COMPLETE MONOTONICITIES OF FUNCTIONS INVOLVING THE GAMMA AND DIGAMMA FUNCTIONS. 1. Introduction

Transcription

1 COMPLETE MONOTONICITIES OF FUNCTIONS INVOLVING THE GAMMA AND DIGAMMA FUNCTIONS FENG QI AND BAI-NI GUO Abstract. In the article, the completely monotonic results of the functions [Γ( + 1)] 1/, [Γ(+α+1)]1/(+α), [Γ(+1)]1/ [Γ(+1)] 1/ (+1) α and [Γ(+1)]1/ α in (1, ) for α R are obtained. In the final, three open problems are posed. 1. Introduction The classical gamma function is usually defined for Re z > 0 by Γ(z) = 0 t z1 e t dt. (1) The psi or digamma function ψ() = Γ () Γ(), the logarithmic derivative of the gamma function, and the polygamma functions can be epressed (See [1, 8] and [12, p. 16]) for > 0 and k N as ψ() = γ + n=0 ψ (k) () = (1) k+1 k! ( n 1 ), (2) + n i=0 1, () ( + i) k+1 where γ = is the Euler-Mascheroni constant. A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I which alternate successively in sign, that is (1) n f (n) () 0 (4) 2000 Mathematics Subject Classification. Primary B15; Secondary 26A48, 26A51. Key words and phrases. complete monotonicity, gamma function, digamma function, psi function. The authors were supported in part by NNSF (# ) of China, SF for the Prominent Youth of Henan Province (# ), SF of Henan Innovation Talents at Universities, Doctor Fund of Jiaozuo Institute of Technology, CHINA. This paper was typeset using AMS-LATEX. 1

2 2 F. QI AND B.-N. GUO for I and n 0. If inequality (4) is strict for all I and for all n 0, then f is said to be strictly completely monotonic. For more information, please refer to [14, 15, 18, 2, 25] and references therein. A function f is said to be logarithmically completely monotonic on an interval I if its logarithm ln f satisfies (1) k [ln f()] (k) 0 (5) for k N on I. If inequality (5) is strict for all I and for all k 1, then f is said to be strictly logarithmically completely monotonic. In this article, using Leibnitz s formula and the formulas (2) and (), the complete monotonicity properties of the functions [Γ( + 1)] 1/, [Γ(+α+1)]1/(+α), [Γ(+1)]1/ [Γ(+1)] 1/ (+1) α and [Γ(+1)]1/ α in (1, ) for α R are obtained. From these, some well known results are deduced, etended and generalized. The main results of this paper are as follows. Theorem 1. The function [Γ( + 1)] 1/ is strictly increasing in (1, ). function ψ(+1) The ln Γ(+1), the logarithmic derivative of [Γ( + 1)] 1/, is strictly 2 completely monotonic in (1, ). The function [Γ(+α+1)]1/(+α) [Γ(+1)] 1/ strictly completely monotonic with (1, ) for α > 0. Theorem 2. For α 1, the function [Γ(+1)]1/ function ln Γ(+1) 2 (+1) α is logarithmically is strictly decreasing and the ψ(+1) + α +1, the logarithmic derivative of (+1) α, is strictly [Γ(+1)] 1/ completely monotonic with (1, ). Let τ(s, t) = 1 s [ t (t + s + 1) ( t t+1) s+1 ] > 0 for (s, t) N (0, ) and τ0 = τ(s 0, t 0 ) > 0 be the maimum of τ(s, t) on the set N (0, ). For a given real number α satisfying α 1 1+τ 0 < 1, the function [Γ(+1)]1/ (+1) is strictly increasing and α the function ψ(+1) ln Γ(+1) α 2 +1 is strictly completely monotonic in (1, ). Theorem. For α 0, the function [Γ(+1)]1/ α function ψ(+1) ln Γ(+1) 2 α completely monotonic in (0, ). decreasing and the function in (0, ). is strictly increasing and the [Γ(+1)]1/, the logarithmic derivative of, is strictly α ln Γ(+1) 2 For α 1, the function [Γ(+1)]1/ ψ(+1) + α α is strictly is strictly completely monotonic For α 0 such that α is real in (1, 0), the function [Γ(+1)]1/ α decreasing and the function ln Γ(+1) 2 ψ(+1) + α is strictly is strictly completely monotonic

