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Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means

Abstract

In this paper, we prove three sharp inequalities as follows: P(a,b)> L 2 (a,b), T(a,b)> L 5 (a,b) and M(a,b)> L 4 (a,b) for all a,b>0 with ab. Here, L r (a,b), M(a,b), P(a,b) and T(a,b) are the r th generalized logarithmic, Neuman-Sándor, first and second Seiffert means of a and b, respectively.

MSC:26E60.

1 Introduction

The Neuman-Sándor mean M(a,b) [1] and the first and second Seiffert means P(a,b) [2] and T(a,b) [3] of two positive numbers a and b are defined by

(1.1)
(1.2)

and

T(a,b)={ a b 2 arctan ( a b a + b ) , a b , a , a = b ,
(1.3)

respectively.

Recently, these means, M, P and T, have been the subject of intensive research. In particular, many remarkable inequalities for M, P and T can be found in the literature [1, 48].

The power mean M p (a,b) of order r of two positive numbers a and b is defined by

M p (a,b)={ ( a p + b p 2 ) 1 / p , p 0 , a b , p = 0 .

The main properties for M p (a,b) are given in [9]. In particular, the function p M p (a,b) (ab) is continuous and strictly increasing on .

The arithmetic-geometric mean AG(a,b) of two positive numbers a and b is defined as the common limit of sequences { a n } and { b n }, which are given by

a 0 =a, b 0 =b, a n + 1 = a n + b n 2 , b n + 1 = a n b n .

Let H(a,b)=2ab/(a+b), G(a,b)= a b , L(a,b)=(ba)/(logbloga), I(a,b)=1/e ( b b / a a ) 1 / ( b a ) , A(a,b)=(a+b)/2, and S(a,b)= ( a 2 + b 2 ) / 2 be the harmonic, geometric, logarithmic, identric, arithmetic and root-square means of two positive numbers a and b with ab, respectively. Then it is well known that the inequalities

H ( a , b ) = M 1 ( a , b ) < G ( a , b ) = M 0 ( a , b ) < L ( a , b ) < AG ( a , b ) < I ( a , b ) < A ( a , b ) = M 1 ( a , b ) < M ( a , b ) < T ( a , b ) < S ( a , b ) = M 2 ( a , b )

hold for all a,b>0 with ab.

For r>0 the r th generalized logarithmic mean L r (a,b) of two positive numbers a and b is defined by

L r (a,b)= L 1 / r ( a r , b r ) ={ [ b r a r r ( log b log a ) ] 1 / r , a b , a , a = b .
(1.4)

It is not difficult to verify that L r (a,b) is continuous and strictly increasing with respect to r(0,+) for fixed a,b>0 with ab.

In [2, 3] Seiffert proved that the double inequalities

L(a,b)<P(a,b)<I(a,b)

and

A(a,b)<T(a,b)<S(a,b)

hold for all a,b>0 with ab.

The following bounds for the first Seiffert mean P(a,b) in terms of power mean were presented by Jagers in [10]:

M 1 / 2 <P(a,b)< M 2 / 3 (a,b)

for all a,b>0 with ab.

Hästö [11, 12] proved that the function xT(1,x)/ M p (1,x) is increasing on (0,+) if p1 and found the sharp lower power mean bound for the Seiffert mean P(a,b) as follows:

P(a,b)> M log 2 / log π (a,b)

for all a,b>0 with ab.

In [13] the authors presented the following best possible Lehmer mean bounds for the Seiffert means P(a,b) and T(a,b):

L ¯ 1 / 6 (a,b)<P(a,b)< L ¯ 0 (a,b)and L ¯ 0 (a,b)<T(a,b)< L ¯ 1 / 3 (a,b)

for all a,b>0 with ab. Here, L ¯ p (a,b)=( a p + 1 + b p + 1 )/( a p + b p ) is the Lehmer mean of a and b.

