JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/517647 517647 Research Article Sharp Power Mean Bounds for the One-Parameter Harmonic Mean Chu Yu-Ming 1 Wu Li-Min 2 Song Ying-Qing 1 Larson David R. 1 School of Mathematics and Computation Sciences Hunan City University Yiyang 413000 China hncu.net 2 Department of Mathematics Huzhou University Huzhou 313000 China hutc.zj.cn 2015 30 4 2015 2015 03 11 2014 27 04 2015 30 4 2015 2015 Copyright © 2015 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 present the best possible parameters α = α ( r ) and β = β ( r ) such that the double inequality M α ( a , b ) < H r ( a , b ) < M β ( a , b ) holds for all r ( 0 , 1 / 2 ) and a , b > 0 with a b , where M p ( a , b ) = [ ( a p + b p ) / 2 ] 1 / p    ( p 0 ) and M 0 ( a , b ) = a b and H r ( a , b ) = 2 [ r a + ( 1 - r ) b ] [ r b + ( 1 - r ) a ] / ( a + b ) are the power and one-parameter harmonic means of a and b , respectively.

1. Introduction

For p R and a , b > 0 , the p th power mean M p ( a , b ) of a and b is defined by (1) M p a , b = a p + b p 2 1 / p p 0 , M 0 a , b = a b .

It is well known that M p ( a , b ) is strictly increasing with respect to p R for fixed a , b > 0 with a b , symmetric and homogeneous of degree 1. Many classical means are special cases of the power mean: for example, M - 1 ( a , b ) = 2 a b / ( a + b ) = H ( a , b ) is the harmonic mean, M 0 ( a , b ) = a b = G ( a , b ) is the geometric mean, M 1 ( a , b ) = ( a + b ) / 2 = A ( a , b ) is the arithmetic mean, and M 2 ( a , b ) = ( a 2 + b 2 ) / 2 = Q ( a , b ) is the quadratic mean. The main properties of the power mean are given in . Recently, the power mean has attracted the attention of many researchers. In particular, many remarkable inequalities for the power mean can be found in the literature .

Let L ( a , b ) = ( b - a ) / ( log b - log a ) , P ( a , b ) = ( a - b ) / [ 2 arcsin ( ( a - b ) / ( a + b ) ) ] , I ( a , b ) = 1 / e ( b b / a a ) 1 / ( b - a ) , T ( a , b ) = ( a - b ) / [ 2 arctan ( ( a - b ) / ( a + b ) ) ] , and C ( a , b ) = ( a 2 + b 2 ) / ( a + b ) be the logarithmic, first Seiffert, identric, second Seiffert, and contraharmonic means of two distinct positive real numbers a and b , respectively. Then it is well known that the inequalities (2) H a , b < G a , b < L a , b < P a , b < I a , b < A a , b < T a , b < Q a , b < C a , b hold for all a , b > 0 with a b .

Lin  proved that the double inequality (3) M p a , b < L a , b < M q a , b holds for all a , b > 0 with a b if and only if p 0 and q 1 / 3 .

In , Pittenger presented the best possible parameters λ = λ ( p ) and μ = μ ( p ) such that the double inequality (4) M λ a , b < L p a , b < M μ a , b holds for all a , b > 0 with a b , where L p ( a , b ) = [ ( b p + 1 - a p + 1 ) / ( ( p + 1 ) ( b - a ) ) ] 1 / p ( p 0 , - 1 ) , L 0 ( a , b ) = I ( a , b ) , and L - 1 ( a , b ) = L ( a , b ) is the generalized logarithmic mean of a and b .

Jagers  and Seiffert  proved that the double inequalities (5) M 1 / 2 a , b < P a , b < M 2 / 3 a , b , M 1 a , b < T a , b < M 2 a , b hold for all a , b > 0 with a b .

In [15, 16], the authors proved that the double inequalities (6) M log 2 / 1 + log 2 a , b < I a , b + L a , b 2 < M 1 / 2 a , b , M 0 a , b < I a , b L a , b < M 1 / 2 a , b

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

Costin and Toader  proved that the double inequality (7) M 4 / 3 a , b < T a , b < M 5 / 3 a , b holds for all a , b > 0 with a b .