3 MONOTONICITIES OF FUNCTIONS INVOLVING GAMMA AND DIGAMMA FUNCTIONS in (1, 0). For α 1 such that α is real in (1, 0), the function [Γ(+1)]1/ α strictly increasing and the function ψ(+1) monotonic in (1, 0). ln Γ(+1) 2 α is is strictly completely Theorem 4. A (strictly) logarithmically completely monotonic function is also (strictly) completely monotonic. As a direct consequence of combining Theorem 1 with Theorem 4, we have the following corollary. Corollary 1. The function [Γ(+α+1)]1/(+α) [Γ(+1)] 1/ (1, ) for α > 0. is strictly completely monotonic with In [] and [4, p. 8], the following result was given: Let f and g be functions such that f g is defined. If f and g are completely monotonic, then f g is also completely monotonic. Thus, from Theorem 1 and Theorem 2 and the fact that the eponential function e is strictly completely monotonic in (, ), the following corollary can be deduced. Corollary 2. The following complete monotonicity properties holds: (1) The function 1 [Γ(+1)] 1/ (2) For α 1, the function [Γ(+1)]1/ (+1) α is strictly completely monotonic in (1, ). is strictly completely monotonic in (1, ). For a given real number α with α 1 1+τ 0 < 1, the function (+1) α [Γ(+1)] 1/ is strictly completely monotonic in (1, ). () For α 0, the function α [Γ(+1)] 1/ (0, ). For α 1, the function [Γ(+1)]1/ α is strictly completely monotonic in is strictly completely monotonic in (0, ). For α 0 such that α is real in (1, 0), the function [Γ(+1)] 1/ α is strictly completely monotonic in (1, 0). For α 1 such that α is real in (1, 0), the function in (1, 0). Proof of Theorem 1. For α > 0, let α [Γ(+1)] 1/ 2. Proofs of theorems strictly completely monotonic for > 1. f α () = [Γ( + α + 1)]1/(+α) [Γ( + 1)] 1/ (6)

4 4 F. QI AND B.-N. GUO By direct calculation and using Leibnitz s formula and formulas (2) and (), we obtain for n N, ln Γ( + α + 1) ln f α () = + α g (n) () = 1 n n+1 k=0 ln Γ( + 1) (1) nk n! k ψ (k1) ( + 1) k! h n() = n ψ (n) ( + 1) > 0, if n is odd and (0, ), 0, g( + α) g(), h n() n+1, (7) if n is odd and (1, 0] or n is even and (1, ), where ψ (1) ( + 1) = ln Γ( + 1) and ψ (0) ( + 1) = ψ( + 1). Hence, the function h n () increases if n is odd and (0, ) and decreases if n is odd and (1, 0) or n is even and (1, ). Since h n (0) = 0, it is easy to see that h n () 0 if n is odd and (1, ) or n is even and (1, 0) and h n () 0 if n is even and (0, ). Then, for (1, ), we have g (n) () 0 if n is odd and g (n) () 0 if n is even. Since lim ψ (k) (+1) n+1 see that lim g (n) () = lim h n() n+1 (8) = 0 for 1 k n, it is easy to = 0. Therefore (1) n+1 g (n) () > 0 with (1, ) for n N. Then the function g () is strictly completely monotonic and [Γ( + 1)] 1/ = ep(g()) is strictly increasing in (1, ). From (1) n+1 g (n) () 0 with (1, ) for n N, it follows that g 2k () increases and g 2k1 () decreases with (1, ) for all k N. This implies that (1) n [ln f α ()] (n) 0, and then the function [Γ(+α+1)]1/(+α) [Γ(+1)] 1/ completely monotonic with (1, ). Proof of Theorem 2. Let for (1, ). Then for n N, ν α () = is logarithmically [Γ( + 1)]1/ ( + 1) α (9) ln Γ( + 1) ln ν α () = α ln( + 1), (10) [ln ν α ()] (n) = 1 [h n+1 n () + (1)n (n 1)!α n+1 ] ( + 1) n µ α,n(), (11) n+1 µ α,n() = n ψ (n) ( + 1) + (1)n (n 1)!α n ( + n + 1) ( + 1) n+1