In [14, 15] the authors proved that the inequalities

and

α 3 T(a,b)+(1 α 3 )G(a,b)<A(a,b)< β 3 T(a,b)+(1 β 3 )G(a,b)

hold for all a,b>0 with ab if and only if α 1 (4π)/[( 2 1)π], β 1 2/3, α 2 2/3, β 2 42logπ/log2, α 3 3/5 and β 3 π/4.

For all a,b>0 with ab, the following inequalities can be found in [16, 17]:

L(a,b)= L 1 (a,b)<AG(a,b)< L 3 / 2 (a,b)< M 1 / 2 (a,b).

Neuman and Sándor [1] established that

P(a,b)<A(a,b)<M(a,b)<T(a,b)

and

π 2 P(a,b)>A(a,b)>arcsinh(1)M(a,b)> π 4 T(a,b)

for all a,b>0 with ab. In particular, the Ky Fan inequalities

G ( a , b ) G ( a , b ) < L ( a , b ) L ( a , b ) < P ( a , b ) P ( a , b ) < A ( a , b ) A ( a , b ) < M ( a , b ) M ( a , b ) < T ( a , b ) T ( a , b )

hold for all 0<a,b1/2 with ab, a =1a and b =1b.

It is the aim of this paper to find the best possible generalized logarithmic mean bounds for the Neuman-Sándor and Seiffert means.

2 Lemmas

In order to establish our main results, we need three lemmas, which we present in this section.

Lemma 2.1 The inequality

( x 4 1 4 log x ) 1 / 2 < x 2 1 4 arctan x π
(2.1)

holds for all x>1.

Proof Let

f(x)= x 4 1 4 log x ( x 2 1 4 arctan x π ) 2 .
(2.2)

Then f(x) can be rewritten as

f(x)= ( x 4 1 ) f 1 ( x ) 4 ( 4 arctan x π ) 2 log x ,
(2.3)

where

f 1 (x)= ( 4 arctan x π ) 2 4 ( x 2 1 ) log x x 2 + 1 .

Simple computations lead to

(2.4)
(2.5)

where

(2.6)
(2.7)

where

(2.8)
(2.9)
(2.10)

and

f 3 (x)= 8 x 3 (x+1)(x1) ( 3 x 4 + 2 x 2 + 3 ) <0
(2.11)

for x>1.

Inequality (2.11) implies that f 3 (x) is strictly decreasing in [1,+), then equation (2.10) leads to the conclusion that f 3 (x) is strictly decreasing in [1,+).

From equations (2.4)-(2.9) and the monotonicity of f 3 (x), we clearly see that

f 1 (x)<0
(2.12)

for x>1.

Therefore, inequality (2.1) follows from equations (2.2) and (2.3) together with inequality (2.12). □

Lemma 2.2 The inequality

( x 5 1 5 log x ) 1 / 5 < x 1 2 arctan x 1 x + 1
(2.13)

holds for all x>1.

Proof Let

g(x)= 1 5 log ( x 5 1 5 log x ) log ( x 1 2 arctan x 1 x + 1 ) .
(2.14)

Then simple computations lead to

(2.15)
(2.16)

where

(2.17)
(2.18)

where

(2.19)
(2.20)
(2.21)
(2.22)
(2.23)

where

(2.24)
(2.25)

Let

g 4 ( x ) = 10 ( 3 , 780 x 4 3 , 360 x 3 + 1 , 170 x 2 + 2 , 900 x + 950 120 x 1 6 x 2 20 x 3 30 x 4 + 120 x 5 30 x 6 360 x 7 630 x 8 ) log x + 2 , 055 x 4 29 , 720 x 3 1 , 845 x 2 + 18 , 060 x + 8 , 510 2 , 500 x 1 1 , 040 x 2 + 190 x 4 2 , 260 x 5 + 445 x 6 + 5 , 220 x 7 + 1 , 845 x 8 .

