Open Access

Optimal bounds for Neuman means in terms of geometric, arithmetic and quadratic means

Journal of Inequalities and Applications20142014:175

https://doi.org/10.1186/1029-242X-2014-175

Received: 16 January 2014

Accepted: 16 April 2014

Published: 12 May 2014

Abstract

In this paper, we present sharp bounds for the two Neuman means S H A and S C A derived from the Schwab-Borchardt mean in terms of convex combinations of either the weighted arithmetic and geometric means or the weighted arithmetic and quadratic means, and the mean generated either by the geometric or by the quadratic mean.

MSC:26E60.

Keywords

Schwab-Borchardt meanNeuman meangeometric meanarithmetic meanquadratic mean

1 Introduction

Let a , b > 0 with a b , then the Schwab-Borchardt mean SB ( a , b ) is defined by
SB ( a , b ) = { b 2 a 2 cos 1 ( a / b ) , a < b , a 2 b 2 cosh 1 ( a / b ) , a > b ,
(1.1)

where cos 1 ( x ) and cosh 1 ( x ) = log ( x + x 2 1 ) are the inverse cosine and inverse hyperbolic cosine functions, respectively.

It is well known that SB ( a , b ) is strictly increasing in both a and b, nonsymmetric and homogeneous of degree 1 with respect to a and b. Many symmetric bivariate means are special cases of the Schwab-Borchardt mean, for example,
P ( a , b ) = a b 2 sin 1 [ ( a b ) / ( a + b ) ] = SB ( G , A ) is the first Seiffert mean , T ( a , b ) = a b 2 tan 1 [ ( a b ) / ( a + b ) ] = SB ( A , Q ) is the second Seiffert mean , M ( a , b ) = a b 2 sinh 1 [ ( a b ) / ( a + b ) ] = SB ( Q , A ) is the Neuman-Sándor mean , L ( a , b ) = a b 2 tanh 1 [ ( a b ) / ( a + b ) ] = SB ( A , G ) is the logarithmic mean ,

where G ( a , b ) = a b , A ( a , b ) = ( a + b ) / 2 and Q ( a , b ) = ( a 2 + b 2 ) / 2 denote the classical geometric mean, arithmetic mean and quadratic mean of a and b, respectively. The Schwab-Borchardt mean SB ( a , b ) was investigated in [1, 2].

Let H ( a , b ) = 2 a b / ( a + b ) , C ( a , b ) = ( a 2 + b 2 ) / ( a + b ) be the harmonic and contraharmonic means of two positive numbers a and b, respectively. Then it is well known that
H ( a , b ) < G ( a , b ) < L ( a , b ) < P ( a , b ) < A ( a , b ) < M ( a , b ) < T ( a , b ) < Q ( a , b ) < C ( a , b )
(1.2)

for a , b > 0 with a b .

Recently, the Schwab-Borchardt mean and its special cases have been the subject of intensive research. Neuman and Sándor [3, 4] proved that the inequalities
P ( a , b ) > 2 π A ( a , b ) , A ( a , b ) log ( 1 + 2 ) > M ( a , b ) > π 4 log ( 1 + 2 ) T ( a , b ) , T ( A ( a , b ) , G ( a , b ) ) < P ( a , b ) , T ( a , b ) > T ( A ( a , b ) , Q ( a , b ) ) , L ( a , b ) < L ( A ( a , b ) , G ( a , b ) ) , M ( a , b ) < L ( A ( a , b ) , Q ( a , b ) ) , L ( a , b ) > H ( P ( a , b ) , G ( a , b ) ) , P ( a , b ) > H ( L ( a , b ) , A ( a , b ) ) , M ( a , b ) > H ( T ( a , b ) , A ( a , b ) ) , T ( a , b ) > H ( M ( a , b ) , Q ( a , b ) ) , G ( a , b ) P ( a , b ) < L 2 ( a , b ) < G 2 ( a , b ) + P 2 ( a , b ) 2 , L ( a , b ) A ( a , b ) < P 2 ( a , b ) < L 2 ( a , b ) + A 2 ( a , b ) 2 , A ( a , b ) T ( a , b ) < M 2 ( a , b ) < A 2 ( a , b ) + T 2 ( a , b ) 2 , M ( a , b ) Q ( a , b ) < T 2 ( a , b ) < M 2 ( a , b ) + Q 2 ( a , b ) 2 , Q 1 / 3 ( a , b ) A 2 / 3 ( a , b ) < M ( a , b ) < 1 3 Q ( a , b ) + 2 3 A ( a , b )
hold for all a , b > 0 with a b . In [5], the author proved that the double inequalities
α Q ( a , b ) + ( 1 α ) A ( a , b ) < M ( a , b ) < β Q ( a , b ) + ( 1 β ) A ( a , b )
and
λ C ( a , b ) + ( 1 λ ) A ( a , b ) < M ( a , b ) < μ C ( a , b ) + ( 1 μ ) A ( a , b )
hold for all a , b > 0 with a b if and only if α [ 1 log ( 1 + 2 ) ] / [ ( 2 1 ) log ( 1 + 2 ) ] = 0.3249  , β 1 / 3 , λ [ 1 log ( 1 + 2 ) ] / log ( 1 + 2 ) = 0.1345 and μ 1 / 6 . Chu and Long [6] found that the double inequality
M p ( a , b ) < M ( a , b ) < q I ( a , b )
holds for all a , b > 0 with a b if and only if p log 2 / log [ 2 log ( 1 + 2 ) ] = 1.224 and q e / [ 2 log ( 1 + 2 ) ] = 1.5420  , where M p ( a , b ) = [ ( a p + b p ) / 2 ] 1 / p ( p 0 ) and M 0 ( a , b ) = a b is the p th power mean of a and b. Zhao et al. [7] presented the least values α 1 , α 2 , α 3 and the greatest values β 1 , β 2 , β 3 such that the double inequalities
α 1 H ( a , b ) + ( 1 α 1 ) Q ( a , b ) < M ( a , b ) < β 1 H ( a , b ) + ( 1 β 1 ) Q ( a , b ) , α 2 G ( a , b ) + ( 1 α 2 ) Q ( a , b ) < M ( a , b ) < β 2 G ( a , b ) + ( 1 β 2 ) Q ( a , b )
and
α 3 H ( a , b ) + ( 1 α 3 ) C ( a , b ) < M ( a , b ) < β 3 H ( a , b ) + ( 1 β 3 ) C ( a , b )

