A COMPLETELY MONOTONIC FUNCTION INVOLVING THE TRI- AND TETRA-GAMMA FUNCTIONS

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1 ao DOI:.2478/s Math. Slovaca 63 (23), No. 3, A COMPLETELY MONOTONIC FUNCTION INVOLVING THE TRI- AND TETRA-GAMMA FUNCTIONS Bai-Ni Guo* Jiao-Lian Zhao** Feng Qi* (Communicated by Ján Borsík ) ABSTRACT. The di-gamma function ψ(x) is defined on (, ) byψ(x) = Γ (x) Γ(x) and ψ (i) (x) fori N denote the polygamma functions, where Γ(x) is the classical Euler s gamma function. In this paper we prove that a function involving the difference between [ψ (x)] 2 ψ (x) and a proper fraction of x is completely monotonic on (, ). c 23 Mathematical Institute Slovak Academy of Sciences. Introduction We recall from [6: Chapter XIII] and [8: Chapter IV] that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and ( ) n f (n) (x) () for x I and n. The famous Bernstein-Widder Theorem (see [8: p. 6, Theorem 2a]) states that a function f(x) on[, ) is completely monotonic if and only if there exists a bounded and non-decreasing function α(t) such that f(x) = e xt dα(t) (2) 2 M a t h e m a t i c s Subject Classification: Primary 26A48, 33B5; Secondary 26A5, 26D, 65R. K e y w o r d s: completely monotonic function, tri-gamma function, tetra-gamma function, polygamma function, inequality. The last two authors were partially supported by the Natural Science Basic Research Plan in Shaanxi Province of China under grant No. 2JM5. Download Date /25/7 5: PM

2 BAI-NI GUO JIAO-LIAN ZHAO FENG QI converges for x [, ). This says that a completely monotonic function f(x) on [, ) is a Laplace transform of the measure α(t). We also recall that the classical Euler gamma function Γ(x) is defined by Γ(x) = t x e t dt, x >. (3) The logarithmic derivative of Γ(x), denoted by ψ(x) = Γ (x) Γ(x), is called the psi or di-gamma function, and the derivatives ψ (i) (x) for i N are respectively called the polygamma functions. In particular, the functions ψ (x) andψ (x) are called the tri-gamma and tetra-gamma functions. In [2: p. 28, (4.39)], it was established that the inequality holds for x>, where [ψ (x)] 2 ψ (x) > p(x) 9x 4 (x ) (4) p(x) =75x 9x 9 484x 8 537x x 6 455x 5 44x 4 297x 3 329x 2 36x 45. For more information on the background, motivation, and history of this topic, please refer to [3, 5, 3, 5], the survey and expository papers [7, 6] and plenty of references cited therein. The aim of this paper is to prove the complete monotonicity of the difference between two functions on both sides of the inequality (4). Our main result may be stated as the following theorem. Theorem. The function g(x) =[ψ (x)] 2 ψ p(x) (x) (6) 9x 4 (x ) is completely monotonic on (, ), wherethefunctionp(x) is defined by (5). Remark. By the definition of completely monotonic functions and the above recited Bernstein-Widder Theorem, it is easy to see that our Theorem is stronger than the inequality (4), so our Theorem generalizes the inequality (4). (5) 2. Proof of Theorem Now we are in a position to verify our Theorem by a simple but effectual approach, which has been used in [3,5,3,5] and others, and by a large amount of calculating derivatives. By the recursion formula ψ (n ) (x )=ψ (n ) (x) ( )n (n )! x n (7) 47 Download Date /25/7 5: PM

3 A COMPLETELY MONOTONIC FUNCTION for x>andn N, see [: pp. 258, 26; 6.3.5, 6.4.6], a direct calculation produces g(x) g(x )= [ ψ (x) ψ (x ) ][ ψ (x)ψ (x ) ] [ ψ (x) ψ (x ) ] [ ] p(x) 9x 4 (x ) p(x ) 9(x ) 4 (x 2) = [ x 2 2ψ (x) ] x 2 2 [ ] x 3 p(x) 9x 4 (x ) p(x ) 9(x ) 4 (x 2) = 2 [ ψ (x) x 2 2x 3 4x (x ) 5 (x 2) 25 2(x ) (x 2) 7 2 6(x ) 7 3 2(x 2) 3 3 9(x ) (x 2) 4 3 8(x ) (x 2) 5 2(x ) 6 6(x 2) 6 8(x ) (x 2) 7 2(x ) 3 8 6(x 2) 8 ] 9(x ) 9 45(x 2) 9 8(x ) 45(x 2) 2 x 2 H(x). Using the formula x = r Γ(r) t r e xt dt (8) for r>andx>, see [: p. 255, 6..], and the integral representations ψ (n) (x) =( ) n t n e t e xt dt (9) for n N and x>, see [: p. 26, 6.4.], gives H(x) = ( t e t t 28 5 e t 5 e 2t 25 2 te t 33 2 te 2t 7 2 t2 e t 7 24 t2 e 2t 3 54 t3 e t t3 e 2t t4 e t t4 e 2t 44 t5 e t 72 t5 e 2t 296 t6 e t t6 e 2t 8 t7 e t t7 e 2t 47 Download Date /25/7 5: PM

