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1 International Mathematical Forum, Vol. 8, 01, no. 1, HIKARI Ltd, The Best Possible Lehmer Mean Bounds for a Convex Combination of Logarithmic and Harmonic Means 1 Zhijun Guo School of Mathematics and Computation Science Hunan City University Yiyang, Hunan, 41000, P. R. China Xuhui Shen College of Nursing, Huzhou Teachers College Zhejiang, Huzhou, 1000, P.R. China Yuming Chu College of Mathematics and Computation Science Hunan City University Yiyang, Hunan, 41000, P.R. China Corresponding author. chuyuming@hutc.zj.cn Copyright c 01 Zhijun Guo, Xuhui Shen and Yuming Chu. 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. Abstract For r R and a, b > 0 the Lehmer mean L r (a, b), logarithmic mean L(a, b), and harmonic mean H(a, b) are defined by L r (a, b) = ar+1 +b r+1 a r +b r, L(a, b) = { b a log b log a, b a, a, b = a, 1 This research was supported by the Natural Science Foundation of China (Grant Nos and ), the Natural Science Foundation of Zhejiang Province (Grant Nos. LY1H and LY1A010004), and the Natural Science Foundation of the Department of Education of Hunan Province (Grant No. 1C0577).

2 1540 Zhijun Guo, Xuhui Shen and Yuming Chu and H(a, b) = ab a+b, respectively. In this paper, we answer the question: For α (0, 1), what are the largest value p and least value q such that the double inequality L q (a, b) >αh(a, b) +(1 α)l(a, b) >L p (a, b) holds for all a, b > 0 with a b? Mathematics Subject Classification: 6E60 Keywords: Lehmer mean, logarithmic mean, harmonic mean 1. Introduction For r R and a, b > 0 the Lehmer mean L r (a, b), logarithmic mean L(a, b), and harmonic mean H(a, b) are defined by L r (a, b) = ar+1 + b r+1, a r + b r (1.1) { b a L(a, b) = log b log a a, b = a, (1.) and H(a, b) = ab a + b. (1.) It is well-known that L r (a, b) is continuous and strictly increasing with respect to r R for fixed a and b with a b. In the recent past, the Lehmer mean has been attracted the attention of many mathematicians [1-14]. Many means are the special cases of the Lehmer mean, for example, A(a, b) = a + b = L 0 (a, b) is the arithmetic mean, G(a, b) = ab = L 1 (a, b) is the geometric mean, H(a, b) = ab a + b = L 1(a, b) is the harmonic mean. Recently, the logarithmic mean has been the subject of intensive research. In particular, many remarkable inequalities for the logarithmic mean can be found in literatures [15-0]. It might be surprising that the logarithmic mean has applications in physics [1], economics [], and even in meteorology []. In [1], the authors study a variant of Jensen s function involving logarithmic mean, which appears in a heat conduction problem. For p R, let { { ( M p (a, b) = ap +b p ) 1 p, p 0, 1 and I(a, b) = ( bb ) 1 e a a b a, b a, ab, p =0 a, b = a

3 Best possible Lehmer mean bounds 1541 be the pth power mean and identric mean of two positive real numbers a and b, respectively. Then it is well known that min{a, b} <H(a, b) =L 1 (a, b) =M 1 (a, b) <G(a, b) = L 1 (a, b) =M 0 (a, b) <L(a, b) <I(a, b) <A(a, b) = L 0 (a, b) =M 1 (a, b) < max{a, b} (1.4) for all a, b > 0 with a b. In [1], Alzer presented the following sharp upper and lower Lehmer mean bounds for identric mean I(a, b): L 1 (a, b) <I(a, b) <L 0 (a, b) 6 for all a, b > 0 with a b. Stolarsky [1] established that M n+1 (a, b) L n (a, b) for all a, b > 0 with a b. In [4, 4, 5], the authors presented bounds for L and I in terms of G and A as follows: G 1 (a, b)a (a, b) <L(a, b) < G(a, b)+1 A(a, b) and I(a, b) > 1 G(a, b)+ A(a, b) for all a, b > 0 with a b. The following inequalities can be found in [6]: G (a, b)a (a, b) <L (a, b)i (a, b) < (L(a, b)+i(a, b)) < 1 (G(a, b)+a(a, b)). The following sharp bounds for L, I, (IL) 1, and 1 (I +L) in terms of power means are proved in [5-7, 6-9]: M 0 (a, b) <L(a, b) <M1(a, b), M (a, b) <I(a, b) <M log (a, b), M 0 (a, b) < I(a, b)l(a, b) <M1(a, b) and 1 (I(a, b)+l(a, b)) <M1 (a, b) for all a, b > 0 with a b, and the given parameters are best possible. Alzer and Qiu [0] proved that M c (a, b) < 1 (L(a, b)+i(a, b)) for all a, b > 0 with a b if and only if c log /(1 + log ). The purpose of this paper is to answer the question: For α (0, 1), what are the largest value p and least value q such that the double inequality L q (a, b) >