hold for all a , b > 0 with a b .

Very recently, the bivariate means S A H , S H A , S C A and S A C derived from the Schwab-Borchardt mean have been defined by Neuman [8, 9] as follows:
S A H = SB ( A , H ) , S H A = SB ( H , A ) , S C A = SB ( C , A ) , S A C = SB ( A , C ) .
(1.3)
We call the means S A H , S H A , S C A and S A C given in (1.3) the Neuman means. Moreover, let v = ( a b ) / ( a + b ) ( 1 , 1 ) , then the following explicit formulas for S A H , S H A , S A C and S C A have been found by Neuman [8]:
S A H = A tanh ( p ) p , S H A = A sin ( q ) q ,
(1.4)
S C A = A sinh ( r ) r , S A C = A tan ( s ) s ,
(1.5)

where p, q, r and s are defined implicitly as sech ( p ) = 1 v 2 , cos ( q ) = 1 v 2 , cosh ( r ) = 1 + v 2 and sec ( s ) = 1 + v 2 , respectively. Clearly, p ( 0 , ) , q ( 0 , π / 2 ) , r ( 0 , log ( 2 + 3 ) ) and s ( 0 , π / 3 ) .

In [8], Neuman proved that the inequalities
H ( a , b ) < S A H ( a , b ) < L ( a , b ) < S H A ( a , b ) < P ( a , b ) ,
(1.6)
T ( a , b ) < S C A ( a , b ) < Q ( a , b ) < S A C ( a , b ) < C ( a , b )
(1.7)

hold for a , b > 0 with a b .

He et al. [10] found the greatest values α 1 , α 2 [ 0 , 1 / 2 ] , α 3 , α 4 [ 1 / 2 , 1 ] and the least values β 1 , β 2 [ 0 , 1 / 2 ] , β 3 , β 4 [ 1 / 2 , 1 ] such that the double inequalities
H ( α 1 a + ( 1 α 1 ) b , α 1 b + ( 1 α 1 ) a ) < S A H ( a , b ) < H ( β 1 a + ( 1 β 1 ) b , β 1 b + ( 1 β 1 ) a ) , H ( α 2 a + ( 1 α 2 ) b , α 2 b + ( 1 α 2 ) a ) < S H A ( a , b ) < H ( β 2 a + ( 1 β 2 ) b , β 2 b + ( 1 β 2 ) a ) , C ( α 3 a + ( 1 α 3 ) b , α 3 b + ( 1 α 3 ) a ) < S C A ( a , b ) < C ( β 3 a + ( 1 β 3 ) b , β 3 b + ( 1 β 3 ) a )
and
C ( α 4 a + ( 1 α 4 ) b , α 4 b + ( 1 α 4 ) a ) < S A C ( a , b ) < C ( β 4 a + ( 1 β 4 ) b , β 4 b + ( 1 β 4 ) a )

hold for all a , b > 0 with a b .

