, b > with a / b Let A a, b a b /2, I a, b 1/e bb /aa √ ab, and H a, b 2ab/ a b be the arithmetic, L a, b b − a / log b − log a , G a, b identric, logarithmic, geometric, and harmonic means of two positive numbers a and b with a / b, respectively Then min{a, b} < H a, b < G a, b < L a, b < I a, b < A a, b < max{a, b} 1.3 For p ∈ R, the power mean Mp a, b of order p of two positive numbers a and b is defined by ⎧ ⎪ ap ⎪ ⎨ Mp a, b bp 1/p , ⎪√ ⎪ ⎩ ab, p / 0, p 1.4 The main properties for power mean are given in 21 In 16 , Jia and Cao presented the inequalities H0 a, b G a, b < L a, b < Hp a, b < Mq a, b , A a, b < Hlog 3/ log a, b 1.5 for all a, b > with a / b, p ≥ 1/2, and q ≥ 2/3 p S´ ndor 22 proved that a I a, b > H1 a, b 1.6 for all a, b > with a / b In 19 , Seiffert established that M1 a, b < T a, b < M2 a, b 1.7 for all a, b > with a / b The purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean T a, b Our main result is the following Theorem 1.1 Theorem 1.1 For all a, b > with a / b, one has Hlog 3/ log π/2 a, b < T a, b < H5/2 a, b , 1.8 and Hlog 3/ log π/2 a, b and H5/2 a, b are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean T a, b , respectively Journal of Inequalities and Applications Lemmas In order to prove our main result, Theorem 1.1, we need two lemmas which we present in this section Lemma 2.1 If k log 3/ log π/2 − 24 k − k − k k k 2.43 and t > 1, then tk 3k k 48k k − k 3k − tk 4k − t8 < 2.1 Proof For t > 1, we clearly see that k − 24 k − k × 3k − tk 3k tk − k k k < t8 −24 k − k k × 3k − t − k 48k k − k 8 4k − t8 3k k k t2 2.2 48k k − k 4k − Let h t −24 k − k k × 3k − t − k 3k k k t2 48k k − k 4k − 2.3 Then h1 68k4 − 281k3 − 1010k2 2072k 2496 −104.992 < 2.4 and h t is strictly decreasing in 1, ∞ because of k k − k 3k − / k − k k 3k < for k log 3/ log π/2 Therefore, Lemma 2.1 follows from 2.2 – 2.4 together with the monotonicity of h t 2− Lemma 2.2 If k log 3/ log π/2 2.43 , t ∈ 1, ∞ , and g t −8t4k−4 8t4k−6 3k 3k 3k−2 3k−4 3k−6 2k 2kt − k t k−4 t 10 − k t − 4k t 4k − t2k − 4k k t t2k−2 4k−1 t2k−4 7−4k t2k−6 10−k tk 2 k−4 tk −2 k tk−2 2ktk−4 2−k tk−6 8t2 −8, then there exists λ ∈ 1, ∞ such that g t > for t ∈ 1, λ and g t < for t ∈ λ, ∞ Journal of Inequalities and Applications g t /t, g2 t t9−k g1 t , g3 t g2 t /2t, g4 t g3 t /2t, g5 t Proof Let g1 t 9−k g5 t /t, and g7 t t g6 t Then elaborated computations lead to g6 t g 0, lim g t 2.5 −∞, t→ ∞ −32 k − t4k−6 3k − t3k−4 k − 10 − k t3k−8 −4 k − 4k 2k t2k−4 k −2 k − k tk−4 t2k 6k2 3k − t2k 4k k − 4k tk × k − 4k tk 2.8 −∞, 2.9 −2 k 3k − k 3k − 3k − t2k k − 10 − k 3k − t2k −8 k−1 k − k − 4k − tk × k − − 4k t k k 10 − k t −2 k − k − k t4 2k k − k − t2 g2 2.10 k−4 144 − 2k > 0, lim g2 t t→ ∞ g3 t −32 k − 2k − 3k 3k − k k −2 k 2 3k 2k k − k − t6 t2k 2−k k−6 k−8 , 2.11 −∞, 2.12 96k k − 2k − t3k−2 t3k 3k − 3k − t2k 6k2 k 3k − t2k 6k k−2 k−4 × 3k − t2k 3k k − 10 − k 3k − t2k−2 × − 4k k tk −4 k − k − k 4k k − k 4k k 5t 2k k 4k − tk × 10 − k t 6k k − k − t − k − k − k 2k k − k − , 2.13 k−2 k−3 k 2k k − k − − 4k tk−2 4 × 4k − tk 2.7 16, 0, 8k k − 4k − tk k − 4k − k k − tk−8 64 k − 2k − t3k k − k − 3k − t2k 4k 4k − t2k−2 2k k − tk−2 2k k − tk−6 lim g1 t × 3k k − 3k − t3k−6 − 4k t2k t→ ∞ 2 t3k k − 4k − t2k−6 g1 −64 k − 2k − t3k − k 3k 10 − k tk × k − t2k−8 g2 t 2.6 16 2k − t4k−8 6k2 t3k−2 − k g1 t g4 t /kt, 4k k 