5 MONOTONICITIES OF FUNCTIONS INVOLVING GAMMA AND DIGAMMA FUNCTIONS 5 [ ] = n ψ (n) ( + 1) + (1)n (n 1)!α ( + 1) n + (1)n n!α ( + 1) n+1 = {(1) n n+1 1 n! ( + i) n+1 [ ] + (1) n 1 (n 1)!α ( + i) n 1 ( + i + 1) n [ ]} + (1) n 1 n!α ( + i) n+1 1 ( + i + 1) n+1 [ = (1) n (n 1)! n α ( + i) n α ( + i + 1) n ] nα n(α 1) + ( + i + 1) n+1 ( + i) n+1 (12) = (n 1)!() n [αy + n(α 1)](y + 1) n+1 α(y + n + 1)y n+1 y n+1 (y + 1) n+1 = (n 1)!() n α[(y + n)(y + 1) n+1 (y + n + 1)y n+1 ] n(y + 1) n+1 y n+1 (y + 1) n+1 = n!() n where y = + i > 0. { [ 1 y n+1 α ( y y (y + n + 1) n y + 1 ) n+1 ] } 1, In [5, p. 28] and [11, p. 154], the Bernoulli s inequality states that if 1 and 0 and if α > 1 or if α < 0 then (1 + ) α > 1 + α. This means that 1 + s+1 t < ( ) s+1 ( t for t > 0, which is equivalent to t (t + s + 1) t s+1 t+1) > 0 for t > 0, and then τ(s, t) > 0 for s 1 and t > 0 and τ(s, 0) = 0. From τ(s, t) > 0, it is deduced that [αy+n(α1)](y+1) n+1 α(y+n+1)y n+1 > 0 for y = + i > 0 and n N if α 1. Therefore, for α 1, we have > 0, if n is even and (1, 0) (0, ) or n is odd and (1, 0), µ α,n() < 0, if n is odd and (0, ), and then µ α,n () is strictly increasing with (1, ) if n is even or with (1, 0) if n is odd and µ α,n () is strictly decreasing with (0, ) if n is odd. Since µ α,n (0) = 0, thus µ α,n () < 0 with > 1 and 0 if n is odd or with (1, 0) if n is even and µ α,n () > 0 with (0, ) if n is even. Therefore, from lim [ln ν α ()] (n) = 0, it is deduced that [ln ν α ()] (n) > 0 if n is even

6 6 F. QI AND B.-N. GUO and [ln ν α ()] (n) < 0 if n is odd, which is equivalent to (1) n [ln ν α ()] (n) > 0 in (1, ) for n N and α 1. Hence, if α 1, then the function [Γ(+1)]1/ (+1) α strictly decreasing and the function monotonic in (1, ). ln Γ(+1) 2 ψ(+1) + α +1 is is strictly completely It is clear that τ 0 > 0. When α 1 1+τ 0 < 1, it follows that µ α,n() < 0 and µ α,n () is decreasing with (1, ) and 0 for n an even integer or with (1, 0) for n an odd integer, and µ α,n() > 0 and µ α,n () is increasing with (0, ) for n an odd integer. Since µ α,n (0) = 0 and lim [ln ν α ()] (n) = 0, we have [ln ν α ()] (n) < 0 for n an even and [ln ν α ()] (n) > 0 for n an odd in (1, ), this implies that (1) n+1 [ln ν α ()] (n) > 0 in (1, ) for n N. Therefore ν α () is strictly increasing and (1) n1 {[ln ν α ()] } (n1) > 0 in (1, ) for n N. Hence, if α 1 1+τ 0, then the function [Γ(+1)]1/ (+1) is strictly increasing and α the function ψ(+1) ln Γ(+1) α 2 +1 is strictly completely monotonic in (1, ). Proof of Theorem. The procedure is same as the one of Theorem 2. Hence, we leave it to the readers. Proof of Theorem 4. It is clear that ep φ() 0. Further, it is easy to see that [ep φ()] = φ () ep φ() 0 and [ep φ()] = {φ () + [f ()] 2 } ep φ() 0. Suppose (1) k [ep φ()] (k) 0 for all nonnegative integers k n, where n N is a given positive integer. By Leibnitz s formula, we have (1) n+1 [ep φ()] (n+1) = (1) n+1 {[ep φ()] } (n) = n i=0 0. = (1) n+1 [φ () ep φ()] (n) = (1) n+1 n i=0 ( ) n φ (i+1) ()[ep φ()] (ni) i ( ) n [(1) i+1 φ (i+1) () ]{ (1) ni [ep φ()] (ni)} i (1) By induction, it is proved that the function ep φ() is completely monotonic.