It follows from (1.2) and (1.6) together with (1.7) that
G ( a , b ) < S H A ( a , b ) < A ( a , b ) < S C A ( a , b ) < Q ( a , b )
(1.8)

for all a , b > 0 with a b .

For fixed a , b > 0 with a b , let x [ 0 , 1 / 2 ] , y [ 1 / 2 , 1 ] ,
f ( x ) = G [ x a + ( 1 x ) b , x b + ( 1 x ) a ] ,
(1.9)
g ( y ) = Q [ y a + ( 1 y ) b , y b + ( 1 y ) a ] .
(1.10)
Then it is not difficult to verify that f ( x ) and g ( y ) are continuous and strictly increasing on [ 0 , 1 / 2 ] and [ 1 / 2 , 1 ] , respectively. Note that
f ( 0 ) = G ( a , b ) < S H A ( a , b ) < A ( a , b ) = f ( 1 / 2 ) ,
(1.11)
g ( 1 / 2 ) = A ( a , b ) < S C A ( a , b ) < Q ( a , b ) = g ( 1 ) .
(1.12)
Motivated by (1.8)-(1.12), in the article we present the best possible parameters α 1 , α 2 , β 1 , β 2 R , α 3 , β 3 [ 0 , 1 / 2 ] and α 4 , β 4 [ 1 / 2 , 1 ] such that the double inequalities
α 1 A ( a , b ) + ( 1 α 1 ) G ( a , b ) < S H A ( a , b ) < β 1 A ( a , b ) + ( 1 β 1 ) G ( a , b ) , α 2 A ( a , b ) + ( 1 α 2 ) Q ( a , b ) < S C A ( a , b ) < β 2 A ( a , b ) + ( 1 β 2 ) Q ( a , b ) , G [ α 3 a + ( 1 α 3 ) b , α 3 b + ( 1 α 3 ) a ] < S H A ( a , b ) < G [ β 3 a + ( 1 β 3 ) b , β 3 b + ( 1 β 3 ) a ] , Q [ α 4 a + ( 1 α 4 ) b , α 4 b + ( 1 α 4 ) a ] < S C A ( a , b ) < Q [ β 4 a + ( 1 β 4 ) b , β 4 b + ( 1 β 4 ) a ]

hold for all a , b > 0 with a b .

Our main results are the following Theorems 1.1-1.4. All numerical computations are carried out using Mathematica software.

Theorem 1.1 The double inequality
α 1 A ( a , b ) + ( 1 α 1 ) G ( a , b ) < S H A ( a , b ) < β 1 A ( a , b ) + ( 1 β 1 ) G ( a , b )

holds for all a , b > 0 with a b if and only if α 1 1 / 3 and β 1 2 / π .

Theorem 1.2 The two-sided inequality
α 2 A ( a , b ) + ( 1 α 2 ) Q ( a , b ) < S C A ( a , b ) < β 2 A ( a , b ) + ( 1 β 2 ) Q ( a , b )

holds true for all a , b > 0 with a b if and only if α 2 1 / 3 and β 2 [ 2 log ( 2 + 3 ) 3 ] / [ ( 2 1 ) log ( 2 + 3 ) ] = 0.2390  .

Theorem 1.3 Let α 3 , β 3 [ 0 , 1 / 2 ] , then the double inequality
G [ α 3 a + ( 1 α 3 ) b , α 3 b + ( 1 α 3 ) a ] < S H A ( a , b ) < G [ β 3 a + ( 1 β 3 ) b , β 3 b + ( 1 β 3 ) a ]

holds for all a , b > 0 with a b if and only if α 3 1 / 2 6 / 6 = 0.09175 and β 3 1 / 2 π 2 4 / ( 2 π ) = 0.1144  .

Theorem 1.4 Let α 4 , β 4 [ 1 / 2 , 1 ] , then the two-sided inequality
Q [ α 4 a + ( 1 α 4 ) b , α 4 b + ( 1 α 4 ) a ] < S C A ( a , b ) < Q [ β 4 a + ( 1 β 4 ) b , β 4 b + ( 1 β 4 ) a ]

holds true for all a , b > 0 with a b if and only if α 4 1 / 2 + 6 / 6 = 0.9082 and β 4 1 / 2 + 3 / [ log ( 2 + 3 ) ] 2 1 / 2 = 0.9271  .

