Chen, Chao-Ping

Qi, Feng

English

Victoria University Eprints Repository

COMPLETELY MONOTONIC FUNCTIONS RELATED TO THEGAMMA FUNCTIONSCHAO-PING CHEN AND FENG QIAbstract. (i) Let a, b > 0 be real numbers, and letfa,b(x) =1xb−a[Γ(bx+ 1)Γ(ax+ 1)]1/x.Then, for x > 0 and n = 1, 2, . . ., (−1)n(ln fa,b(x))(n) ≷ 0 according as b ≷ a.(ii) Let p > 0 be a real number, and let fp(x) = θ(px)− pθ(x), whereθ(x) =∫ ∞0(tet − 1 − 1 +t2)e−xtt2dt, x > 0is remainder of Binet’s formula. Then, for x > 0 and n = 0, 1, 2, . . .,(−1)nf (n)p (x) ≷ 0 according as p ≶ 1.1. IntroductionThe Euler gamma function Γ and its logarithmic derivative ψ, the so-calleddigamma function, are defined for Re z > 0 byΓ(z) =∫ ∞0tz−1e−t dt and ψ(z) =Γ′(z)Γ(z).There exists a very extensive literature on these functions. In particular, inequal-ities, monotonicity and complete monotonicity properties for these functions havebeen published, we refer to the paper [1] and [2], and the references given therein.We recall that a function f is said to be completely monotonic on an interval I, iff has derivatives of all orders on I and satisfies(−1)nf (n)(x) ≥ 0 (x ∈ I;n = 0, 1, 2, . . .). (1)If the inequality (1) is strict, then f is said to be strictly completely monotonicon I. Completely monotonic functions have remarkable applications in differentbranches. For instance, they play a role in potential theory [4], probability theory[6, 8, 10], physics [7], numerical and asymptotic analysis [9, 15], and combinatorics[3]. A detailed collection of the most important properties of completely monotonicfunctions can be found in [14, Chapter IV], and in an abstract in [5].In a recent paper [12], the terminology “(strictly) logarithmically completelymonotonic function” was introduced. It was also shown in this paper that a(strictly) logarithmically completely monotonic function is also (strictly) completely2000 Mathematics Subject Classification. 33B15; 26A48.Key words and phrases. Gamma function, psi function, complete monotonicity, logarithmicallycomplete monotonicity, Binet’s formula, remainder.The authors were supported in part by SF of Henan Innovation Talents at Universities, China.12 CH.-P. CHEN AND F. QImonotonic. For convenience, we recall that a positive function f is said to be log-arithmically completely monotonic on an interval I if its logarithm ln f satisfies(−1)n(ln f(x))(n) ≥ 0 (x ∈ I;n = 1, 2, . . .). (2)If inequality (2) is strict, then f is said to be strictly logarithmically completelymonotonic.In 2003, J. Sa´ndor [13] showed thatlimx→∞1xb−a[Γ(bx+ 1)Γ(ax+ 1)]1/x=bbaaeb−a. (3)Our first theorem considers logarithmically complete monotonicity property ofthe function in (3).Theorem 1. Let a, b > 0 be real numbers, and letfa,b(x) =1xb−a[Γ(bx+ 1)Γ(ax+ 1)]1/x.Then, for x > 0 and n = 1, 2, . . ., (−1)n(ln fa,b(x))(n) ≷ 0 according as b ≷ a.If we denote byI(a, b) =1e(bbaa)1/(b−a), a > 0, b > 0, a 6= b,the so-called identric mean, then, we yield from (3) and the monotonicity of thefunction fa,b that, for x > 0,1xb−a[Γ(bx+ 1)Γ(ax+ 1)]1/x≷ [e2I(a, b)]b−a according as b ≷ a. (4)Binet’s formula [16, p. 103] states that for x > 0,ln Γ(x) =(x− 12)lnx− x+ ln√2pi + θ(x),whereθ(x) =∫ ∞0(tet − 1 − 1 +t2)e−xtt2dt. (5)Let p > 0 be a real number. Our second theorem considers complete monotonic-ity property of the function x 7→ θ(px)− pθ(x) on (0,∞).Theorem 2. Let p > 0 be a real number, and let fp(x) = θ(px) − pθ(x), whereθ(x) is defined by (5). Then, for x > 0 and n = 0, 1, 2, . . .,(−1)nf (n)p (x) ≷ 0 according as p ≶ 1.2. Proofs of TheoremsProof of Theorem 1. Using Leibniz’ rule[u(x)v(x)](n) =n∑k=0(nk)u(k)(x)v(n−k)(x),we obtain(ln fa,b(x))(n) = − (b− a)(−1)n−1(n− 1)!xnCOMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTIONS 3+n∑k=0(nk)(1x)(n−k)[ln Γ(bx+ 1)− ln Γ(ax+ 1)](k)= − (b− a)(−1)n−1(n− 1)!xn+(−1)nn!xn+1[ln Γ(bx+ 1)− ln Γ(ax+ 1)]+(−1)nn!xn+1n∑k=1(−1)kk!xk[bkψ(k−1)(bx+ 1)− akψ(k−1)(ax+ 1)].Define for x > 0,ga,b(x) =(−1)nxn+1n!