Research Article Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

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1 Applied Mathematics Volume 2012, Article ID , 8 pages doi: /2012/ Research Article Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means Wei-Mao Qian 1 and Bo-Yong Long 2 1 School of Distance Education, Huzhou Broadcast and TV University, Huzhou , China 2 School of Mathematics Science, Anhui University, Hefei , China Correspondence should be addressed to Wei-Mao Qian, qwm661977@126.com Received 29 January 2012; Revised 19 February 2012; Accepted 12 March 2012 Academic Editor: Yuri Sotskov Copyright q 2012 W.-M. Qian and B.-Y. Long. 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 sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means. 1. Introduction For p R the generalized logarithmic mean L p a, b of two positive numbers a and b is defined by a, a b, ] 1/p, a p 1 b p 1 ( ) p/ 0, p/ 1, a/ b, p 1 a b L p a, b ( ) 1/ b a 1 b b e a a, p 0, a/ b, b a log b log a, p 1, a / b. 1.1 It is well-known that L p a, b is continuous and strictly increasing with respect to p R for fixed a and b with a / b. In the recent past, the generalized logarithmic mean has been the subject of intensive research. In particular, many remarkable inequalities for L p can be

2 2 Applied Mathematics found in the literature The generalized logarithmic mean has applications in convex function, economics, physics, and even in meteorology In 26 the authors study a variant of Jensen s functional equation involving L p, which appear in a heat conduction problem. Let A a, b a b /2, I a, b 1/e b b /a a 1/ b a, L a, b b a / log b log a, G a, b ab, andh a, b 2ab/ a b be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers a and b with a / b, respectively. Then it is well known that min{a, b} <H a, b <G a, b L 2 a, b <L a, b L 1 a, b <I a, b L 0 a, b <A a, b L 1 a, b < max{a, b}. 1.2 In 28 30, the authors present bounds for L and I in terms of G and A. Proposition 1.1. For all positive real numbers a and b with a / b, one has A 1/3 a, b G 2/3 a, b <L a, b < 1 3 A a, b 2 G a, b, G a, b 2 A a, b <I a, b The proof of the following Proposition 1.2 can be found in 31. Proposition 1.2. For all positive real numbers a and b with a / b, we have G a, b A a, b < L a, b I a, b < 1 2 L a, b I a, b < 1 G a, b A a, b For r R the rth power mean M r a, b of two positive numbers a and b is defined by ( a r b r ) 1/r, r/ 0, M r a, b ab, r 0. The main properties of these means are given in 32. Several authors discussed the relationship of certain means to M r. The following sharp bounds for L, I, IL 1/2,and I L /2 in terms of power means are proved in 31, Proposition 1.3. For all positive real numbers a and b with a / b one has M 0 a, b <L a, b <M 1/3 a, b, M 2/3 a, b <I a, b <M log 2 a, b, M 0 a, b <I 1/2 a, b L 1/2 a, b <M 1/2 a, b, 1 2 I a, b L a, b <M 1/2 a, b. 1.6 The following three results were established by Alzer and Qiu in 38.

3 Applied Mathematics 3 Proposition 1.4. The inequalities αa a, b 1 α G a, b <I a, b <βa a, b ( 1 β ) G a, b 1.7 hold for all positive real numbers a and b with a / b if and only if α 2 3, β 2 e Proposition 1.5. Let a and b be real numbers with a / b.if0 <a, b e,then G a, b A a,b < L a, b I a,b < A a, b G a,b. 1.9 And, if a, b e,then A a, b G a,b < I a, b L a,b < G a, b A a,b Proposition 1.6. For all positive real numbers a and b with a / b, one has M c a, b < 1 L a, b I a, b with the best possible parameter c log 2/ 1 log In 39 the authors presented inequalities between the generalized logarithmic mean and the product A α a, b G β a, b H γ a, b for all a, b > 0witha / b and α, β > 0withα β<1. It is the aim of this paper to give a solution to the problem: for α 0, 1, what are the greatest value p and the least value q, such that the inequality L p a, b G α a, b H 1 α a, b L q a, b 1.12 holds for all a, b >0? 2. Main Result Theorem 2.1. For α 0, 1 and all a, b > 0, one has the following: 1 L 3α 5 a, b G α a, b H 1 α a, b L 2/α a, b for α 2/3, 2 L 3α 5 a, b G α a, b H 1 α a, b L 2/α a, b for 0 <α<2/3, and L 3α 5 a, b G α a, b H 1 α a, b L 2/α a, b for 2/3 <α<1, with equality if and only if a b, and the parameters 3α 5 and 2/α in each inequality cannot be improved. Proof. 1 If α 2/3 anda b, then 1.1 implies that L 3α 5 a, b G α a, b H 1 α a, b L 2/α a, b a.