2 Two lemmas

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

Lemma 2.1 Let p R and
f ( x ) = ( 1 p ) x 3 + ( 2 p 2 + 5 p 1 ) x 2 + ( 2 p 2 + p 1 ) x + p 1 .
(2.1)
Then the following statements are true:
  1. (1)

    If p = 1 / 3 , then f ( x ) < 0 for all x ( 0 , 1 ) and f ( x ) > 0 for all x ( 1 , 2 ) ;

     
  2. (2)

    If p = 2 / π , then there exists λ 1 ( 0 , 1 ) such that f ( x ) < 0 for x ( 0 , λ 1 ) and f ( x ) > 0 for x ( λ 1 , 1 ) ;

     
  3. (3)

    If p = [ 2 log ( 2 + 3 ) 3 ] / [ ( 2 1 ) log ( 2 + 3 ) ] , then there exists λ 2 ( 1 , 2 ) such that f ( x ) < 0 for x ( 1 , λ 2 ) and f ( x ) > 0 for x ( λ 2 , 2 ) .

     
Proof For part (1), if p = 1 / 3 , then (2.1) becomes
f ( x ) = 2 9 ( x 1 ) ( 3 x 2 + 5 x + 3 ) .
(2.2)

Therefore, part (1) follows easily from (2.2).

For part (2), if p = 2 / π , then simple computations lead to
2 p 2 + 5 p 1 = π 2 + 10 π 8 π 2 > 0 ,
(2.3)
2 p 2 + p 1 = π 2 + 2 π + 8 π 2 > 0 ,
(2.4)
f ( 0 ) = π 2 π < 0 ,
(2.5)
f ( 1 ) = 2 ( 6 π ) π > 0 ,
(2.6)
f ( x ) = 3 ( 1 p ) x 2 + 2 ( 2 p 2 + 5 p 1 ) x + ( 2 p 2 + p 1 ) .
(2.7)

It follows from (2.3) and (2.4) together with (2.7) that f ( x ) is strictly increasing on ( 0 , 1 ) . Therefore, part (2) follows from (2.5) and (2.6) together with the monotonicity of f ( x ) .

For part (3), if p = [ 2 log ( 2 + 3 ) 3 ] / [ ( 2 1 ) log ( 2 + 3 ) ] = 0.2390  , then numerical computations lead to
2 p 2 + 5 p 1 = 0.0810 > 0 ,
(2.8)
f ( 1 ) = 0.5656 < 0 ,
(2.9)
f ( 2 ) = 0.6388 > 0 .
(2.10)
It follows from (2.7) and (2.8) that
f ( x ) > 3 ( 1 p ) + 2 ( 2 p 2 + 5 p 1 ) + ( 2 p 2 + p 1 ) = 2 p ( 4 p ) > 0
(2.11)

for x ( 1 , 2 ) .

Therefore, part (3) follows easily from (2.9)-(2.11). □

Lemma 2.2 Let p R and
g ( x ) = ( 2 p 1 ) 4 x 3 + ( 256 p 6 + 768 p 5 1 , 008 p 4 + 736 p 3 296 p 2 + 56 p 3 ) x 2 + ( 512 p 6 1 , 536 p 5 + 1 , 776 p 4 992 p 3 + 248 p 2 8 p 1 ) x + ( 256 p 6 + 768 p 5 784 p 4 + 288 p 3 24 p 2 + 8 p 1 ) .
(2.12)
Then the following statements are true:
  1. (1)

    If p = 1 / 2 6 / 6 , then g ( x ) < 0 for all x ( 0 , 1 ) ;

     
  2. (2)

    If p = 1 / 2 + 6 / 6 , then g ( x ) > 0 for all x ( 1 , 2 ) ;

     
  3. (3)

    If p = 1 / 2 π 2 4 / ( 2 π ) , then there exists λ 3 ( 0 , 1 ) such that g ( x ) < 0 for x ( 0 , λ 3 ) and g ( x ) > 0 for x ( λ 3 , 1 ) ;

     
  4. (4)

    If p = 1 / 2 + 3 / [ log ( 2 + 3 ) ] 2 1 / 2 , then there exists λ 4 ( 1 , 2 ) such that g ( x ) < 0 for x ( 1 , λ 4 ) and g ( x ) > 0 for x ( λ 4 , 2 ) .

     
Proof For parts (1) and (2), if p = 1 / 2 6 / 6 or p = 1 / 2 + 6 / 6 , then (2.12) becomes
g ( x ) = 4 27 ( x 1 ) ( 3 x 2 + 4 x + 2 ) .
(2.13)

Therefore, parts (1) and (2) follow from (2.13).