4 154 Zhijun Guo, Xuhui Shen and Yuming Chu αh(a, b)+(1 α)l(a, b) >L p (a, b) holds for all a, b > 0 with a b.. Lemmas In order to prove our main result, we need several lemmas which we present in this section. Lemma.1. Suppose that α (0, 1) and x (1, ). If p = α+1, then x p+ +(1 α)x p+1 +(1 α)x +1> 0. (.1) Proof. Let h(x) =x p+ +(1 α)x p+1 +(1 α)x+1, then simple computations lead to lim h(x) = 4(1 α) > 0, (.) h (x) =(p +)x p+1 +(1 α)(p +1)x p +1 α, (.) lim h (x) =(p + )(1 α) > 0, (.4) h (x) =(p +1)x p 1 h 1 (x), (.5) where h 1 (x) =(p +)x +(1 α)p. Note that lim 1(x) =α (α 1) > 0, (.6) h 1(x) =p +> 0. (.7) Therefore, inequality (.1) follows from (.)-(.7). Lemma.. If α (0, 1), then 76α α 5 908α 4 + 8α 5111α + 665α > 0. Proof. We clearly see that 76α α 5 908α 4 + 8α 5111α + 665α = α (76α 4 908α + 81) α 5 + 8α 59α +665α > α (76α 4 908α + 81) + 8α 59α + 665α + 565α = α (76α 4 908α + 81) + α(8α 59α + 40). (.8) The discriminants Δ 1 and Δ of the quadratic functions 76t 908t + 81 and 8t 59t + 40 satisfy and Δ 1 =( 908) = 800 < 0 (.9) Δ =( 59) = < 0 (.10) respectively. Therefore, Lemma. follows from (.8)-(.10).

5 Best possible Lehmer mean bounds 154 Lemma.. If α (0, 1), then 808α α α + 14α 14α > 0. Proof. We clearly see that 808α α α + 14α 14α > 1750α + 150α 150α = 50(5α +7α 6α + 1). (.11) It is not difficult to verify that min (5α +7α 6α + 1) α (0,1) = 5( ) + 7( ) 6( ) = 0....>0. (.1) Therefore, Lemma. follows from (.11) and (.1). Lemma.4. If α (0, 1), then 04α 5 +74α 4 488α +89α 61α+87 > 0. Proof. We clearly see that 04α α 4 488α + 89α 61α + 87 > 74α 4 488α + 488α + 51α 61α + 87 > 74α α 4 61α + 87 = 1075α 4 61α (.1) It is easy to verify that min α (0,1) (1075α4 61α + 87) = 1075( )4 61( 61 ) = >0. (.14) Therefore, Lemma.4 follows from (.1) and (.14).. Main Result Theorem.1. If α (0, 1), then L 0 (a, b) >αh(a, b) +(1 α)l(a, b) > L α+1 (a, b) for all a, b > 0 with a b, and L 0 (a, b) and L α+1 (a, b) are the best possible upper and lower Lehmer mean bounds for the sum αh(a, b)+ (1 α)l(a, b). Proof. Without loss of generality, we assume that a>b. From (1.4) we clearly see that L 0 (a, b) >αh(a, b)+(1 α)l(a, b). Next, we prove that αh(a, b)+(1 α)l(a, b) >L α+1 (a, b). (.1)