In , the authors proved that the double inequalities (8) M p a , b < P a , b < M q a , b , M r a , b < T a , b < M s a , b hold for all a , b > 0 with a b if and only if p log π / log 2 , q 2 / 3 , r log 2 / ( log π - log 2 ) , and s 5 / 3 .

Čizmesija  proved that p = - α / 2 and q = log 2 / [ log 4 - log ( 1 - α ) ] are the best possible parameters such that the double inequality M p ( a , b ) < [ H ( a , b ) + α G ( a , b ) + ( 1 - α ) A ( a , b ) ] / 2 < M q ( a , b ) holds for all α ( 0,1 ) and a , b > 0 with a b .

In [22, 23], the authors proved that the inequalities (9) M p K r , E r > π 2 , M λ K r , K 1 - r 2 K 2 2 M μ K r , K 1 - r 2 hold for all r ( 0,1 ) if and only if p - 1 / 2 , λ log 2 / ( log π - log 2 ) - log [ K ( 2 / 2 ) ] = - 4.18 , , and μ 1 - 4 K 4 ( 2 / 2 ) / π 2 = - 3.789 , , where K ( r ) = 0 π / 2 ( 1 - r 2 sin 2 θ ) - 1 / 2 d θ and E ( r ) = 0 π / 2 ( 1 - r 2 sin 2 θ ) 1 / 2 d θ are, respectively, the complete elliptic integrals of the first and second kinds.

Let p [ 0,1 ] and N be the bivariate symmetric mean. Then, the one-parameter mean N p ( a , b ) was defined by Neuman  as follows: (10) N p a , b = N p a + 1 - p b , p b + 1 - p a .

Let p 1 , q 1 , p 2 , q 2 , λ 1 , λ 2 , λ 3 , λ , μ ( 0,1 / 2 ) and p 3 , q 3 ( 1 / 2,1 ) . Then, the authors in  proved that the inequalities (11) H p 1 a , b < P a , b < H q 1 a , b , L p 2 a , b < P a , b < L q 2 a , b , H λ 1 a , b > G a , b , H λ 2 a , b > L a , b , G λ 3 a , b > L a , b , C p 3 a , b < T a , b < C q 3 a , b , G λ a , b < I a , b < G μ a , b hold for all a , b > 0 with a b if and only if p 1 1 / 2 - 1 - 2 / π / 2 , q 1 1 / 2 - 6 / 12 , p 2 λ 0 , q 2 1 / 2 - 2 / 4 , λ 1 1 / 2 - 2 / 4 , λ 2 1 / 2 - 3 / 6 , λ 3 1 / 2 - 6 / 6 , p 3 1 / 2 + 4 / π - 1 / 2 , q 3 1 / 2 + 3 / 6 , λ 1 / 2 - 1 - 4 / e 2 / 2 , and μ 1 / 2 - 3 / 6 , where λ 0 is the unique solution of the equation log [ ( 1 - λ ) / λ ] = π ( 1 - 2 λ ) .

The main purpose of this paper is to present the best possible parameters α = α ( r ) and β = β ( r ) such that the double inequality M α ( a , b ) < H r ( a , b ) < M β ( a , b ) holds for all r ( 0,1 / 2 ) and a , b > 0 with a b .

2. Lemmas

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

Lemma 1.

The inequality (12) - 8 r 2 + 8 r - 1 + log 2 log 2 r 1 - r < 0 holds for all r ( 0,1 / 2 ) .

Proof.

It is not difficult to verify that log [ 2 r ( 1 - r ) ] < 0 for all r ( 0,1 / 2 ) . Therefore, we only need to prove that f ( r ) > 0 for r ( 0,1 / 2 ) , where f ( r ) = ( - 8 r 2 + 8 r - 1 ) log [ 2 r ( 1 - r ) ] + log 2 . Simple computations lead to (13) f 1 2 = 0 , f r = 1 - 2 r g r , where (14) g r = 8 log 2 r 1 - r - 1 r 1 - r + 8 , (15) g 1 2 = 4 1 - log 4 < 0 , g r = 1 - 2 r 8 r 1 - r + 1 r 2 1 - r 2 > 0 for all r ( 0,1 / 2 ) .