(ln f(x))(n)=(b− a)xn+ lnΓ(bx+ 1)− ln Γ(ax+ 1)+n∑k=1(−1)kk!xk[bkψ(k−1)(bx+ 1)− akψ(k−1)(ax+ 1)].Using the representations(n− 1)!xn=∫ ∞0tn−1e−xt dt, (x > 0),ψ(n)(x) = (−1)n+1∫ ∞0tn1− e−t e−xt dt, (x > 0, n = 1, 2, . . .),see [11, p. 16], we implyn!xng′a,b(x) =(b− a)(n− 1)!xn+ (−1)n[bn+1ψ(n)(bx+ 1)− an+1ψ(n)(ax+ 1)]= (b− a)∫ ∞0tn−1e−xt dt−∫ ∞0bn+1tnet − 1 e−bxt dt+∫ ∞0an+1tnet − 1 e−axt dt= (b− a)∫ ∞0tn−1e−xt dt−∫ ∞0tnet/b − 1e−xt dt+∫ ∞0tnet/a − 1e−xt dt=∫ ∞0[(tet/a − 1 − a)−(tet/b − 1 − b)]tn−1e−xt dt.For fixed t > 0, we define the functionht(a) =tet/a − 1 − a (a > 0).Differentiation yieldsh′t(a) =(t/a)2et/a − (et/a − 1)2(et/a − 1)2 .Now we are in a position to prove h′t(a) < 0 for a > 0, which is equivalent to(t/a)et/(2a) < et/a − 1,i.e.,(t/a) < et/(2a) − e−t/(2a).4 CH.-P. CHEN AND F. QIUsing power series expansion, we haveet/(2a) − e−t/(2a) − (t/a) = 2∞∑n=21(2n− 1)!(t2a)2n−1> 0for a > 0. Hence h′t(a) < 0 for a > 0, and then, for x > 0, g′a,b(x) ≷ 0 and ga,b(x) ≷ga,b(0) = 0 according as b ≷ a. This implies that for x > 0 and n = 1, 2, . . .,(−1)n(ln fa,b(x))(n) ≷ 0 according as b ≷ a. The proof is complete. Proof of Theorem 2. By (5), we implyfp(x) =∫ ∞0(ueu − 1 − 1 +u2)e−pxuu2du− p∫ ∞0(tet − 1 − 1 +t2)e−xtt2dt= p∫ ∞0[tp(et/p − 1) − 1 +t2p]e−xtt2dt− p∫ ∞0(tet − 1 − 1 +t2)e−xtt2dt= p∫ ∞0[tp(et/p − 1) −1et − 1 +1− pp]e−xtt2dt=∫ ∞0δp(t)2(et/p − 1)(et − 1)te−xt dtand therefore,(−1)nf (n)p (x) =∫ ∞0tn−1δp(t)2(et/p − 1)(et − 1)e−xt dt.whereδp(t) = (1 + p)et − (1 + p)et/p + (1− p)e[(1+p)/p]t + p− 1=∞∑k=3[pk − 1 + (1− p)(1 + p)k−1] (1 + p)tkpk · k! .It is easy to see thatpk − 1 + (1− p)(1 + p)k−1 = (p− 1)k−1∑m=0pm + (1− p)k−1∑m=0(k − 1m)pm= (p− 1)k−2∑m=1[1−(k − 1m)]pm ≷ 0 according as p ≶ 1.This implies for x > 0 and n ≥ 0,(−1)nf (n)p (x) ≷ 0 according as p ≶ 1.The proof is complete. References[1] H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), 337–346.[2] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997),373–389.[3] K. Ball, Completely monotonic rational functions and Hall’s marriage theorem, J. Comb. Th.,Ser. B 61 (1994), 118–124.[4] C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergebnisse derMath. 87, Springer, Berlin, 1975.COMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTIONS 5[5] C. Berg, J. P. R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Pos-itive Definite and related Functions, raduate Texts in Mathematics 100, Springer, Berlin-Heidelberg-New York, 1984.[6] L. Bondesson, Generalized Gamma Convolutions and related Classes of Distributions andDensities, Lecture Notes in Statistics 76, Springer, New York, 1992.[7] W. A. Day, On monotonicity of the relaxation functions of viscoelastic meterial,Proc.Cambridge Philos. Soc. 67 (1970), 503–508.[8] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, NewYork, 1966.[9] 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.[10] C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequat. Math. 10(1974), 152–164.[11] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functionsof mathematical physics, Springer, Berlin, 1966.[12] F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math.Anal. Appl. 296 (2004), no. 2, 603–607.[13] J. Sa´ndor, On a limit involving the Euler-gamma function, Octogon Mathematical Magazine,11 (2004), no. 1, 262–264.[14] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, NJ, 1941.[15] J. Wimp, Sequence Transformations and their Applications, Academic Press, Nex York, 1981.[16] Zh.-X. Wang and D.-R. Guo, Introduction to Special Function, The Series of AdvancedPhysics of Peking University, Peking University Press, Beijing, China, 2000. (Chinese)(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, chenchaoping@sohu.com(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

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