For part (3), if p = 1 / 2 π 2 4 / ( 2 π ) , then numerical computations show that
256 p 6 + 768 p 5 1 , 008 p 4 + 736 p 3 296 p 2 + 56 p 3 = 3 π 6 + 56 π 4 240 π 2 + 256 π 6 > 0 ,
(2.14)
512 p 6 1 , 536 p 5 + 1 , 776 p 4 992 p 3 + 248 p 2 8 p 1 = π 6 8 π 4 + 240 π 2 512 π 6 > 0 ,
(2.15)
g ( 0 ) = π 6 + 8 π 4 16 π 2 + 256 π 6 < 0 ,
(2.16)
g ( 1 ) = 4 ( 12 π 2 ) π 2 > 0 ,
(2.17)
g ( x ) = 3 ( 2 p 1 ) 4 x 2 + 2 ( 256 p 6 + 768 p 5 1 , 008 p 4 + 736 p 3 296 p 2 + 56 p 3 ) x g ( x ) = + ( 512 p 6 1 , 536 p 5 + 1 , 776 p 4 992 p 3 + 248 p 2 8 p 1 ) .
(2.18)

From (2.14), (2.15) and (2.18) we clearly see that g ( x ) is strictly increasing on ( 0 , 1 ) . Therefore, part (3) follows from (2.16) and (2.17) together with the monotonicity of g ( x ) .

For part (4), if p = 1 / 2 + 3 / [ log ( 2 + 3 ) ] 2 1 / 2 , then numerical computations lead to
256 p 6 + 768 p 5 1 , 008 p 4 + 736 p 3 296 p 2 + 56 p 3 = 0.2329 < 0 ,
(2.19)
512 p 6 1 , 536 p 5 + 1 , 776 p 4 992 p 3 + 248 p 2 8 p 1 = 0.6027 < 0 ,
(2.20)
g ( 1 ) = 0.7567 < 0 ,
(2.21)
g ( 2 ) = 1.6692 > 0 ,
(2.22)
48 p 4 + 96 p 3 68 p 2 + 20 p 1 = 0.1322 > 0 .
(2.23)
It follows from (2.18), (2.19), (2.20) and (2.23) that
g ( x ) > 3 ( 2 p 1 ) 4 x 2 + 2 ( 256 p 6 + 768 p 5 1 , 008 p 4 + 736 p 3 296 p 2 + 56 p 3 ) x 2 + ( 512 p 6 1 , 536 p 5 + 1 , 776 p 4 992 p 3 + 248 p 2 8 p 1 ) x 2 = 4 ( 48 p 4 + 96 p 3 68 p 2 + 20 p 1 ) x 2 > 0
(2.24)

for x ( 1 , 2 ) .

Therefore, part (4) follows from (2.21) and (2.22) together with (2.24). □

3 Proofs of Theorems 1.1-1.4

Proof of Theorem 1.1 Without loss of generality, we assume that a > b . Let v = ( a b ) / ( a + b ) , λ = v 2 v 2 , x = 1 λ 2 4 and p { 1 / 3 , 2 / π } . Then v , λ , x ( 0 , 1 ) and (1.4) leads to
S H A ( a , b ) G ( a , b ) A ( a , b ) G ( a , b ) = λ ( 1 λ 2 ) 1 / 4 sin 1 ( λ ) [ 1 ( 1 λ 2 ) 1 / 4 ] sin 1 ( λ ) ,
(3.1)
S H A ( a , b ) [ p A ( a , b ) + ( 1 p ) G ( a , b ) ] = A ( a , b ) [ λ sin 1 ( λ ) ( 1 p ) ( 1 λ 2 ) 1 / 4 p ] = A ( a , b ) [ p + ( 1 p ) ( 1 λ 2 ) 1 / 4 ] sin 1 ( λ ) F ( x ) ,
(3.2)
where
F ( x ) = 1 x 4 ( 1 p ) x + p sin 1 ( 1 x 4 ) , F ( 0 ) = 1 p π 2 ,
(3.3)
F ( 1 ) = 0
(3.4)
and
F ( x ) = 1 x 1 x 4 [ ( 1 p ) x + p ] 2 f ( x ) ,
(3.5)

where f ( x ) is defined as in Lemma 2.1.

We divide the proof into two cases.

Case 1: p = 1 / 3 . Then from Lemma 2.1(1) and (3.5) we clearly see that F ( x ) is strictly decreasing on ( 0 , 1 ) . Therefore,
S H A ( a , b ) > 1 3 A ( a , b ) + 2 3 G ( a , b )
(3.6)

for all a , b > 0 with a b follows from (3.2) and (3.4) together with the monotonicity of F ( x ) .

Case 2: p = 2 / π . Then from (3.3), (3.5) and Lemma 2.1(2) we know that
F ( 0 ) = 0
(3.7)
and there exists λ 1 ( 0 , 1 ) such that F ( x ) is strictly decreasing on ( 0 , λ 1 ] and strictly increasing on [ λ 1 , 1 ) . Therefore,
S H A ( a , b ) < 2 π A ( a , b ) + ( 1 2 π ) G ( a , b )
(3.8)

for all a , b > 0 with a b follows from (3.2) and (3.4) together with (3.7) and the piecewise monotonicity of F ( x ) .