4 = BAI-NI GUO JIAO-LIAN ZHAO FENG QI t8 e t 844 t8 e 2t t9 e t ) t9 e 2t e xt dt [ 63296(t 2)e 3t e 2t( t 9 8t 8 e t 648t 7 54t t t t t t ) e t( 5t 9 54t t t t 5 972t t t t ) 4 ( t 9 9t 8 72t 7 8t t t t t t )] e (x2)t dt e t θ(t)e (x2)t dt. A straightforward computation yields θ (t) = 63296(3t 5)e 3t ( t t t 3 458t 4 648t 5 466t 6 44t 7 27t 8 2t 9) e 2t ( t 87856t t t t t t 7 9t 8 5t 9) e t 36 ( t 674t t 3 26t 4 672t 5 546t 6 8t 7 t 8), θ (t) = (3t 4)e 3t 4 ( t t t t 4 836t t 6 774t 7 9t 8 t 9) e 2t ( t t 2 332t t t t 6 396t 7 36t 8 5t 9) e t 44 ( t 5922t 2 26t 3 84t 4 89t 5 4t 6 2t 7), θ (3) (t) = (t )e 3t 4 ( t t t t t t 6 62t 7 9t 8 2t 9) e 2t ( t t t t t 5 Download Date /25/7 5: PM

5 A COMPLETELY MONOTONIC FUNCTION 24488t t 7 8t 8 5t 9) e t 8 ( t 54t 2 48t 3 585t 4 2t 5 2t 6), θ (4) (t) = (3t 2)e 3t 6 ( t t t t t t 6 828t 7 t 9) e 2t ( t 27224t t t t 5 592t 6 324t 7 26t 8 5t 9) e t 296 ( 4 9t 2t 2 95t 3 5t 4 t 5), θ (5) (t) = (3t )e 3t 6 ( t t t t t t 6 656t 7 9t 8 2t 9) e 2t ( t t t t t 5 736t 6 26t 7 7t 8 5t 9) e t 648 ( 8 48t 7t 2 4t 3 t 4), θ (6) (t) = te 3t 64 ( t t t t t 5 268t 6 8t 7 9t 8 t 9) e 2t ( t t t t t t 6 648t 7 26t 8 5t 9) e t 296 ( 24 7t 6t 2 2t 3), θ (7) (t) = ( 3t)e 3t 64 ( t t t t t 5 486t 6 548t 7 27t 8 2t 9) e 2t ( t t t t t 5 98t 6 8t 7 26t 8 5t 9) e t ( 2t t ), θ (8) (t) = (2 3t)e 3t 28 ( t 73898t t 3 52t t t 6 44t 7 36t 8 2t 9) e 2t ( t t t t t t 6 368t 7 36t 8 5t 9) e t 4552(t ), θ (9) (t) = ( t)e 3t 28 ( t t t 3 782t 4 546t t Download Date /25/7 5: PM

6 BAI-NI GUO JIAO-LIAN ZHAO FENG QI 2592t 7 9t 8 4t 9) e 2t ( t t t t t t 6 566t 7 35t 8 5t 9) e t 4552, θ () (t) =e t[ t t t t t 5 296t t 7 396t 8 5t 9 52 ( t t t t 4 926t 5 63t 6 6t 7 54t 8 2t 9) e t (4 3t)e 2t] e t θ (t), θ (t) = ( 6t)e2t 52 ( t t t t 4 34t 5 782t 6 684t 7 72t 8 2t 9) e t 9 ( t t t 3 466t 4 864t t 6 352t 7 5t 8), θ (t) =8[ (7 3t)e 2t 64 ( t 764t t t t 5 756t 6 8t 7 9t 8 2t 9) e t 9 ( t t t 3 54t 4 494t 5 38t 6 5t 7)], θ (3) (t) =8[ (7 6t)e 2t 64 ( t 82574t t t t t 6 62t 7 8t 8 2t 9) e t 63 ( t 9882t 2 288t 3 35t 4 264t 5 5t 6)], θ (4) (t) =6[ ( 3t)e 2t 32 ( t t t t t t 6 476t 7 26t 8 2t 9) e t 945 ( t 288t 2 468t 3 44t 4 t 5)], θ (5) (t) =6[ (23 6t)e 2t 32 ( t t t t t 5 674t t 7 44t 8 2t 9) e t 945 ( t 44t 2 76t 3 5t 4)], θ (6) (t) =64[ (3 3t)e 2t 8 ( Download Date /25/7 5: PM