6 1544 Zhijun Guo, Xuhui Shen and Yuming Chu Let x = a b > 1 and p = α+1, then (1.1)-(1.) lead to αh(a, b)+(1 α)l(a, b) L p (a, b) = b[xp+ +(1 α)x p+1 +(1 α)x +1] f(x), (1 + x)(1 + x p ) log x (.) where f(x) = (1 α)(x p+ x p +x 1) log x. Note that x p+ +(1 α)x p+1 +(1 α)x+1 f (x) = lim f(x) =0, (.) f 1 (x) x[x p+ +(1 α)x p+1 +(1 α)x +1], (.4) where f 1 (x) = x p+4 +(α + α 1)x p+ (4α α 1)x p+ +(α α + 1)x p+1 p(1 α)x p+4 +α(α 1)x p+ (4α α + αp p)x p+ +α(α 1)x p+1 p(1 α)x p +(α α +1)x (4α α 1)x +(α + α 1)x 1, lim f 1(x) =0. (.5) Let f (x) = x 4 p f (4) 1 (x), f 1 (x) = f (p+1) (x), f 4(x) = 1f (x), f 5(x) = x 4 p f 4 1 (x), f 6 (x) = f 1 (p+1)(p+) 5(x), and f 7 (x) = f (p+)(p+) 6(x), then simple computations yield that f 1 (x) = (p +)xp+ +(p + )(α + α 1)x p+ (p +1) (4α α 1)x p+1 +(p + 1)(α α +1)x p p(p + 4)(1 α)x p+ +α(p + )(α 1)x p+ (p +) (4α α + αp p)x p+1 +α(p + 1)(α 1)x p p (1 α)x p 1 + (α α +1)x (4α α 1)x +α + α 1, lim f 1(x) =0, (.6) f 1 (x) = (p + )(p +)xp+ +(p + 1)(p + )(α + α 1)x p+1 (p + 1)(p + 1)(4α α 1)x p +p(p +1) (α α +1)x p 1 p(p + )(p + 4)(1 α)x p+ +α(p + )(p + )(α 1)x p+1 (p + 1)(p +) (4α α + αp p)x p +αp(p + 1)(α 1)x p 1 p (p 1)(1 α)x p + 6(α α +1)x (4α α 1), lim f 1 (x) =0, (.7)

7 Best possible Lehmer mean bounds 1545 f 1 (x) = 4(p + 1)(p + )(p +)x p+1 +(p + 1)(p + 1)(p +) (α + α 1)x p 4p(p + 1)(p + 1)(4α α 1)x p 1 +p(p 1)(p + 1)(α α +1)x p p(p + )(p +) (p + 4)(1 α)x p+1 +α(p + 1)(p + )(p + )(α 1)x p p(p + 1)(p + )(4α α + αp p)x p 1 +αp(p 1) (p + 1)(α 1)x p p (p 1)(p )(1 α)x p +6(α α +1), lim f 1 (x) =0, (.8) f (x) = 4(p + 1)(p + )(p + 1)(p +)x p+4 +4p(p + 1)(p +1) (p + )(α + α 1)x p+ 4p(p + 1)(p 1)(p +1) (4α α 1)x p+ +4p(p 1)(p 1)(p +1) (α α +1)x p+1 p(p + 1)(p + )(p + )(p + 4)(1 α)x 4 +αp(p + 1)(p + )(p + )(α 1)x p(p 1)(p + 1)(p +) (4α α + αp p)x +αp(p 1)(p )(p + 1)(α 1)x p (p 1)(p )(p )(1 α), lim f (x) = (1 α) (10α +5α +1) > 0, (.9) 7 f (x) = (p + )(p + 4)(p + 1)(p +)x p+ +p(p + )(p +1) (p + )(α + α 1)x p+ p(p + )(p 1)(p +1) (4α α 1)x p+1 +p(p 1)(p 1)(p +1) (α α +1)x p p(p + )(p + )(p + 4)(1 α)x +αp(p + )(p + )(α 1)x p(p 1)(p +) (4α α + αp p)x + αp(p 1)(p )(α 1), lim f (x) = 64(1 α) (10α +5α +1) > 0, (.10) 7 f 4 (x) = (p + )(p + )(p + 4)(p + 1)(p +)x p+ + p(p + )(p +) (p + 1)(p + )(α + α 1)x p+1 p(p + 1)(p + )(p 1) (p + 1)(4α α 1)x p + p (p 1)(p 1)(p +1) (α α +1)x p 1 p(p + )(p + )(p + 4)(1 α)x +αp(p + )(p + )(α 1)x p(p 1)(p +) (4α α + αp p), lim f 4(x) = 7 (1 α)[48α α + 786α(1 α) + 41α + 65] > 0, (.11) f 4(x) = (p +) (p + )(p + 4)(p + 1)(p +)x p+1 + p(p + 1)(p +) (p + )(p + 1)(p + )(α + α 1)x p p (p + 1)(p +) (p 1)(p + 1)(4α α 1)x p 1 + p (p 1) (p 1) (p + 1)(α α +1)x p 6p(p + )(p + )(p +4) (1 α)x +αp(p + )(p + )(α 1),