Note that
lim λ 0 + λ ( 1 λ 2 ) 1 / 4 sin 1 ( λ ) [ 1 ( 1 λ 2 ) 1 / 4 ] sin 1 ( λ ) = 1 3
(3.9)
and
lim λ 1 λ ( 1 λ 2 ) 1 / 4 sin 1 ( λ ) [ 1 ( 1 λ 2 ) 1 / 4 ] sin 1 ( λ ) = 2 π .
(3.10)

Therefore, Theorem 1.1 follows from (3.6) and (3.8) together with the following statements.

  • If α > 1 / 3 , then equations (3.1) and (3.9) imply that there exists small enough δ > 0 such that S H A ( a , b ) < α A ( a , b ) + ( 1 α ) G ( a , b ) for all a > b > 0 with b / a ( 1 δ , 1 ) .

  • If β < 2 / π , then equations (3.1) and (3.10) imply that there exists large enough M > 1 such that S H A ( a , b ) > β A ( a , b ) + ( 1 β ) G ( a , b ) for all a > b > 0 with a / b ( M , + ) .

 □

Proof of Theorem 1.2 Without loss of generality, we assume that a > b . Let v = ( a b ) / ( a + b ) , μ = v 2 + v 2 , x = 1 + μ 2 4 and p { [ 2 log ( 2 + 3 ) 3 ] / [ ( 2 1 ) log ( 2 + 3 ) ] , 1 / 3 } . Then v ( 0 , 1 ) , μ ( 0 , 3 ) , x ( 1 , 2 ) and (1.5) leads to
S C A ( a , b ) Q ( a , b ) A ( a , b ) Q ( a , b ) = μ ( 1 + μ 2 ) 1 / 4 sinh 1 ( μ ) [ 1 ( 1 + μ 2 ) 1 / 4 ] sinh 1 ( μ ) ,
(3.11)
S C A ( a , b ) [ p A ( a , b ) + ( 1 p ) Q ( a , b ) ] = A ( a , b ) [ μ sinh 1 ( μ ) ( 1 p ) ( 1 + μ 2 ) 1 / 4 p ] = A ( a , b ) [ ( 1 p ) ( 1 + μ 2 ) 1 / 4 + p ] sinh 1 ( μ ) G ( x ) ,
(3.12)
where
G ( x ) = x 4 1 ( 1 p ) x + p sinh 1 ( x 4 1 ) , G ( 1 ) = 0 ,
(3.13)
G ( 2 ) = 3 2 ( 2 1 ) p log ( 2 + 3 ) ,
(3.14)
G ( x ) = x 1 x 4 1 [ ( 1 p ) x + p ] 2 f ( x ) ,
(3.15)

where f ( x ) is defined as in Lemma 2.1.

We divide the proof into two cases.

Case 1: p = [ 2 log ( 2 + 3 ) 3 ] / [ ( 2 1 ) log ( 2 + 3 ) ] = 0.2390  . Then from (3.14) and (3.15) together with Lemma 2.1(3) we clearly see that there exists λ 2 ( 1 , 2 ) such that G ( x ) is strictly decreasing on ( 1 , λ 2 ] and strictly increasing on [ λ 2 , 2 ) , and
G ( 2 ) = 0 .
(3.16)
Therefore,
S C A ( a , b ) < 2 log ( 2 + 3 ) 3 ( 2 1 ) log ( 2 + 3 ) A ( a , b ) + 3 log ( 2 + 3 ) ( 2 1 ) log ( 2 + 3 ) Q ( a , b )
(3.17)

for all a , b > 0 with a b follows easily from (3.12) and (3.13) together with (3.16) and the piecewise monotonicity of G ( x ) .

Case 2: p = 1 / 3 . Then Lemma 2.1(1) and (3.15) lead to the conclusion that G ( x ) is strictly increasing on ( 1 , 2 ) . Therefore,
S C A ( a , b ) > 1 3 A ( a , b ) + 2 3 Q ( a , b )
(3.18)

for all a , b > 0 with a b follows from (3.12) and (3.13) together with the monotonicity of G ( x ) .

