In the article, some strictly Logarithmically completely monotonic

ratios of mean values are presented

Chen, Chao-Ping

Qi, Feng

English

In the article, some strictly Logarithmically completely monotonic\ud
ratios of mean values are presented

Name not available

Logarithmically Completely Monotonic Ratios of Mean Values and an Application\ud

Victoria University Eprints Repository

LOGARITHMICALLY COMPLETELY MONOTONIC RATIOS OFMEAN VALUES AND AN APPLICATIONCHAO-PING CHEN AND FENG QIAbstract. In the article, some strictly Logarithmically completely monotonicratios of mean values are presented.A function f is said to be completely monotonic on an interval I, if f hasderivatives of all orders on I and satisfies(−1)nf (n)(x) ≥ 0 (1)for x ∈ I and n ≥ 0. If inequality (1) is strict, then f is said to be strictly completelymonotonic on I.Completely monotonic functions have remarkable applications in different math-ematical branches. For instance, they play a role in potential theory [3], probabilitytheory [4, 7, 10], physics [5], numerical and asymptotic analysis [8, 18], and combi-natorics [1]. A detailed collection of the most important properties of completelymonotonic functions can be found in [17, Chapter IV], and in an abstract in [2].A positive function f is said to be logarithmically completely monotonic on aninterval I if its logarithm ln f satisfies(−1)n[ln f(x)](n) ≥ 0 (2)for x ∈ I and n ∈ N. If inequality (2) is strict, then f is said to be strictlylogarithmically completely monotonic.The terminology “(strictly) logarithmically completely monotonic function” wasnamed first by F. Qi, B.-N. Guo and Ch.-P. Chen in [11, 12, 13]. It was also showedin these papers that a (strictly) logarithmically completely monotonic function isalso (strictly) completely monotonic.The generalized logarithmic mean or Stolarsky mean Lr(a, b) of two positivenumbers a and b was introduced in [9, 15, 16] and [6, p. 6] for a = b by Lr(a, b) = aand for a 6= b byLr(a, b) =(br+1 − ar+1(r + 1)(b− a))1/r, r 6= −1, 0; (3)L−1(a, b) =b− aln b− ln a ; (4)L0(a, b) =1e(bbaa)1/(b−a). (5)2000 Mathematics Subject Classification. Primary 26A48, Secondary 30E20.Key words and phrases. Completely monotonic function, mean, ratio, integral representation.The authors were supported in part by NSF (#10001016) of China, SF for the ProminentYouth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, DoctorFund of Henan Polytechnic University, China.12 CH.-P. CHEN AND F. QIHere L−1(a, b) , L(a, b) and L0(a, b) , I(a, b) are the logarithmic and identricmeans, respectively. When a 6= b, Lr(a, b) is a strictly increasing function of r.Further,L1(a, b) , A(a, b), L−2(a, b) , G(a, b), (6)where A and G are the arithmetic and geometric means, respectively.For a 6= b, the following well known inequalities holdH(a, b) < G(a, b) < L(a, b) < I(a, b) < A(a, b), (7)where H is the harmonic mean.In this paper, the (logarithmically) complete monotonicity of some ratios of meanvalues are obtained. Our main results are as follows.Theorem 1. The ratiosA(x, x+ 1)I(x, x+ 1),A(x, x+ 1)G(x, x+ 1)=G(x, x+ 1)H(x, x+ 1), (8)A(x, x+ 1)H(x, x+ 1),I(x, x+ 1)G(x, x+ 1),I(x, x+ 1)H(x, x+ 1)(9)of mean values A, G, H and I are strictly logarithmically completely monotonic in(0,∞) and the ratioA(x, x+ 1)L(x, x+ 1)(10)is strictly completely monotonic in (0,∞).Proof. Define for x > 0φA/I(x) = lnA(x, x+ 1)I(x, x+ 1)= lnx+ 1/2(x+ 1)(1 + 1/x)x. (11)Differentiating directly, using the following representations for x > 0, s ≥ 0 andn ∈ Nlnx =∫ ∞0e−t − e−xttdt, (12)1(x+ s)n=1(n− 1)!