The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

Volume 2010, Article ID 436457,7pages doi:10.1155/2010/436457

Research Article

The Optimal Convex Combination

Bounds of Arithmetic and Harmonic Means for

the Seiffert’s Mean

Yu-Ming Chu,

_{Ye-Fang Qiu,}

_{Miao-Kun Wang,}

and Gen-Di Wang

1_{Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China}

2_{Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China}

Correspondence should be addressed to Yu-Ming Chu,[email protected]

Received 28 December 2009; Revised 17 April 2010; Accepted 22 April 2010

Academic Editor: Andrea Laforgia

Copyrightq2010 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.

We find the greatest valueαand least valueβsuch that the double inequalityαAa, b 1−

αHa, b< Pa, b< βAa, b1−βHa, bholds for alla, b >0 witha /b. HereAa, b,Ha, b, andPa, bdenote the arithmetic, harmonic, and Seiffert’s means of two positive numbersaand

b, respectively.

1. Introduction

Fora, b >0 witha /bthe Seiffert’s meanPa, bwas introduced by Seiffert1as follows:

Pa, b a−b 4 arctana/b−π

. _1.1

Recently, the inequalities for means have been the subject of intensive research 2–

11. In particular, many remarkable inequalities for the Seiffert’s mean can be found in the literature12–17. The Seiffert’s meanPa, bcan be rewritten assee14,2.4

Pa, b a−b

a , andLa, b b−a/logb−logabe the arithmetic, geometric, harmonic, identric, and logarithmic means of two positive real numbersaandbwitha /b. Then

min{a, b}< Ha, b< Ga, b< La, b< Ia, b< Aa, b<max{a, b}. 1.3

In1, Seiffert proved that

La, b< Pa, b< Ia, b 1.4

for alla, b >0 witha /b.

Later, Seiffert18established that

Pa, b> 3Aa, bGa, b Aa, b 2Ga, b,

Pa, b> Aa, bGa, b La, b ,

Pa, b> 2

πAa, b

1.5

for alla, b >0 witha /b.

In19, S´andor proved that

1

2Aa, b Ga, b< Pa, b<

Aa, b

1

2Aa, b Ga, b,

A2/3_{a, bG}1/3_{a, b< Pa, b<} 2

3Aa, b 1 3Ga, b

1.6

for alla, b >0 witha /b.

The following bounds for the Seiffert’s mean Pa, b in terms of the power mean Mra, b arbr/21/r r /0were presented by Jagers in17:

M1/2a, b< Pa, b< M2/3a, b 1.7

for alla, b >0 witha /b.

H¨ast ¨o13found the sharp lower power bound for the Seiffert’s mean as follows:

Mlog 2/logπa, b< Pa, b 1.8

for alla, b >0 witha /b.

2. Main Result

Theorem 2.1. The double inequalityαAa, b1−αHa, b< Pa, b< βAa, b1−βHa, b

holds for alla, b >0witha /bif and only ifα≤2/πandβ≥5/6.

Proof. Firstly, we prove that

Pa, b< 5 6Aa, b

1

6Ha, b, 2.1

Pa, b> 2 πAa, b

1− 2

π

Ha, b 2.2

for alla, b >0 witha /b.

Without loss of generality, we assumea > b. Lett a/b > 1 andp ∈ {5/6,2/π}. Then1.1leads to

Pa, b− pAa, b 1−pHa, b

b

pt2₁2_41−pt2

2t2_{14 arctan}t−π

2t4₋₁

pt4_4−2pt2_p −4 arctantπ

.

2.3

Let

ft 2

t4₋₁

pt4₄−2pt2p −4 arctantπ. 2.4

Then simple computations lead to

lim

t→1ft 0, 2.5

lim

t→∞ft 2

p−π, 2.6

f_t 4t−12

t2_{1 pt}4_4−2pt2_p2gt, 2.7

where

gt −p2_t6_−2p2_−2p4t5_p2_−12p8t4

4p2_−20p16t3_p2_−12p8t2

−2p2_−2p4t−p2_.

Case 1. Ifp5/6, then it follows from2.8that

gt −1

36

25t4_16t3_54t2_16t25t−12_{<0 2.9}

fort >1.

Therefore, inequality2.1follows from2.3–2.5and2.7together with2.9.