Note that
lim μ 0 + μ ( 1 + μ 2 ) 1 / 4 sinh 1 ( μ ) [ 1 ( 1 + μ 2 ) 1 / 4 ] sinh 1 ( μ ) = 1 3
(3.19)
and
lim μ 3 μ ( 1 + μ 2 ) 1 / 4 sinh 1 ( μ ) [ 1 ( 1 + μ 2 ) 1 / 4 ] sinh 1 ( μ ) = 2 log ( 2 + 3 ) 3 ( 2 1 ) log ( 2 + 3 ) .
(3.20)

Therefore, Theorem 1.2 follows from (3.11) and (3.17)-(3.20). □

Proof of Theorem 1.3 Without loss of generality, we assume that a > b . Let v = ( a b ) / ( a + b ) , λ = v 2 v 2 , x = 1 λ 2 and p [ 0 , 1 / 2 ] . Then v , λ , x ( 0 , 1 ) and (1.4) leads to
G [ p a + ( 1 p ) b , p b + ( 1 p ) a ] S H A ( a , b ) = A ( a , b ) [ 1 ( 1 2 p ) 2 ( 1 1 λ 2 ) λ sin 1 ( λ ) ] = A ( a , b ) 1 ( 1 2 p ) 2 ( 1 1 λ 2 ) sin 1 ( λ ) H ( x ) ,
(3.21)
where
H ( x ) = sin 1 ( 1 x 2 ) 1 x 2 ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 , H ( 1 ) = 0 ,
(3.22)
H ( 0 ) = π 2 1 1 ( 1 2 p ) 2
(3.23)
and
H ( x ) = h ( x ) 2 1 x 2 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 ,
(3.24)
where
h ( x ) = ( 1 2 p ) 2 x 2 + 2 [ 1 ( 1 2 p ) 2 ] x + ( 1 2 p ) 2 2 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 = ( x 1 ) g ( x ) ( 1 2 p ) 2 x 2 + 2 [ 1 ( 1 2 p ) 2 ] x + ( 1 2 p ) 2 + 2 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 ,
(3.25)

where g ( x ) is defined as in Lemma 2.2.

We divide the proof into four cases.

Case 1: p = 1 / 2 6 / 6 . Then Lemma 2.2(1) and (3.24) together with (3.25) lead to the conclusion that H ( x ) is strictly increasing on ( 0 , 1 ) . Therefore,
S H A ( a , b ) > G [ ( 1 2 6 6 ) a + ( 1 2 + 6 6 ) b , ( 1 2 6 6 ) b + ( 1 2 + 6 6 ) a ]

for all a , b > 0 with a b follows easily from (3.21) and (3.22) together with the monotonicity of H ( x ) .

Case 2: 1 / 2 6 / 6 < p 1 / 2 . Let q = ( 1 2 p ) 2 and λ 0 + , then 0 q < 2 / 3 and power series expansions lead to
1 ( 1 2 p ) 2 ( 1 1 λ 2 ) λ sin 1 λ = 1 q ( 1 1 λ 2 ) sin 1 λ λ sin 1 λ = 1 sin 1 λ [ ( 1 6 q 4 ) λ 3 + o ( λ 3 ) ] .
(3.26)

Equations (3.21) and (3.26) imply that there exists small enough δ 1 > 0 such that S H A ( a , b ) < G [ p a + ( 1 p ) b , p b + ( 1 p ) a ] for all a , b > 0 with b / a ( 1 δ 1 , 1 ) .

Case 3: p = 1 / 2 π 2 4 / ( 2 π ) . Then from Lemma 2.2(3) and (3.23)-(3.25) we clearly see that there exists λ 3 ( 0 , 1 ) such that H ( x ) is strictly increasing on ( 0 , λ 3 ] and strictly decreasing on [ λ 3 , 1 ) , and
H ( 0 ) = 0 .
(3.27)
Therefore,
S H A ( a , b ) < G [ ( 1 2 π 2 4 2 π ) a + ( 1 2 + π 2 4 2 π ) b , ( 1 2 π 2 4 2 π ) b + ( 1 2 + π 2 4 2 π ) a ]

for all a , b > 0 with a b follows easily from (3.21) and (3.22) together with (3.27) and the piecewise monotonicity of H ( x ) .

Case 4: 0 p < 1 / 2 π 2 4 / ( 2 π ) . Then
lim λ 1 [ 1 ( 1 2 p ) 2 ( 1 1 λ 2 ) λ sin 1 ( λ ) ] = 1 ( 1 2 p ) 2 2 π < 0 .
(3.28)

Equation (3.21) and inequality (3.28) imply that there exists large enough M 1 > 1 such that S H A ( a , b ) > G [ p a + ( 1 p ) b , p b + ( 1 p ) a ] for all a , b > 0 with a / b ( M 1 , + ) . □