Inequality (15) implies that g ( r ) < 0 for all r ( 0,1 / 2 ) . Then, from (13) we clearly see that f ( r ) > 0 for all r ( 0,1 / 2 ) .

Lemma 2.

The inequality (16) - 16 r 2 + 16 r - 2 - 8 r 2 - 8 r - 1 log 2 log 2 r 1 - r + log 2 log 2 r 1 - r 2 < 0 holds for all r ( 0,1 / 2 ) .

Proof.

Let p = - log 2 / log [ 2 r ( 1 - r ) ] and q = - 8 r 2 + 8 r - 1 . Then, it is not difficult to verify that (17) 0 < p < 1 for all r ( 0,1 / 2 ) .

It follows from Lemma 1 that (18) p > q for all r ( 0,1 / 2 ) .

Inequalities (17) and (18) lead to (19) - 16 r 2 + 16 r - 2 - 8 r 2 - 8 r - 1 log 2 log 2 r 1 - r + log 2 log 2 r 1 - r 2 = 2 q - q + 2 p + p 2 = - p - q 2 - p < 0 for all r ( 0,1 / 2 ) .

Lemma 3.

The inequality (20) - 6 r 2 + 6 r - 12 r 2 - 12 r + 1 log 2 log 2 r 1 - r + log 2 log 2 r 1 - r 2 > 0 holds for all r ( 0,1 / 2 ) .

Proof.

Let p = - log 2 / log [ 2 r ( 1 - r ) ] and q = - 8 r 2 + 8 r - 1 . Then, it follows from (17) and (18) that (21) - 6 r 2 + 6 r - 12 r 2 - 12 r + 1 log 2 log 2 r 1 - r + log 2 log 2 r 1 - r 2 = 3 4 q + 1 + 1 - 3 2 q + 1 p + p 2 = 3 4 q + 1 1 - 2 p + p p + 1 > 3 4 q + 1 1 - 2 p + p q + 1 = 1 4 q + 1 3 - 2 p = 2 r 1 - r 3 - 2 p > 0 for all r ( 0,1 / 2 ) .

3. Main Results Theorem 4.

The double inequality (22) M α a , b < H r a , b < M β a , b holds for all r ( 0,1 / 2 ) and a , b > 0 with a b if and only if α - 8 r 2 + 8 r - 1 and β - log 2 / log [ 2 r ( 1 - r ) ] .

Proof.

Without loss of generality, we assume that a = x ( 1 , ) and b = 1 . Let log H r ( x , 1 ) - log M p ( x , 1 ) = f ( x ) , where (23) f x = log r 1 - r x 2 + 1 - 2 r + 2 r 2 x + r 1 - r 1 + x + 1 + p log 2 - log 1 + x p p . Then, simple computations lead to (24) f 1 = 0 , (25) f x = g x r 1 - r x 2 + 1 - 2 r + 2 r 2 x + r 1 - r 1 + x 1 + x p x 1 - p , where (26) g x = - r 2 + r x 3 - p + - 2 r 2 + 2 r x 2 - p + 3 r 2 - 3 r + 1 x 1 - p + - 3 r 2 + 3 r - 1 x 2 + 2 r 2 - 2 r x + r 2 - r , (27) g 1 = 0 , (28) g 1 = - 8 r 2 + 8 r - 1 - p , (29) g ′′ 1 = - 16 r 2 + 16 r - 2 + p 8 r 2 - 8 r - 1 + p 2 , (30) g ′′′ x = 1 - p x - p - 2 h x , where (31) h x = - r 2 + r 3 - p 2 - p x 2 - - 2 r 2 + 2 r p 2 - p x + 3 r 2 - 3 r + 1 p p + 1 , (32) h 1 = 6 - r 2 + r + p 12 r 2 - 12 r + 1 + p 2 , (33) h x = 2 r - r 2 2 - p 3 - p x - p .

We divide the proof into two cases.

Case 1 ( p = - 8 r 2 + 8 r - 1 ). We divide the discussion into two subcases.