2 t2 Journal of Inequalities and Applications g3 72 5k − − 2k > 0, 2.14 lim g3 t 2.15 −∞, t→ ∞ g4 t −48k k − 2k − 3k × 3k − t3k−4 6k2 k 3k − k k k 3k − t2k k × 3k − t2k 48k k − 2k − t3k−2 6k k −2 k k tk 2k k − k −2 k − k − k k 4k − tk−2 × k 12k k × 3k − 3k − t3k−6 ×k k k − 4k tk k−2 k−3 k × − 4k tk−6 g5 48 k −1038k3 48 k − 2k − t3k−4 k k 3k 3k − t2k − k k−1 k × 4k − tk − k − k − k 10 − k t2 k 2 3k − k k k 2.19 tk−2 k−2 k−3 k−4 24 k − k − , 3549k2 − 3360k lim g5 t k 4k k 4k − tk−4 t→ ∞ 2.17 2.18 10 − k 3k − t2k−6 2, 304.99 > 0, 12 k − k − k − k k−1 k−2 k −∞, 2−k k × 3k − 3k − t2k−2 ×t2k−4 2.16 2k k − k − 885k2 − 210k − 72 −318k3 12k k k 12k k − k − t2 − k − k − k −48 k − 2k − 3k − 3k 3k − 4k − tk tk 4k lim g4 t ×t2k k k k t→ ∞ g5 t k k − k − k − − 4k tk−4 10 − k t4 g4 k − k − 3k − t2k−2 3k k − k − 10 − k 3k − t2k−4 × − 4k k t2k 3k 2196 −∞, 323.50 > 0, 2.20 2.21 Journal of Inequalities and Applications g6 t −48 k − 2k − 3k − 3k − 3k × 2k − 3k − 3k − t3k−8 ×k k t2k 3k × 3k − t2k−2 − k − k 24 k − k − 2 × k−2 12 − k k 24k2 k k−4 k − 4k tk k ×k 4k − tk−2 − k − k − tk−4 k−2 k−2 k−3 k−4 g6 1 k k 28092k − 6768 k k × k k 3k − tk k × 3k − 3k − tk k ×t − k k k −2933.37 < 0, t2k × 4−k 6−k k 2 k k k k 3−k 4−k k − k − k 10 − k − 3k k k k 4k − t 4k − t6 k−1 t4 − k − 4k 4k − t 2k 3−k k−2 3−k k−4 2 × − k 4k − − k From the expression of g7 t and Lemma 2.1, we get g7 t < − 144 k − k − 2k − 3k − 3k − 3k × k−2 3−k 4−k k × k k × 4k k k k 2.22 48k2 k − k − 16 k − k − k 48 k − k − 4−k k tk 3k − 24 k − k − × k−1 k−2 k × k−2 10 − k , 2k − 3k − 3k − − 3k t2k − 24k k − × k × 3k − tk 4 96 k −144 k − k − 2k − 3k − 3k − 3k −144 k − 2 k 4k − tk−6 k − − 4k tk−8 2 k k−3 k−4 k 16233k3 − 30204k2 −3348k4 g7 t 12 k − 2k k − k 3k − 3k − t2k−4 k k − 10 − k 3k − t 2 k 3k − t2k−6 ×k × 4k 2 2k−8 k k k k 144 k − 2 t3k−6 4k − 1 3k − 2k k − k − k k−1 k−2 k−2 3−k k−4 −24 k − k × k−1 k 48 k − 2 4−k k k − k − k 4k − t2k k 3k 3k − tk tk − k 48k k k 4k − t8 2.23 Journal of Inequalities and Applications 140k7 − 9353k6 −131968k 52543k5 − 103636k4 54016 t2k × −24 k − k − k k k k −20221.36 t2k × −24 k − k −k k 3k k tk 88448k2 48k k − k 3k − tk 48k k − k 3k − tk 4k − t8 k k k k k k 51700k3 k 3k 2 tk 4k − t8 < 2.24 From 2.24 , we know that g6 t is strictly decreasing in 1, ∞ Then 2.22 implies that g5 t is strictly decreasing in 1, ∞ From 2.20 and 2.21 together with the monotonicity of g5 t , we clearly see that there exists λ1 ∈ 1, ∞ such that g4 t is strictly increasing in 1, λ1 and strictly decreasing in λ1 , ∞ Inequality 2.17 and 2.18 together with the piecewise monotonicity of g4 t imply that there exists λ2 ∈ 1, ∞ such that g3 t is strictly increasing in 1, λ2 and strictly decreasing in λ2 , ∞ The piecewise monotonicity of g3 t together with 2.14 and 2.15 leads to the fact that there exists λ3 ∈ 1, ∞ such that g2 t is strictly increasing in 1, λ3 and strictly decreasing in λ3 , ∞ From 2.11 and 2.12 together with the piecewise monotonicity of g2 t , we conclude that there exists λ4 ∈ 1, ∞ such that g1 t is strictly increasing in 1, λ4 and strictly decreasing in λ4 , ∞ Equations 2.8 and 2.9 together with the piecewise monotonicity of g1 t imply that there exists λ5 ∈ 1, ∞ such that g t is strictly increasing in 1, λ5 and strictly decreasing in λ5 , ∞ Therefore, Lemma 2.2 follows from 2.5 and 2.6 together with the piecewise