8 1546 Zhijun Guo, Xuhui Shen and Yuming Chu lim f 4 (x) = 4 (1 α)[α(1 α)(75α α ) +748α + 105α ] > 0, (.1) f 4 (x) = (p + 1)(p +) (p + )(p + 4)(p + 1)(p +)x p + p (p +1) (p + )(p + )(p + 1)(p + )(α + α 1)x p 1 p (p 1) (p + 1)(p + )(p 1)(p + 1)(4α α 1)x p + p (p 1) (p )(p 1)(p + 1)(α α +1)x p 6p(p + )(p + )(p + 4)(1 α), lim f 4 (x) = 8 4 (1 α)(76α α 5 908α 4 + 8α (.1) 5111α + 665α + 565), lim (x) = 6p(p + )(p + )(p + 4)(1 α) > 0 (.14) f 4 x + f 5 (x) = p(p + 1)(p +) (p + )(p + 4)(p + 1)(p +)x + p (p 1) (p + 1)(p + )(p + )(p + 1)(p + )(α + α 1)x p (p 1)(p )(p + 1)(p + )(p 1)(p + 1)(4α α 1)x +p (p 1) (p )(p )(p 1)(p + 1)(α α +1), lim f 5(x) = (1 α)(1 4α)(α + 1)[808α5 + 86α α + 14α 14α α ] (.15) f 6 (x) = p(p + )(p + )(p + 4)(p + 1)(p +)x +p (p 1) (p + )(p + 1)(p + )(α + α 1)x p (p 1) (p )(p 1)(p + 1)(4α α 1), lim f 6(x) = 4 79 (1 4α)(α + 1)(04α5 + 74α 4 488α (.16) +89α 61α + 87) f 7 (x) = p(p + )(p + 4)(p +1)x + p (p 1)(p + 1)(α + α 1) lim f 7(x) = 1 81 (1 4α)(α + 1)(169 8α4 4α α 90α), (.17) f 7 (x) = (1 4α)p(p + )(p +4). (.18) We divide the proof of inequality (.1) into two cases. Case 1. Ifα (0, 1 ], then from Lemmas.-.4 and (.1) together with 4 (.15)-(.18) we clearly see that lim f 4 (x) > 0, (.19) lim f 5(x) > 0, (.0)

9 Best possible Lehmer mean bounds 1547 lim 6(x) > 0, (.1) lim 7(x) > 0, (.) f 7 (x) > 0. (.) It easily follows from (.)-(.1) and (.19)-(.) that f(x) > 0 (.4) for x>1. Therefore inequality (.1) follows from (.) and (.4) together with Lemma.1. Case. Ifα ( 1, 1), then from Lemmas.-.4 and (.1) together with 4 (.15)-(.18) we clearly see that (.19) again holds and lim f 5(x) < 0, (.5) lim 6(x) < 0, (.6) lim 7(x) < 0, (.7) f 7 (x) < 0. (.8) Inequalities (.5)-(.8) imply that f 4 (x) is strictly decreasing in (1, ). Then from (.14) and (.19) together with the monotonicity of f 4 (x) we know that f 4 (x) > 0 (.9) for x (1, ). Therefore, inequality (.1) follows from (.)-(.1) and (.9) together with Lemma.1. Finally, we prove that L 0 (a, b) and L α+1 (a, b) are the best possible upper and lower Lehmer mean bounds for the sum αh(a, b) +(1 α)l(a, b), respectively. For any ε>0 and x>0, we have and = L α+1 +ε ((1 + x), 1) [αh((1 + x), 1) + (1 α)l((1 + x), 1)] J(x) [1 + (1 + x) ε 1 α ][1 + (1 + x) ] log (1 + x) lim x + [αh(x, 1) + (1 α)l(x, 1) L ε(x, 1)] = lim [ αx x + 1+x = +, + (1 α)(x 1) log x x1 ε +1 x ε +1 ] (.0) (.1)