Then

(2.26)
(2.27)

where

(2.28)

for x>1.

Equation (2.27) and inequality (2.28) lead to the conclusion that g 4 (x) is strictly decreasing in [1,+). Then equation (2.26) implies that

g 4 (x)<0
(2.29)

for x>1.

Inequalities (2.25) and (2.29) imply that g 3 (x)<0. Then equation (2.24) shows that

g 3 (x)<0
(2.30)

for x>1.

From equations (2.17)-(2.23) and inequality (2.30), we clearly see that

g 1 (x)<0
(2.31)

for x>1.

Therefore, inequality (2.13) follows easily from equations (2.14)-(2.16) and inequality (2.31). □

Lemma 2.3 The inequality

arcsinh 4 ( x 1 x + 1 ) ( x 1 ) 4 log x 4 ( x 4 1 ) <0
(2.32)

holds for all x>1.

Proof Let

h(x)=log [ arcsinh 4 ( x 1 x + 1 ) ] log [ ( x 1 ) 4 log x 4 ( x 4 1 ) ] .
(2.33)

Then simple computations lead to

(2.34)
(2.35)

where

(2.36)
(2.37)

where

(2.38)
(2.39)

where

(2.40)
(2.41)
(2.42)
(2.43)

where

(2.44)
(2.45)
(2.46)

where

h 6 ( x ) = 8 ( 105 x 10 + 21 x 9 45 x 8 + 20 x 7 + 3 x 6 36 x 5 + 3 x 4 + 20 x 3 45 x 2 + 21 x + 105 ) log x ( x 2 1 ) ( 463 x 8 + 346 x 7 14 x 6 + 298 x 5 + 6 x 4 + 298 x 3 14 x 2 + 346 x + 463 ) < 0
(2.47)

for all x>1.

Equations (2.45) and (2.46) together with inequality (2.47) imply that h 5 (x) is strictly decreasing in [1,+). Then equation (2.44) leads to

h 5 (x)<0
(2.48)

for all x>1.

From equations (2.36)-(2.43) and inequality (2.48), we clearly see that

h 1 (x)<0
(2.49)

for all x>1.

Therefore, inequality (2.32) follows from equations (2.33)-(2.35) and inequality (2.49). □

3 Main results

Theorem 3.1 The inequality

P(a,b)> L 2 (a,b)

holds for all a,b>0 with ab, and L 2 (a,b) is the best possible lower generalized logarithmic mean bound for the first Seiffert mean P(a,b).

Proof From (1.2) and (1.4), we clearly see that both P(a,b) and L r (a,b) are symmetric and homogenous of degree one. Without loss of generality, we assume that b=1 and a= x 2 >1. Then (1.2) and (1.4) lead to

L 2 ( x 2 , 1 ) P ( x 2 , 1 ) = ( x 4 1 4 log x ) 1 / 2 x 2 1 4 arctan x π .
(3.1)

Therefore, P( x 2 ,1)> L 2 ( x 2 ,1) follows from Lemma 2.1 and equation (3.1).

Next, we prove that L 2 (a,b) is the best possible lower generalized logarithmic mean bound for the first Seiffert mean P(a,b).

For any ϵ>0 and x>0, from (1.2) and (1.4), one has

L 2 + ϵ (1+x,1)P(1+x,1)= [ ( 1 + x ) 2 + ϵ 1 ( 2 + ϵ ) log ( 1 + x ) ] 1 / ( 2 + ϵ ) x 4 arctan 1 + x π .
(3.2)

Letting x0 and making use of Taylor expansion, we get

(3.3)

Equations (3.2) and (3.3) imply that for any ϵ>0, there exists δ 1 = δ 1 (ϵ)>0 such that L 2 + ϵ (1+x,1)>P(1+x,1) for x(0, δ 1 ). □

Remark 3.1 It follows from (1.2) and (1.4) that

lim x + P ( x , 1 ) L λ ( x , 1 ) = lim x + λ 1 / λ ( 1 1 / x ) log 1 / λ x π ( 1 x λ ) 1 / λ =+
(3.4)

for all λ>0.

