Let c > b > a > 0 be real numbers. Then the function f(r) =

Lr(a,b)/Lr(a,c) is strictly decreasing on (−∞,∞), where Lr(a, b) denotes the generalized (extended) logarithmic mean of two positive numbers a and b

Chen, Chao-Ping

Qi, Feng

English

Victoria University Institutional Repository (VUIR)

Monotonicity and Inequalities for Ratio of the Generalized Logarithmic Means

Victoria University Eprints Repository

MONOTONICITY AND INEQUALITIES FOR RATIO OF THEGENRALIZED LOGARITHMIC MEANSFENG QI AND CHAO-PING CHENAbstract. Let c > b > a > 0 be real numbers. Then the function f(r) =Lr(a,b)Lr(a,c)is strictly decreasing on (−∞,∞), where Lr(a, b) denotes the general-ized (extended) logarithmic mean of two positive numbers a and b.1. IntroductionIf −∞ < p < ∞ and a, b are two positive numbers, the generalized (extended)logarithmic mean Lp(a, b) of a and b is defined for a = b by Lp(a, b) = a and fora 6= b byLp(a, b) =(bp+1 − ap+1(p+ 1)(b− a))1/p, p 6= −1, 0;b− aln b− ln a, p = −1;1e(bbaa)1/(b−a), p = 0.(1)the case p = −1 is called the logarithmic mean of a and b, and will be writtenL(a, b); while the case p = 0 is the identric mean of a and b, written I(a, b).This definition of the generalized logarithmic mean can be found in [2, p. 6] and[33, 34].It is well known that if r > 0 is a real number, then for all natural numbers nnn+ 1<(1nn∑i=1ir/1n+ 1n+1∑i=1ir)1/r<n√n!n+1√(n+ 1)!. (2)2000 Mathematics Subject Classification. Primary 26D15.Key words and phrases. Monotonicity, inequality, ratio, the generalized (extended) logarithmicmean, identric (exponential) mean.The authors were supported in part by NNSF (#10001016) of China, SF for the ProminentYouth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, DoctorFund of Jiaozuo Institute of Technology, CHINA.This paper was typeset using AMS-LATEX.12 F. QI AND CH.-P. CHENThe first inequality in (2) is called H. Alzer’s inequality [1], and the second onein (2) J. S. Martins’ inequality [11]. The inequality between two ends of (2) iscalled Minc-Sathre’s inequality [12].There exists a very rich literature on inequality (2). Alzer’s inequality has beengeneralized and extended, for example, in [4, 5, 6, 7, 10, 14, 15, 16, 17, 22, 24, 25,30, 31, 32, 35, 36, 37]. So does Martins’s inequality in [3, 5, 17, 21, 23, 25, 26,27, 29, 35, 37, 38] and Minc-Sathre’s inequality in [1, 5, 9, 18, 19, 20, 25, 27, 28],respectively.Recently, F. Qi and B.-N. Guo proved in [15, 23] the following double inequality:Let b > a > 0 and δ > 0, then for any positive real number r,bb+ δ<(1b−a∫ baxr dx1b+δ−a∫ b+δaxr dx)1/r<[bb/aa]1/(b−a)[(b+ δ)b+δ/aa]1/(b+δ−a). (3)The upper and lower bounds in (3) are the best possible, or more accurately say,limr→∞(1b−a∫ baxr dx1b+δ−a∫ b+δaxr dx)1/r=bb+ δ, (4)limr→0(1b−a∫ baxr dx1b+δ−a∫ b+δaxr dx)1/r=[bb/aa]1/(b−a)[(b+ δ)b+δ/aa]1/(b+δ−a). (5)Inequality (3) can be taken for an integral form of (2).It is easy to see that inequality (3) can be written for r > 0 asbb+ δ<Lr(a, b)Lr(a, b+ δ)<I(a, b)I(a, b+ δ). (6)In this short note, we are about to extend the result presented by (3) to (5)which are established in [15, 23] by F. Qi and B.-N. Guo, and obtain the followingTheorem 1. Let c > b > a > 0 be real numbers. Then the functionf(r) =Lr(a, b)Lr(a, c)(7)is strictly decreasing with r ∈ (−∞,∞).The following corollary is straightforward.Corollary 1. Let c > b > a > 0 be real numbers.MONOTONICITY AND INEQUALITIES FOR RATIO OF MEANS 3(1) For any real number r ∈ R,bc<Lr(a, b)Lr(a, c)< 1. (8)The both bounds in (8) are the best possible.