monotonicity of g t Proof of Theorem 1.1 Proof of Theorem 1.1 Without loss of generality, we assume that a > b We first prove that T a, b < H5/2 a, b Let t a/b > 1, then from 1.1 and 1.2 we have log T a, b − log H5/2 a, b log t4 − arctan t4 − / t4 − t10 t5 log 3.1 Journal of Inequalities and Applications Let f t log t4 − arctan t4 − / t4 t10 t5 log − 1 3.2 Then simple computations lead to limf t 0, t→1 f t where f1 t t5 2t3 2t6 t4 − t10 arctan t4 − / t4 t5 t arctan t4 − / t4 − t4 − t10 limf1 t − t2 t8 t t5 / t8 f1 t , t5 2t6 t Note that 0, t→1 f1 t 3.3 2t6 t−1 t5 t 3.4 f t , 2 where f2 t t18 2t17 4t16 6t7 − 8t6 21t8 6t15 − 8t12 6t3 4t2 6t11 2t 21t10 log t2 − arctan t2 − / t2 3.5 1>0 for t > Therefore, T a, b < H5/2 a, b follows from 3.1 – 3.5 Next, we prove that T a, b > Hlog 3/ log π/2 a, b Let k a/b > 1, then 1.1 and 1.2 lead to and t log T a, b − log Hk a, b 28t9 log 3/ log π/2 − t2k tk log k 2.43 3.6 Let F t log t2 − arctan t2 − / t2 − t2k tk log k 3.7 Then simple computations lead to lim F t t→1 F t t2 −1 t2k lim F t t→ ∞ 0, 2t2k−1 tk tk−1 2t tk arctan t2 − / t2 3.8 F1 t , 3.9 Journal of Inequalities and Applications where F1 t that arctan t2 − / t2 − t2 − t2k tk / t4 2t2k−2 tk tk−2 Note lim F1 t 0, 3.10 π − < 0, 3.11 t→1 lim F1 t t→ ∞ 2t3 F t t4 2t2k−2 tk tk−2 2 F2 t , 3.12 where F2 t −8t4k−4 8t4k−6 k − t3k−4 − k t3k −2 k tk−2 2kt3k − k 10 − k t3k−6 4k − t2k − 4k − 4k t2k−6 − 4k t2k t2k−2 10 − k tk 2ktk−4 2 t3k−2 2 4k − t2k−4 3.13 k − tk − k tk−6 8t2 − From 3.12 and 3.13 together with Lemma 2.2, we clearly see that there exists λ ∈ 1, ∞ such that F1 t is strictly increasing in 1, λ and strictly decreasing in λ, ∞ Equations 3.9 – 3.11 and the piecewise monotonicity of F1 t imply that there exists μ ∈ 1, ∞ such that F t is strictly increasing in 1, μ and strictly decreasing in μ, ∞ Then from 3.8 we get F t >0 3.14 for t > Therefore, T a, b > Hlog 3/ log π/2 a, b follows from 3.6 and 3.7 together with 3.14 At last, we prove that Hlog 3/ log π/2 a, b and H5/2 a, b are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean T a, b , respectively For any < ε < k log 3/ log π/2 2.43 and x > 0, from 1.1 and 1.2 , one has T 1, x 5/2−ε − H5/2−ε 1, lim x→ ∞ where J x 3x5/2−ε − x 5/2−ε 25/2−ε π −1/ k ·3 Hk ε x, T x, x J x 5/2−ε x 5/4−ε/2 ε > arctan x/ x π −1/k ·3 2 5/2−ε 1, arctan x/ x , 3.15 3.16 5/2−ε 10 Journal of Inequalities and Applications Let x → 0, making use of Taylor extension, we get J x 3x 5/2−ε × −2 5/2−ε x x2 − 15 − ε x ε − 2ε x9/2−ε x3 12 5/2−ε o x 13 − ε ε − x o x2 3.17 o x9/2−ε Equations 3.15 and 3.17 together with inequality 3.16 imply that for any < ε < log 3/ log π/2 , there exist δ δ ε > and X X ε > such that T 1, x > H5/2−ε 1, x for x ∈ 0, δ and Hlog 3/ log π/2 ε 1, x > T 1, x for x ∈ X, ∞ Acknowledgments This work was supported by the Natural Science Foundation of China under Grant no 11071069, the Natural Science Foundation of Zhejiang Province under Grant no Y7080106, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no T200924 References M.-K Wang, Y.-M Chu, and Y.-F Qiu, “Some comparison inequalities for generalized Muirhead and identric means,” Journal of Inequalities and Applications, vol 2010, Article ID 295620, 10 pages, 2010 B.-Y Long and Y.-M Chu, “Optimal inequalities for generalized logarithmic, 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