10 1548 Zhijun Guo, Xuhui Shen and Yuming Chu where J(x) = [1 + (1 + x) +ε α ][1 + (1 + x) ] log(1 + x) 6α(1 + x) [1 + (1 + x) ε 1 α ] log (1 + x) (1 α)[(1 + x) 6 1][1 + (1 + x) ε 1 α ]. Let x 0, making use of the Taylor expansion, one has J(x) = x[ + ( + ε α)x + 1 ( + ε α)(1 + ε α)x +o(x )][ + x +x + o(x )][1 1 x + 1 x + o(x )] 6αx[1 + x +x + o(x )][ + (ε 1 α)x + 1 (ε α 1)(ε α )x + o(x )][1 1 x + 1 x +o(x )] (1 α)x[6 + 15x +0x + o(x )][ + (ε 1 α)x + 1 (ε α 1)(ε α )x + o(x )] = 7εx + o(x ). (.) Equations (.0)-(.) imply that for any ε>0 there exist δ = δ(ε) > 0 and X = X(ε) > 1 such that αh((1 + x), 1) + (1 α)l((1 + x), 1) < L α+1 +ε ((1 + x), 1) for x (0,δ) and L ε (x, 1) <αh(x, 1) + (1 α)l(x, 1) for x (X, ). References [1] M. K. Wang, Y. M. Chu and G. D. Wang, A sharp double inequality between the Lehmer and arithmetic-geometric means, Pac. J. Appl. Math., 4(01), no. 1, 1-5. [] Y. M. Chu, M. K. Wang and Y. F. Qiu, Optimal Lehmer mean bounds for the geometric and arithmetic combinations of arithmetic and Seiffert means, Azerb. J. Math., (01), no. 1, -9. [] Y. M. Chu and M. K. Wang, Optimal Lehmer mean bounds for the Toader mean, Results Math., 61(01), no. -4, -9. [4] M. K. Wang, Y. M. Chu and G. D. Wang, A sharp double inequality between the Lehmer and arithmetic-geometric means, Pac. J. Appl. Math., (011), no. 4, [5] Y. F. Qiu, M. K. Wang, Y. M. Chu and G. D. Wang, Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J. Math. Inequal., 5(011), no., [6] M. K. Wang, Y. F. Qiu and Y. M. Chu, Sharp bounds for Seiffert means in terms of Lehmer means, J. Math. Inequal., 4(010), no. 4,