∫ ∞0tn−1e−(x+s)t dt (13)and the power series expansion of tet/2 − et + 1 at 0, we conclude that(−1)nφ(n)A/I(x) = −∫ ∞0(tet/2 − et + 1)tn−2e−(x+1)t dt=∞∑k=3(1k− 12k−1)1(k − 1)!∫ ∞0tn+k−2e−(x+1)t dt> 0.(14)This means that the ratio A(x,x+1)I(x,x+1) is strictly logarithmically monotonic in (0,∞).Define for x > 0φA/G(x) = lnA(x, x+ 1)G(x, x+ 1)= ln(x+12)− 12lnx− 12ln(x+ 1), (15)LOGARITHMICALLY COMPLETELY MONOTONIC RATIOS OF MEAN VALUES 3then, by argument as above, we have for any nonnegative integer n(−1)nφ(n)A/G(x) =12∫ ∞0(et + 1− 2et/2)tn−1e−(x+1)t dt=12∞∑k=2(1− 12k−1)1k!∫ ∞0tn+k−1e−(x+1)t dt> 0.(16)This reveals that the ratio A(x,x+1)G(x,x+1) is strictly logarithmically completely monotonicin (0,∞).Define for x > 0φA/H(x) = lnA(x, x+ 1)H(x, x+ 1)= 2 ln(x+12)− lnx− ln(x+ 1), (17)then we have for nonnegative integer n(−1)nφ(n)A/H(x) =∫ ∞0(et − 2et/2 + 1)tn−1e−(x+1)t dt=∞∑k=2(1− 12k−1)1k!∫ ∞0tn+k−1e−(x+1)t dt> 0.(18)Therefore, it follows that the ratio A(x,x+1)H(x,x+1) is strictly logarithmically completelymonotonic in (0,∞).Define for x > 0φI/G(x) = lnI(x, x+ 1)G(x, x+ 1)= x ln(1 +1x)+12ln(x+ 1)− 12lnx− 1. (19)By standard argument above, differentiation for nonnegative integer n yields(−1)nφ(n)I/G(x) =12∫ ∞0(2et − tet − t− 2)tn−2e−(x+1)t dt=12∞∑k=3k − 2k!∫ ∞0tn+k−2e−(x+1)t dt> 0.(20)This shows that the ratio I(x,x+1)G(x,x+1) is also strictly logarithmically completely mono-tonic in (0,∞).Define for x > 0φI/H(x) = lnI(x, x+ 1)H(x, x+ 1)= x ln(1 +1x)+ ln(x+12)− lnx− 1. (21)By the same procedure as above, we obtain for n ∈ N(−1)nφ(n)I/H(x) =∫ ∞0(et + tet/2 − tet − t− 1)tn−2e−(x+1)t dt=∞∑k=3(1− 1k− 12k−1)1(k − 1)!∫ ∞0tn+k−2e−(x+1)t dt> 0.(22)4 CH.-P. CHEN AND F. QIThus, it is proved that I(x,x+1)H(x,x+1) is also strictly logarithmically completely monotonicin (0,∞).Define for x > 0φA/L(x) =A(x, x+ 1)L(x, x+ 1)=(x+12)ln(1 +1x). (23)Straightforward differentiating, using formulas (12) and (13) and expanding thefunction 2et − tet − t− 2 at 0 yields(−1)nφ(n)A/L(x) = −12∫ ∞0(2et − tet − t− 2)tn−2e−(x+1)t dt=12∞∑k=3k − 2k!∫ ∞0tn+k−2e−(x+1)t dt> 0.(24)This tell us that the ratio A(x,x+1)L(x,x+1) is strictly completely monotonic in (0,∞).The proof is complete. In the final, as an applications of Theorem 1, we give the following remark.Remark 1. As stated above, a strictly logarithmically completely monotonic func-tion is also strictly completely monotonic. As a result, we deduce from Theorem 1thate(1− 12x+ 1)< e√xx+ 1<(1 +1x)x< e(1− 12x+ 2)(25)for x > 0. Inequality (25) can be found in [14, 19].By using the right-hand side of (25), Yang in [19] obtained a strengthenedHardy’s inequality:∞∑n=1λn+1(aλ11 aλ22 · · · aλnn)1/Λn< e∞∑n=1[1− λn2(Λn + λn)]λnan, (26)where 0 < λn+1 ≤ λn, Λn =∑nm=1 λm and an ≥ 0 for n ∈ N, 0 <∑∞n=1 λnan <∞.In particular, if setting λn ≡ 1, then (26) becomes the following strengthenedCarleman’s inequality [19]:∞∑n=1(a1a2 · · · an)1/n < e∞∑n=1[1− 12(n+ 1)]an. (27)References[1] K. Ball, Completely monotonic rational functions and Hall’s marriage theorem, J. Combin.Theory Ser. B 61 (1994), 118-124.