Equation (3.4) implies that λ>0 such that L λ (a,b)>P(a,b) for all a,b>0 does not exist.

Theorem 3.2 The inequality

T(a,b)> L 5 (a,b)

holds for all a,b>0 with ab, and L 5 (a,b) is the best possible lower generalized logarithmic mean bound for the second Seiffert mean T(a,b).

Proof Without loss of generality, we assume that b=1 and a=x>1. Then (1.3) and (1.4) lead to

L 5 (x,1)T(x,1)= ( x 5 1 5 log x ) 1 / 5 x 1 2 arctan x 1 x + 1 .
(3.5)

Therefore, T(x,1)> L 5 (x,1) follows from Lemma 2.2 and equation (3.5).

Next, we prove that L 5 (a,b) is the best possible lower generalized logarithmic mean bound for the second Seiffert mean T(a,b).

For any ϵ>0 and x>0, from (1.3) and (1.4), one has

L 5 + ϵ (1+x,1)T(1+x,1)= [ ( 1 + x ) 5 + ϵ 1 ( 5 + ϵ ) log ( 1 + x ) ] 1 / ( 5 + ϵ ) x 2 arctan x 2 + x .
(3.6)

Letting x0 and making use of Taylor expansion, we have

(3.7)

Equations (3.6) and (3.7) imply that for any ϵ>0, there exists δ 2 = δ 2 (ϵ)>0 such that L 5 + ϵ (1+x,1)>T(1+x,1) for x(0, δ 2 ). □

Remark 3.2 It follows from (1.3) and (1.4) that

lim x + T ( x , 1 ) L μ ( x , 1 ) = lim x + 2 μ 1 / μ ( 1 1 / x ) log 1 / μ x π ( 1 x μ ) 1 / μ =+
(3.8)

for all μ>0.

Equation (3.8) implies that μ>0 such that L μ (a,b)>T(a,b) for all a,b>0 does not exist.

Theorem 3.3 The inequality

M(a,b)> L 4 (a,b)

holds for all a,b>0 with ab, and L 4 (a,b) is the best possible lower generalized logarithmic mean bound for the Neuman-Sándor mean M(a,b).

Proof Without loss of generality, we assume that b=1 and a=x>1. Then (1.1) and (1.4) lead to

(3.9)

Therefore, M(x,1)> L 4 (x,1) follows from Lemma 2.3 and equation (3.9).

Next, we prove that L 4 (a,b) is the best possible lower generalized logarithmic mean bound for the Neuman-Sándor mean M(a,b).

For any ϵ>0 and x>0, from (1.1) and (1.4), one has

L 4 + ϵ (1+x,1)T(1+x,1)= [ ( 1 + x ) 4 + ϵ 1 ( 4 + ϵ ) log ( 1 + x ) ] 1 / ( 4 + ϵ ) x 2 arcsinh ( x 2 + x ) .
(3.10)

Letting x0 and making use of Taylor expansion, we have

(3.11)

Equations (3.10) and (3.11) imply that for any ϵ>0, there exists δ 3 = δ 3 (ϵ)>0 such that L 4 + ϵ (1+x,1)>M(1+x,1) for x(0, δ 3 ). □

Remark 3.3 It follows from (1.1) and (1.4) that

lim x + M ( x , 1 ) L ν ( x , 1 ) = lim x + ν 1 / ν ( 1 1 / x ) log 1 / ν x 2 arcsinh ( 1 ) ( 1 x ν ) 1 / ν =+
(3.12)

for all ν>0.

Equation (3.12) implies that ν>0 such that L ν (a,b)>M(a,b) for all a,b>0 does not exist.