7 MONOTONICITIES OF FUNCTIONS INVOLVING GAMMA AND DIGAMMA FUNCTIONS 7. Remarks Remark 1. In [10, 1], among other things, the following monotonicity results were obtained [Γ(1 + k)] 1/k < [Γ(2 + k)] 1/(k+1), k N; [ ( Γ 1 + )] 1 decreases with > 0. These are etended and generalized in [16]: The function [Γ(r)] 1/(r1) is increasing in r > 0. Clearly, Theorem 1 generalizes and etends these results for the range of the argument. Remark 2. It is proved in [19] that the function 1 ln Γ( + 1) ln + 1 is strictly completely monotonic on (0, ) and tends to + as 0 and to 0 as. A similar result was found in [24]: The function ln Γ( + 1) ln( + 1) is strictly completely monotonic on (1, ) and tends to 1 as 1 and to 0 as. Our main results generalize these ones. Remark. From our main results, the following can be deduced: Let n be natural number. Then the sequence n n! n+k (n+k+1)! are increasing with n N. Remark 4. A function f is logarithmic conve on an interval I if f is positive and ln f is conve on I. Since f() = ep[ln f()], it follows that a logarithmic conve function is conve. Remark 5. Straightforward computation shows that the maimum τ 2 of τ(2, t) in (0, ) is τ ( 2, ) = ( ) ( ) ( ) = (14) and the maimum τ of τ(, t) in (0, ) is ( τ, ) = (15) 4 If α 1 1+τ 2 = , then µ α,2() 0 and µ α,2 () decreases in (1, ). Since µ α,2 (0) = 0 and lim [ln ν α ()] (2) = 0, it is obtained that

8 8 F. QI AND B.-N. GUO [ln ν α ()] (2) < 0. Therefore the function ν α () = [Γ(+1)]1/ (+1) α and strictly logarithmically concave for α 1 1+τ 2 in (1, ). is strictly increasing If α 1 1+τ = , then µ α,() < 0 and µ α, () decreases in (1, 0) and µ α,() > 0 and µ α, () increases in (0, ). Thus µ α, () 0 and then [ln ν α ()] () > 0 in (1, ). Hence [ln ν α ()] (2) is strictly increasing in (1, ) if α 1 1+τ. Mathematica shows that τ 0 > Remark 6. The motivation of this paper has been eposited in detail in [21] and a lot of literature is listed therein. Please also refer to [2, 6, 7, 9, 17, 20, 22]. 4. Open problems A function f(t) is said to be absolutely monotonic on an interval I if it has derivatives of all orders and f (k) (t) 0 for t I and k N. A function f(t) is said to be regularly monotonic if it and its derivatives of all orders have constant sign (+ or ; not all the same) on (a, b). A function f(t) is said to be absolutely conve on (a, b) if it has derivatives of all orders and f (2k) (t) 0 for t (a, b) and k N. The function [Γ(+α+1)]1/(+α) can be epressed as [Γ(+1)] 1/ +α t 0 +α e t dt, (16) t 0 e t dt where 0 e t dt = 1. Then we propose the following Open Problem 1. Let w() 0 be a nonnegative weight defined on a domain Ω with w() d = 1. Find conditions about w() and f() 0 such that the ratio Ω between two power means Q(t) = [ Ω w()f t+α () d ] 1/(t+α) [ Ω w()f t () d ] 1/t is completely (absolutely, regularly) monotonic (conve) with t R for a given number α > 0. Open Problem 2. Find conditions about α and β such that the ratio F() = (17) [Γ( + 1)]1/ ( + β) α (18) is completely (absolutely, regularly) monotonic (conve) with > 1.