4 4 Applied Mathematics If α 2/3anda / b, then 1.1 leads to L 3α 5 a, b L 2/α a, b L 3 a, b a 2 b 2 2 b a ] 1/3 ( ) 2ab 1/3 ab 1/3 G 2/3 a, b H 1/3 a, b G α a, b H 1 α a, b. a b If a b, then from 1.1 we clearly see that L 3α 5 a, b G α a, b H 1 α a, b L 2/α a, b a for any α 0, 1. If a / b, without loss of generality, we assume a>b.leta/b t>1and ] f t log L 3α 5 a, b log G α a, b H 1 α a, b. 2.2 Then 1.1 and simple computations yield f t 1 3α 5 log t 3α 4 1 3α 4 t 1 α log t 1 α log 2t 2 1 t, 2.3 lim t 1 f t 0, t 4 3α f t t t 2 1 ( t 4 3α 1 )g t, 2.4 where g t 2 α/2 t 3α 2 2 α 2 3α /5 3α t 3α 3 1 α 2 3α /2 5 3α t 3α 4 1 α 2 3α /2 5 3α t 2 2 α 2 3α / 5 3α t 2 α /2, g 1 0, g 2 α 3α 2 t t 3α α 2 3α α 1 t 3α α 1 α 2 3α 3α 4 t 3α 5 1 α 2 3α t 2 5 3α 5 3α 2 α 2 3α, 5 3α g 1 0, g 3 2 α 3α 2 α 1 t t 3α α 2 3α α 1 3α α 1 α 2 3α 3α 4 t 3α 6 1 α 2 3α, 2 5 3α g 1 0, g t α 2 α 4 3α 3α 2 t3α 7 t 1 2. t 3α

5 Applied Mathematics 5 If 0 <α<2/3, then 2.7 implies g t < for t>1. From and 2.8 we know that f t > 0fort>1. If 2/3 <α<1, then 2.7 leads to g t > for t>1. Therefore f t < 0fort>1follows from and 2.9. Let ] h t log L 2/α a, b log G α a, b H 1 α a, b 2.10 for t a/b > 1; then 1.1 and elementary calculations lead to h t α 2 log t α 2 /α 1 α 2 /α t 1 α log t 1 α log 2t 2 1 t, 2.11 lim t 1 h t 0, t 2 α /α h t t t 2 1 ( t 2 α /α 1 )v t, 2.12 where v t 2 α /2 t 3α 2 /α 3α 2 /2 t 2α 2 /α 3α 2 /2 t 2 α /2, v 1 0, v t v 1 0, 2 α 3α 2 t 2α 2 /α 2α 3α 2 α 1 α t α 2 /α 3α 2, v t 2 α 1 α 2 3α α 2 t 2/α t If α 0, 2/3, then 2.15 implies v t > for t>1. From and 2.16 we know that h t < 0fort>1. If α 2/3, 1, then 2.15 leads to v t < for t>1. Therefore, h t > 0fort>1 follows from and 2.17.