[2] C. Berg, J. P. R. Christensen and P. Ressel, Harmonic Analysis on Semigroups. Theoryof Positive Definite and Related Functions, Graduate Texts in Mathematics 100, Springer,Berlin-Heidelberg-New York, 1984.[3] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergebnisse derMath. 87, Springer, Berlin, 1975.[4] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions andDensities, Lecture Notes in Statistics 76, Springer, New York, 1992.[5] W. A. Day, On monotonicity of the relaxation functions of viscoelastic meterial, Proc. Cam-bridge Philos. Soc. 67 (1970), 503–508.[6] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure andApplied Mathematics 97, Addison Wesley Longman Limited, 1998.LOGARITHMICALLY COMPLETELY MONOTONIC RATIOS OF MEAN VALUES 5[7] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, NewYork, 1966.[8] C. L. Frenzen, Error bounds for asymptotic expansions of the ratio of two gamma functions,SIAM J. Math. Anal. 18 (1987), no. 3, 890–896.[9] L. Galvani, Dei limiti a cui tendono alcune media, Boll. Un. Mat. Ital. (1927), no. 6, 173–179.[10] C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math.10 (1974), 152–164.[11] F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math.Anal. Appl. 296 (2004), no. 2, 603–607. A complete monotonicity of the gamma function,RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 1. Available online at http://rgmia.vu.edu.au/v7n1.html.[12] F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma anddigamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8. Available online athttp://rgmia.vu.edu.au/v7n1.html.[13] F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving thegamma and polygamma functions, J. Austral. Math. Soc. Ser. A (2004), in press. RGMIA Res.Rep. Coll. 7 (2004), no. 1, Art. 5. Available online at http://rgmia.vu.edu.au/v7n1.html.[14] J. Sa´ndor, On certain inequalies involving the constant e and their applications, J. Math.Anal. Appl. 249 (2000), 569–582.[15] K. B. Stolarsky, Generalizations of the logarithmic mean, Mag. Math. 48 (1975), 87–92.[16] K. B. Stolarsky, The power and generalized logarithmic means, Amer. Math. Monthly 87(1980), 545–548.[17] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.[18] J. Wimp, Sequence Transformations and Their Applications, Academic Press, Nex York,1981.[19] B.-Ch. Yang, Note on Hardy’s inequality, J. Math. Anal. Appl. 234 (1999), 717–722.(Ch.-P. Chen) Department of Applied Mathematics and Informatics, Research In-stitute of Applied Mathematics, Henan Polytechnic University, Jiaozuo City, Henan454010, ChinaE-mail address: chenchaoping@hpu.edu.cn(F. Qi) Department of Applied Mathematics and Informatics, Research Institute ofApplied Mathematics, Henan Polytechnic University, Jiaozuo City, Henan 454010, ChinaE-mail address: qifeng@hpu.edu.cn, fengqi618@member.ams.orgURL: http://rgmia.vu.edu.au/qi.html, http://dami.hpu.edu.cn/qifeng.html

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Logarithmically Completely Monotonic Ratios of Mean Values and an Application

Victoria University Institutional Repository (VUIR)

Logarithmically Completely Monotonic Ratios of Mean Values and an Application

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