(2) For any positive real number r > 0,bc<Lr(a, b)Lr(a, c)<I(a, b)I(a, c). (9)The both bounds in (9) are also the best possible.Remark 1. It is worthwhile pointing out that inequalities (3) and (9) are equivalenteach other.In [29] it was conjectured that the function(1n∑ni=1 ir1n+1∑n+1i=1 ir)1/r(10)is decreasing with r. Now it is still keep open. We can regard Theorem 1 as asolution to an integral form of the conjecture above.2. Proof of Theorem 1In order to verify Theorem 1, we shall make use of the following elementarylemma which can be found in [8, p. 395].Lemma 1 ([8, p. 395]). Let the second derivative of φ(x) be continuous withx ∈ (−∞,∞) and φ(0) = 0. Defineg(x) =φ(x)x, x 6= 0;φ′(0), x = 0.(11)Then φ(x) is (strictly) convex if and only if g(x) is (strictly) increasing with x ∈(−∞,∞).Remark 2. In [13, p. 18] a general conclusion was given: A function f is convex on[a, b] if and only if f(x)−f(x0)x−x0 is nondecreasing on [a, b] for every point x0 ∈ [a, b].4 F. QI AND CH.-P. CHENProof of Theorem 1. Define for r ∈ (−∞,∞)ϕ(r) =ln(c− ab− a ·br+1 − ar+1cr+1 − ar+1), r 6= −1;ln(c− ab− a ·ln b− ln aln c− ln a), r = −1.(12)Then we haveln f(r) =ϕ(r)r, r 6= 0,ϕ′(0), r = 0.(13)In order to prove that ln f(r) is strictly decreasing it suffices to show that ϕ isstrictly concave in (−∞,∞). Easy computation reveals thatϕ(−1− r) = ϕ(r − 1) + r ln cb, (14)which implies that ϕ′′(−r − 1) = ϕ′′(r − 1), and then ϕ(r) has the same concavityon both (−∞,−1) and (−1,∞). Hence, it is sufficient to prove that ϕ is strictlyconcave on (−1,∞).A simple computation yieldsϕ′′(r) =(a/c)r+1[ln(a/c)]2[1− (a/c)r+1]2 −(a/b)r+1[ln(a/b)]2[1− (a/b)r+1]2 . (15)Define for 0 < t < 1ω(t) =t(ln t)2(1− t)2 . (16)Differentiation yields(1− t)t ln tω′(t)ω(t)= (1 + t) ln t+ 2(1− t) = −∞∑n=2n− 1n(n+ 1)tn+1 < 0, (17)which means that ω′(t) > 0 for 0 < t < 1. As a result of applying this conclusionin (15), we obtain ϕ′′(r) < 0 for r > −1. Thus ϕ(r) is strictly concave in (−1,∞).The proof is complete. References[1] H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Anal. Appl. 179 (1993),396–402.[2] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure andApplied Mathematics 97, Addison Wesley Longman Limited, 1998.MONOTONICITY AND INEQUALITIES FOR RATIO OF MEANS 5[3] T. H. Chan, P. Gao and F. Qi, On a generalization of Martins’ inequality, Monatsh. Math.(2003), in press. RGMIA Res. Rep. Coll. 4 (2001), no. 1, Art. 12, 93–101. 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Appl. 44 (2002),no. 3-4, 365–373.(F. Qi) Department of Applied Mathematics and Informatics, Jiaozuo Institute ofTechnology, Jiaozuo City, Henan 454000, CHINAE-mail address: qifeng@jzit.edu.cn or fengqi618@member.ams.orgURL: http://rgmia.vu.edu.au/qi.html(Ch.-P. Chen) Department of Applied Mathematics and Informatics, Jiaozuo Instituteof Technology, Jiaozuo City, Henan 454000, CHINA

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Let c > b > a > 0 be real numbers. Then the function f(r) =\ud
Lr(a,b)/Lr(a,c) is strictly decreasing on (−∞,∞), where Lr(a, b) denotes the generalized (extended) logarithmic mean of two positive numbers a and b

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Monotonicity and Inequalities for Ratio of the Generalized Logarithmic Means

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