6 6 Applied Mathematics Next, we prove that the parameters 2/α and 3α 5 in either case cannot be improved. The proof is divided into two cases. Case 1 α 0, 2/3. For any ɛ>0andx 0, 1, from 1.1 one has ] 5 3α ɛ G α 1, 1 x H 1 α 1, 1 x L3α 5 ɛ 1, 1 x 5 3α ɛ f 1 x ], 1 x/2 1 α 5 3α ɛ 1 x 4 3α ɛ where f 1 x 1 x 1 α/2 5 3α ɛ 1 x 4 3α ɛ 1 4 3α ɛ x 1 x 4 3α ɛ 1 x/2 1 α 5 3α ɛ. Let x 0; making use of the Taylor expansion, we get f 1 x ɛ 4 3α ɛ 5 3α ɛ ( x 3 o x 3) Equations 2.18 and 2.19 imply that for any α 0, 2/3 and ɛ > 0 there exists δ δ ɛ, α 0, 1, such that L 3α 5 ɛ 1, 1 x <G α 1, 1 x H 1 α 1, 1 x for x 0,δ. On the other hand, for any ɛ 0, 2/α 1 we have L 2/α ɛ 1,t G α 1,t H 1 α 1,t ] α/ 2 ɛα t α/ 2 ɛα 1 t 2/α ɛ 1 t ɛα2 /2 2 ɛα 2/α ɛ 1 1 1/t ] α/ 2 ɛα 1 t 2/α ɛ 1 lim t ɛα2 /2 2 ɛα t 2/α ɛ 1 1 1/t ( 2t 1 t) 1 α ( ) 2t 1 α 1 t, 2.20 ( ) 2 α/ 2 ɛα α ɛ 1 > 0. From 2.20 we know that for any α 0, 2/3 and ɛ 0, 2/α 1 there exists T T ɛ, α > 1, such that L 2/α ɛ 1,t >G α 1,t H 1 α 1,t for t T,. Case 2 α 2/3, 1. For any ɛ 0, 4 3α and x 0, 1, from 1.1 one has ] 5 3α ɛ L 3α 5 ɛ 1, 1 x 5 3α ɛ G α 1, 1 x H 1 α 1, 1 x f 2 x 1 x/2 1 α 5 3α ɛ ], 1 x 4 3α ɛ where f 2 x 4 3α ɛ x 1 x 4 3α ɛ 1 x/2 1 α 5 3α ɛ 1 x 1 α/2 5 3α ɛ 1 x 4 3α ɛ 1. Let x 0; making use of the Taylor expansion, we have f 2 x ɛ ( α ɛ 5 3α ɛ x3 o x 3). 2.22

7 Applied Mathematics 7 Equations 2.21 and 2.22 imply that for any α 2/3, 1 and ɛ 0, 4 3α there exists δ δ ɛ, α 0, 1, such that L 3α 5 ɛ 1, 1 x >G α 1, 1 x H 1 α 1, 1 x for x 0,δ. On the other hand, for any ɛ>0, we have G α 1,t H 1 α 1,t L 2/α ɛ 1,t ( ) 2t 1 α t α/2 t ɛα2 /2 2 ɛα 1 t ( ) 2t 1 α lim t ɛα2 /2 2 ɛα t 1 t 1 t 2/α ɛ 1 2/α ɛ 1 1 1/t 1 t 2/α ɛ 1 2/α ɛ 1 1 1/t ] α/ 2 ɛα, ] α/ 2 ɛα 21 α > From 2.23 we know that for any α 2/3, 1 and ɛ>0 there exists T T ɛ, α > 1, such that L 2/α ɛ 1,t <G α 1,t H 1 α 1,t for t T,. Acknowledgment This work was supported by the Natural Science Foundation of Zhejiang Broad-cast and TV University under Grant XKT-09G21. References 1 K. B. Stolarsky, The power and generalized logarithmic means, The American Mathematical Monthly, vol. 87, no. 7, pp , F. Qi and B.-N. Guo, An inequality between ratio of the extended logarithmic means and ratio of the exponential means, Taiwanese Mathematics, vol. 7, no. 2, pp , C.-P. Chen and F. Qi, Monotonicity properties for generalized logarithmic means, The Australian Mathematical Analysis and Applications, vol. 1, no. 2, article 2, p. 4, X. Li, C.-P. Chen, and F. Qi, Monotonicity result for generalized logarithmic means, Tamkang Journal of Mathematics, vol. 38, no. 2, pp , F. Qi, S.-X. Chen, and C.-P. Chen, Monotonicity of ratio between the generalized logarithmic means, Mathematical Inequalities & Applications, vol. 10, no. 3, pp , C.-P. Chen, The monotonicity of the ratio between generalized logarithmic means, Mathematical Analysis and Applications, vol. 345, no. 1, pp , B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numerical Algorithms, vol. 52, no. 1, pp , Y.-M. Chu and W.-F. Xia, Inequalities for generalized logarithmic means, Inequalities and Applications, vol. 2009, Article ID , 7 pages, M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, Optimal inequalities among various means of two arguments, Abstract and Applied Analysis, vol. 2009, Article ID , 10 pages, W.-F. Xia and Y.-M. Chu, Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means, Revue d Analyse Numérique et de Théorie de l Approximation, vol. 39, no. 2, pp , Y.-M. Chu and W.-F. Xia, Two optimal double inequalities between power mean and logarithmic mean, Computers & Mathematics with Applications, vol. 60, no. 1, pp , W.-F. Xia, Y.-M. Chu, and G.-D. Wang, The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means, Abstract and Applied Analysis, vol. 2010, Article ID , 9 pages, B.-Y. Long and Y.-M. Chu, Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, Inequalities and Applications, vol. 2010, Article ID , 10 pages, 2010.

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