9 MONOTONICITIES OF FUNCTIONS INVOLVING GAMMA AND DIGAMMA FUNCTIONS 9 Open Problem. For (s, t) N (0, ), find the maimum of the following τ(s, t) = 1 [ ( t ) ] s+1 t (t + s + 1). (19) s t + 1 References [1] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, [2] H. Alzer, On some inequalities for the gamma and psi function, Math. Comp. 66 (1997), [] H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, [4] S. Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley-Los Angeles, [5] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman Limited, [6] Ch.-P. Chen and F. Qi, Inequalities relating to the gamma function, J. Inequal. Pure Appl. Math. (2004), accepted. [7] Ch.-P. Chen and F. Qi, Monotonicity results for the gamma function, J. Inequal. Pure Appl. Math. (200), no. 2, Art. 44. Available online at html. RGMIA Res. Rep. Coll. 5 (2002), suppl., Art. 16. Available online at vu.edu.au/v5(e).html. [8] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, McGraw-Hill ook Company, New York-Toronto-London, 195. [9] B.-N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (200), no. 2, Available online at tw/tjm/abstract/006/tjm006_6.html. [10] D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 119 (1985), [11] J.-Ch. Kuang, Chángyòng Bùděngshì (Applied Inequalities), 2nd edition, Hunan Education Press, Changsha, China, 199. (Chinese) [12] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, [1] H. Minc and L. Sathre, Some inequalities involving (r!) 1/r, Proc. Edinburgh Math. Soc. 14 (1965/66), [14] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 199. [15] F. Qi, Logarithmic conveity of etended mean values, Proc. Amer. Math. Soc. 10 (2002), no. 6, Available online at

10 10 F. QI AND B.-N. GUO S X. RGMIA Res. Rep. Coll. 2 (1999), no. 5, Art. 5, Available online at [16] F. Qi, Monotonicity results and inequalities for the gamma and incomplete gamma functions, Math. Inequal. Appl. 5 (2002), no. 1, RGMIA Res. Rep. Coll. 2 (1999), no. 7, Art. 7, Available online at [17] F. Qi, On a new generalization of Martins inequality, RGMIA Res. Rep. Coll. 5 (2002), no., Art. 1, Available online at [18] F. Qi, The etended mean values: definition, properties, monotonicities, comparison, conveities, generalizations, and applications, Cubo Mathematica Educational 6 (2004), no. 1, in press. RGMIA Res. Rep. Coll. 5 (2002), no. 1, Art. 5. Available online at [19] F. Qi and Ch.-P. Chen, A complete monotonicity of the gamma function, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 1. Available online at [20] F. Qi and Ch.-P. Chen, Monotonicity and conveity results for functions involving the gamma function, RGMIA Res. Rep. Coll. 6 (200), no. 4, Art. 10. Available online at vu.edu.au/v6n4.html. [21] F. Qi and B.-N. Guo, Monotonicity and conveity of the function Γ(+1) +α Γ(+α+1), RGMIA Res. Rep. Coll. 6 (200), no. 4, Art. 16. Available online at html. [22] F. Qi and B.-N. Guo, Some inequalities involving the geometric mean of natural numbers and the ratio of gamma functions, RGMIA Res. Rep. Coll. 4 (2001), no. 1, Art. 6, Available online at [2] F. Qi and S.-L. Xu, The function (b a )/: Inequalities and properties, Proc. Amer. Math. Soc. 126 (1998), no. 11, Available online at journal-getitem?pii=s [24] H. Vogt and J. Voigt, A monotonicity property of the Γ-function, J. Inequal. Pure Appl. Math. (2002), no. 5, Art. 7. Available online at [25] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, (F. Qi) Department of Applied Mathematics and Informatics and Institute of Applied Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan , CHINA address: qifeng@jzit.edu.cn, fengqi618@member.ams.org URL: (B.-N. Guo) Department of Applied Mathematics and Informatics and Institute of Applied Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan , CHINA address: guobaini@jzit.edu.cn