Case 2. Ifp2/π, then from2.8we have

g1 85−6p8

5−12 π

>0, 2.10

lim

t→∞gt −∞, 2.11

g_{t −6p}2_t5_−10p2_−10p20t4_4p2_−48p32t3

12p2_−60p48t2_2p2_{−24p16t−2p}2_−2p4,

2.12

g_{1 245−6p245−} 12 π

>0, 2.13

lim

t→∞g

_{t −∞, 2.14}

g_{t −30p}2_t4_−40p2_−40p80t3_12p2_−144p96t2

24p2_−120p96t2p2_−24p16,

2.15

g_{1 836−41p−4p}2₈

36−82

π − 16 π2

>0, 2.16

lim

t→∞g

_{t −∞, 2.17}

g_{t −120p}2_t3_−120p2_−120p240t2

24p2_{−288p192t24p}2_−120p96,

2.18

g_{1 4811−11p−4p}2₄₈

11− 22

π − 16 π2

>0, 2.19

lim

t→∞g

_{t −∞, 2.20}

g4_{t −360p}2_t2_−240p2_{−240p480t24p}2_{−288p192, 2.21}

g4_{1 4814−11p−12p}2₄₈

14−22

π − 48 π2

lim

t→∞g

4_{t −∞, 2.23}

g5_{t −720p}2_t−240p2_{−240p480, 2.24}

g5_{1 2402−p−4p}2₂₄₀

2− 2

π − 16 π2

<0. 2.25

From2.24and2.25we clearly see thatg5_{t<0 fort≥1, henceg}4_{tis strictly} decreasing in1,∞. It follows from 2.22 and 2.23together with the monotonicity of g4_{tthat there existsλ}

1>1 such thatg4t>0 fort∈1, λ1andg4t<0 fort∈λ1,∞,

hencegtis strictly increasing in1, λ1and strictly decreasing inλ1,∞.

From2.19and2.20together with the monotonicity ofgtwe know that there existsλ2 >1 such thatgt>0 fort∈1, λ2andgt<0 fort ∈λ2,∞, hence,gtis

strictly increasing in1, λ2and strictly decreasing inλ2,∞.

From2.16and2.17together with the monotonicity ofgtwe clearly see that there existsλ3>1 such thatgtis strictly increasing in1, λ3and strictly decreasing inλ3,∞. It

follows from2.13and2.14together with the monotonicity ofgtthat there existsλ4 >1

such that gtis strictly increasing in 1, λ4 and strictly decreasing inλ4,∞. Then2.7,

2.10and2.11imply that there existsλ5 > 1 such thatftis strictly increasing in1, λ5

and strictly decreasing inλ5,∞.

Note that2.6becomes

lim

t→∞ft 0 2.26

forp2/π.

It follows from2.5and2.26together with the monotonicity offtthat

ft>0 2.27

fort >1.

Therefore, inequality2.2follows from2.3and2.4together with2.27.

Secondly, we prove that5/6Aa, b 1/6Ha, bis the best possible upper convex combination bound of arithmetic and harmonic means for the Seiffert’s meanPa, b.

For anyt >1 andβ∈R, from1.1we have

P1, t2_{−βA1, t}2_{1−βH1, t}2 t2−1

4 arctant−π − β 2

1t2−21−β t

2

1t2

ht

4 arctant−π1t2,

2.28

where

ht t4₋₁₋β

2

h1 h1 h1 0, 2.30

h_{1 45−6β. 2.31}

Ifβ <5/6, then2.31leads to

h_{1>0. 2.32}

From2.32and the continuity ofhtwe clearly see that there existsδ δβ > 0 such that

h_{t>0 2.33}

fort∈1,1δ. Then2.30and2.33imply that

ht>0 2.34

fort∈1,1δ.

Therefore,P1, t2_{> βA1, t}2 _{1−βH1, t}2_{fort∈1,1δfollows from2.28and}

2.34.

Finally, we prove that2/πAa, b1−2/πHa, bis the best possible lower convex combination bound of arithmetic and harmonic means for the Seiffert’s meanPa, b.

Forα >2/π, then from1.1one has

lim

x→∞

αA1, x 1−αH1, x P1, x

π

2α >1. 2.35

Inequality2.35implies that for anyα > 2/π there exists X Xα > 1 such that αA1, x 1−αH1, x> P1, xforx∈X,∞.

Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China Grant 60850005, N S Foundation of Zhejiang Province

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