References

  1. Neuman E, Sándor J: On the Schwab-Borchardt mean. Math. Pannon. 2003, 14(2):253–266.

    MATH  MathSciNet  Google Scholar 

  2. Seiffert H-J: Problem 887. Nieuw Arch. Wiskd. 1993, 11: 176.

    Google Scholar 

  3. Seiffert H-J: Aufgabe β 16. Die Wurzel 1995, 29: 221–222.

    Google Scholar 

  4. Chu Y-M, Qiu Y-F, Wang M-K: Sharp power mean bounds for combination of Seiffert and geometric means. Abstr. Appl. Anal. 2010., 2010: Article ID 108920

    Google Scholar 

  5. Chu Y-M, Qiu Y-F, Wang M-K, Wang G-D: The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert’s means. J. Inequal. Appl. 2010., 2010: Article ID 436457

    Google Scholar 

  6. Liu H, Meng X-J: The optimal convex combination bounds for the Seiffert’s mean. J. Inequal. Appl. 2011., 2011: Article ID 686834

    Google Scholar 

  7. Sándor J: On certain inequalities for means III. Arch. Math. 2001, 76(1):30–40. 10.1007/s000130050538

    Article  MathSciNet  Google Scholar 

  8. Seiffert H-J: Ungleichunen für einen bestimmten Mittelwert. Nieuw Arch. Wiskd. 1995, 13(2):195–198.

    MATH  MathSciNet  Google Scholar 

  9. Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities. Reidel, Dordrecht; 1988.

    Book  MATH  Google Scholar 

  10. Jagers AA: Solution of problem 887. Nieuw Arch. Wiskd. 1994, 12: 230–231.

    Google Scholar 

  11. Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. J. Inequal. Pure Appl. Math. 2002., 3(5): Article ID 71

  12. Hästö PA: Optimal inequalities between Seiffert’s mean and power mean. Math. Inequal. Appl. 2004, 7(1):47–53.

    MATH  MathSciNet  Google Scholar 

  13. Wang M-K, Qiu Y-F, Chu Y-M: Sharp bounds for Seiffert means in terms of Lehmer means. J. Math. Inequal. 2010, 4(4):581–586.

    Article  MATH  MathSciNet  Google Scholar 

  14. Chu Y-M, Wang M-K, Gong W-M: Two sharp double inequalities for Seiffert mean. J. Inequal. Appl. 2010., 2010: Article ID 44

    Google Scholar 

  15. Chu Y-M, Zong C, Wang G-D: Optimal convex combination bounds of Seiffert and geometric means for the arithmetic mean. J. Math. Inequal. 2011, 5(3):429–434.

    Article  MATH  MathSciNet  Google Scholar 

  16. Borwein JM, Borwein PB: Inequalities for compound mean iterations with logarithmic asymptotes. J. Math. Anal. Appl. 1993, 177(2):572–582. 10.1006/jmaa.1993.1278

    Article  MATH  MathSciNet  Google Scholar 

  17. Vamanamurthy MK, Vuorinen M: Inequalities for means. J. Math. Anal. Appl. 1994, 183(1):155–166. 10.1006/jmaa.1994.1137

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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Correspondence to Yu-Ming Chu.

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Authors’ contributions

YMC provided the main idea and carried out the proof of Remarks 3.1-3.3 in this article. BYL carried out the proof of Lemma 2.1 and Theorem 3.1 in this article. WMG carried out the proof of Lemma 2.2 and Theorem 3.2 in this article. YQS carried out the proof of Lemma 2.3 and Theorem 3.3 in this article. All authors read and approved the final manuscript.

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Chu, YM., Long, BY., Gong, WM. et al. Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means. J Inequal Appl 2013, 10 (2013). https://doi.org/10.1186/1029-242X-2013-10

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Keywords

  • generalized logarithmic mean
  • Neuman-Sándor mean
  • first Seiffert mean
  • second Seiffert mean