Proof of Theorem 1.4 Without loss of generality, we assume that a > b . Let v = ( a b ) / ( a + b ) , μ = v 2 + v 2 , x = 1 + μ 2 and p [ 1 / 2 , 1 ] . Then v ( 0 , 1 ) , μ ( 0 , 3 ) , x ( 1 , 2 ) and (1.5) leads to
Q [ p a + ( 1 p ) b , p b + ( 1 p ) a ] S C A ( a , b ) = A ( a , b ) [ 1 + ( 1 2 p ) 2 ( 1 + μ 2 1 ) μ sinh 1 ( μ ) ] = A ( a , b ) 1 + ( 1 2 p ) 2 ( 1 + μ 2 1 ) sinh 1 ( μ ) J ( x ) ,
(3.29)
where
J ( x ) = sinh 1 ( x 2 1 ) x 2 1 ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 , J ( 1 ) = 0 ,
(3.30)
J ( 2 ) = log ( 2 + 3 ) 3 1 + ( 1 2 p ) 2 ,
(3.31)
J ( x ) = 2 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 [ ( 1 2 p ) 2 x 2 + 2 ( 1 ( 1 2 p ) 2 ) x + ( 1 2 p ) 2 ] 2 x 2 1 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 J ( x ) = 1 2 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 + [ ( 1 2 p ) 2 x 2 + 2 ( 1 ( 1 2 p ) 2 ) x + ( 1 2 p ) 2 ] J ( x ) = × x 1 2 x 2 1 [ ( 1 2 p ) 2 x ( 1 2 p ) 2 + 1 ] 3 / 2 g ( x ) ,
(3.32)

where g ( x ) is defined as in Lemma 2.2.

We divide the proof into four cases.

Case 1: p = 1 / 2 + 6 / 6 . Then Lemma 2.2(2) and (3.32) lead to the conclusion that J ( x ) is strictly increasing on ( 1 , 2 ) . Therefore,
S C A ( a , b ) > Q [ ( 1 2 + 6 6 ) a + ( 1 2 6 6 ) b , ( 1 2 + 6 6 ) b + ( 1 2 6 6 ) a ]

for all a , b > 0 with a b follows easily from (3.29) and (3.30) together with the monotonicity of J ( x ) .

Case 2: 1 / 2 + 6 / 6 < p 1 . Let q = ( 1 2 p ) 2 and μ 0 + , then 1 q > 2 / 3 and power series expansions lead to
1 + ( 1 2 p ) 2 ( 1 + μ 2 1 ) μ sinh 1 ( μ ) = 1 + q ( 1 + μ 2 1 ) sinh 1 ( μ ) μ sinh 1 ( μ ) = 1 sinh 1 ( μ ) [ ( 1 4 q 1 6 ) μ 3 + o ( μ 3 ) ] .
(3.33)

Equations (3.29) and (3.33) imply that there exists small enough δ 2 > 0 such that S C A ( a , b ) < Q [ p a + ( 1 p ) b , p b + ( 1 p ) a ] for all a , b > 0 with b / a ( 1 δ 2 , 1 ) .

Case 3: p = 1 / 2 + 3 / [ log ( 2 + 3 ) ] 2 1 / 2 . Then (3.31) and (3.32) together with Lemma 2.2(4) lead to the conclusion that there exists λ 4 ( 1 , 2 ) such that J ( x ) is strictly increasing on ( 1 , λ 4 ] and strictly decreasing on [ λ 4 , 2 ) , and
J ( 2 ) = 0 .
(3.34)
Therefore,
S C A ( a , b ) < Q [ p a + ( 1 p ) b , p b + ( 1 p ) a ]

for all a , b > 0 with a b follows easily from (3.29) and (3.30) together with (3.34) and the piecewise monotonicity of J ( x ) .

Case 4: 1 / 2 p < 1 / 2 + 3 / [ log ( 2 + 3 ) ] 2 1 / 2 . Then
lim μ 3 [ 1 + ( 1 2 p ) 2 ( 1 + μ 2 1 ) μ sinh 1 ( μ ) ] = 1 + ( 2 p 1 ) 2 3 log ( 2 + 3 ) < 0 .
(3.35)

Equation (3.29) and inequality (3.35) imply that there exists large enough M 2 > 1 such that S C A ( a , b ) > Q [ p a + ( 1 p ) b , p b + ( 1 p ) a ] for all a , b > 0 with a / b ( M 2 , + ) . □

Declarations

Acknowledgements

The authors would like to express their deep gratitude to the referees for giving many valuable suggestions. The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307, the Natural Science Foundation of the Open University of China under Grant Q1601E-Y and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-13Z04.

Authors’ Affiliations

(1)
School of Distance Education, Huzhou Broadcast and TV University
(2)
School of Mathematics and Computation Science, Hunan City University

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© Qian and Chu; licensee Springer. 2014

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