Subcase 1.1 ( p = - 8 r 2 + 8 r - 1 and r ( 0 , ( 2 - 2 ) / 4 ) ( 2 - 2 ) / 4,1 / 2 ) ). Then, we clearly see that p ( - 1,0 ) ( 0,1 ) , and (28), (29), (32), and (33) lead to (34) g 1 = g ′′ 1 = 0 , (35) h 1 = 1 4 p + 1 3 - 2 p > 0 , (36) h x = 1 4 p + 1 2 - p 3 - p x - p > 1 4 p + 1 2 - p 3 - 2 p > 0 for x ( 1 , ) .

It follows easily from (24), (25), (27), (30), and (34)–(36) that (37) H r x , 1 > M - 8 r 2 + 8 r - 1 x , 1 for all r ( 0,1 / 2 ) .

Subcase 1.2 ( p = - 8 r 2 + 8 r - 1 and r = ( 2 - 2 ) / 4 ). Then, p = 0 , and (37) follows from (38) H r x , 1 - M - 8 r 2 + 8 r - 1 x , 1 = x 2 + 6 x + 1 4 1 + x - x = x - 1 2 4 1 + x .

Case 2 ( p = - log 2 / log [ 2 r ( 1 - r ) ] ). Then, we clearly see that p ( 0,1 ) , and Lemmas 13 and (23), (26), (28), (29), and (32) lead to (39) lim x + f x = 0 , (40) lim x + g x = + , (41) lim x + g x = + , (42) g 1 < 0 , (43) lim x + g ′′ x = + , (44) g ′′ 1 < 0 , (45) h 1 > 0 and (36) again holds.

It follows from (30), (36), and (45) that g ′′ is strictly increasing on ( 1 , ) . Then, (43) and (44) lead to the conclusion that there exists μ 0 > 1 such that g is strictly decreasing on ( 1 , μ 0 ] and strictly increasing on [ μ 0 , ) .

From (41) and (42) together with the piecewise monotonicity of g we clearly see that there exists μ 1 > 1 such that g is strictly decreasing on ( 1 , μ 1 ] and strictly increasing on [ μ 1 , ) . Then, (25), (27), and (40) lead to the conclusion that there exists μ 2 > 1 such that f is strictly decreasing on ( 1 , μ 2 ] and strictly increasing on [ μ 2 , ) . Therefore, (46) H r x , 1 < M - log 2 / log 2 r 1 - r x , 1 for all r ( 0,1 / 2 ) follows from (24) and (39) together with the piecewise monotonicity of f .

Next, we prove that α = - 8 r 2 + 8 r - 1 and β = - log 2 / log [ 2 r ( 1 - r ) ] are the best possible parameters such that the double inequality (47) M α a , b < H r a , b < M β a , b holds for all r ( 0,1 / 2 ) and a , b > 0 with a b .

Let r ( 0,1 / 2 ) , α 0 = - 8 r 2 + 8 r - 1 , β 0 = - log 2 / log [ 2 r ( 1 - r ) ] , ε ( 0 , β 0 ) , and t > 0 . Then, we have (48) H r 1,1 + t - M α 0 + ε 1,1 + t = 2 1 - r 1 + t + r r 1 + t + 1 - r t + 2 - 1 + 1 + t α 0 + ε 2 1 / α 0 + ε , (49) lim t + H r 1 , t M β 0 - ε 1 , t = 2 ε / β 0 β 0 - ε > 1 .

Let t 0 + and make use of the Taylor expansion; then, (48) leads to (50) H r 1,1 + t - M α 0 + ε 1,1 + t = - ε 8 x 2 + o x 2 .

Inequality (49) and equation (50) imply that for any r ( 0,1 / 2 ) and ε ( 0 , β 0 ) there exist T = T ( ε ) > 1 and δ = δ ( ε ) ( 0,1 ) such that H r ( 1 , t ) > M β 0 - ε ( 1 , t ) for t > T and H r ( 1,1 + t ) < M α 0 + ε ( 1,1 + t ) for t ( 0 , δ ) .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11371125, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of Hunan Province under Grant 14JJ2127.

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