11 Best possible Lehmer mean bounds 1549 [7] W. F. Xia and Y. M. Chu, The Schur harmonic convexity of Lehmer means, Int. Math. Forum, 4(009), no , [8] S. Toader and G. Toader, Complementaries of Greek means with respect to Lehmer means, Automat. Comput. Appl. Math., 15(006), no., [9] I. Costin and G. Toader, Generalized inverses of Lehmer means, Automat. Comput. Appl. Math., 14(005), no. 1, [10] S. R. Wassell, Rediscovering a family of means, Math. Intelligencer, 4(00), no., [11] Zh. Liu, Remark on inequalities between Hölder and Lehmer means, J. Math. Anal. Appl., 47(000), no. 1, [1] K. B. Stolarsky, Hölder means, Lehmer means, and x 1 log cosh x, J. Math. Anal. Appl., 0(1996), no., [1] H. Alzer, Bestmögliche Abschätzungen für spezielle Mittelwerte, Zb. Rad. Prirod. -Mat. Fak. Ser. Mat., (199), no. 1, [14] H. Alzer, Über Lehmers Mittelwertfamilie, Elem. Math., 4(1988), no., [15] H. N. Hu, S. S. Wang and Y. M. Chu, Optimal upper power mean bound for the convex combination of Harmonic and logarithmic means, Pac. J. Appl. Math., 4(01), no. 1, [16] Y. M. Li, B. Y. Long and Y. M. Chu, Sharp bounds for the Neuman- Sándor mean in terms of generalized logarithmic mean, J. Math. Inequal., 6(01), no. 4, [17] Y. M. Chu, M. K. Wang and Z. K. Wang, Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means, Math. Inequal. Appl., 15(01), no., [18] Y. M. Chu, S. W. Shou and W. M. Gong, Inequalities between logarithmic, harmonic, arithmetic and centroidal means, J. Math. Appl., (011), no., 1-5. [19] Y. M. Chu and M. K. Wang, Optimal inequalities between harmonic, geometric, logarithmic, and arithmetic-geometric means, J. Appl. Math., 011(011), Article ID 61899, 9 pages.

12 1550 Zhijun Guo, Xuhui Shen and Yuming Chu [0] Y. M. Chu, S. S. Wang and C. Zong, Optimal lower power mean bound for convex combination of harmonic and logarithmic means, Abstr. Appl. Anal., 011(011), Article ID 50648, 9 pages. [1] W. F. Xia and Y. M. Chu, Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means, Rev. Anal. Numér. Théor. Approx.,, 9(010), no., [] 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, Abstr. Appl. Anal., 010(010), Article ID , 9 pages. [] Y. M. Chu and W. F. Xia, Two optimal double inequalities between power mean and logarithmic mean, Comput. Math. Appl., 60(011), no. 1, [4] B. C. Carlson, The logarithmic mean, Amer. Math. Monthly, 79(197), [5] T. P. Lin, The power mean and the logarithmic mean, Amer. Math. Monthly, 81(1974), [6] H. Alzer, Ungleichungen für Mittelwerte, Arch. Math., 47(1986), no. 5, [7] F. Burk, Notes: The geometric, logarithmic, and arithmetic mean Inequality, Amer. Math. Monthly, 94(1987), no. 6, [8] J. Sándor, On the identric and logarithmic means, Aequations Math., 40(1990), no. -, [9] G. Allasia, C. Giordano and J. Pe carić, On the arithmetic and logarithmic means with applications to stirling s formula, Atti Sem. Mat. Fis. Univ. Modena, 47(1999), no., [0] H. Alzer and S. L. Qiu, Inequalities for means in two variables, Arch. Math., 80(00), no., [1] P. Kahlig and J. Matkowski, Functional equations involving the logarithmic mean, Z. Angew. Math. Mech., 76(1996), no. 7, [] A. O. Pittenger, The logarithmic mean in n variables, Amer. Math. Monthly, 9(1985), no., [] G. Polya and G. Szegö, Isoperimetric Inequalities in Mathematical physics, Princeton University Press, Princeton, 1951.

13 Best possible Lehmer mean bounds 1551 [4] E. B. Leach and M. C. Sholander, Extended mean values II, J. Math. Anal. Appl., 9(198), no. 1, 07-. [5] J. Sándor, A note on some inequalities for means, Arch. Math., 56(1991), no. 5, [6] H. Alzer, Ungleichungen für (e/a) a (b/e) b, Elem. Math., 40(1985), [7] A. O. Pittenger, Inequalities between arithmetic and logarithmic means, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1980), no , [8] A. O. Pittenger, The symmetric, logarithmic and power means, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1980), no , 19-. [9] K. B. Stolarsky, The power and generalized logarithmic means, Amer. Math. Monthly, 87(1980), no. 7, Received: July 5, 01

国学思想与大学数学

国学思想与大学数学 Pure Mathematics 理论数学, 2013, 3, 201-206 http://dx.doi.org/10.12677/pm.2013.33030 Published Online May 2013 (http://www.hanspub.org/journal/pm.html) Chinese Traditional Culture and College Mathematics