7 A COMPLETELY MONOTONIC FUNCTION t t t t t t t 7 62t 8 2t 9) e t 945 ( 44 72t 32t 2 5t 3)], θ (7) (t) =64[ (29 6t)e 2t 8 ( t t t t t 5 89t t 7 8t 8 2t 9) e t 2835 ( t 5t 2)], θ (8) (t) = 28[ (6 3t)e 2t 4 ( t t t t t t t 7 98t 8 2t 9) e t 2835(44 5t) ], θ (9) (t) = 28[ (35 6t)e 2t 4 ( t t t 3 664t t t t 7 26t 8 2t 9) e t 475 ], θ () (t) = 52e t[ (9 3t)e t t t t t t 5 592t t 7 234t 8 2t 9] 52e t θ 2 (t), θ 2 (t) =9[ t t t 3 87t t t 6 28t 7 2t (22 3t)e t], θ 2 (t) =72[ t t t 3 483t t 5 82t 6 2t (25 3t)e t], θ (3) 2 (t) = 54[ t 495t 2 276t 3 435t 4 56t 5 2t (28 3t)e t], θ (4) 2 (t) = 648[ t 69t 2 345t 3 65t 4 t (3 3t)e t], θ (5) 2 (t) = 324[ t 87t 2 52t 3 t (34 3t)e t], θ (6) 2 (t) = 648[ 62882(37 3t)e t 38 87t 78t 2 2t 3], θ (7) 2 (t) = 844[ (4 3t)e t t 2t 2], 475 Download Date /25/7 5: PM

8 BAI-NI GUO JIAO-LIAN ZHAO FENG QI θ (8) 2 (t) = 72576[ (43 3t)e t 3 t ], θ (9) 2 (t) = 72576[ (46 3t)e t ]. It is easy to calculate that θ () =, θ () =, θ (3) () =, θ (4) () =, θ (5) () = 63296, θ (6) () = , θ (7) () = 63284, θ (8) () = 62792, θ (9) () = , θ () () = , θ () = , θ () = , θ (3) θ (5) θ (7) () = , θ(4) () = , () = , θ(6) () = , () = , θ(8) () = , θ (9) () = , θ() () = , () = , θ () = , θ 2 θ (3) 2 θ (5) 2 θ (7) 2 () = , θ(4) 2 () = , () = , θ(6) 2 () = , () = , θ(8) 2 () = , θ (9) 2 () = Since θ (9) 2 (t) is increasing, so θ(9) 2 (t) > on(, ), which means that θ(8) 2 (t) is increasing and positive on (, ). By the same argument, it is derived that the functions θ (i) 2 (t) for i 9, θ(i) (t) andθ(i) (t) for i are increasing and positive on (, ). Therefore, the function θ(t) is increasing and positive on (, ), which implies that the function H(x) is completely monotonic on (, ). Because the function 2 x is completely monotonic on (, ) and the product of finitely many completely 2 monotonic functions are also completely monotonic, we obtain that the function g(x) g(x ) is completely monotonic on (, ), which is equivalent to ( ) k [g(x) g(x )] (k) =( ) k g (k) (x) ( ) k g (k) (x ) for k on(, ). By induction, we have ( ) k g (k) (x) ( ) k g (k) (x ) ( ) k g (k) (x 2) ( ) k g (k) (x m) lim m [( )k g (k) (x m)] = for k on(, ). The proof of Theorem is complete Download Date /25/7 5: PM

9 A COMPLETELY MONOTONIC FUNCTION Remark 2. An easy simplification yields that the function H(x) =ψ (x) is completely monotonic on (, ), where for x (, ). Q(x) 8x 2 ( x) (2 x) () Q(x) = x x x x x x x x x x x x x x x x x x x 9 549x 2 8x 2 Remark 3. This paper is a revised version of the preprint [2] and has been further developed in [3 5,8,3 5,7,9]. Acknowledgement. The authors appreciate the editor and anonymous referees for their valuable comments on this manuscript. REFERENCES [] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (9th ed.) (M. Abramowitz, I. A. Stegun, eds). Applied Mathematics Series Vol. 55, National Bureau of Standards, Washington, DC, 972. [2] ALZER, H.: Sharp inequalities for the digamma and polygamma functions, ForumMath. 6 (24), 8 22; [3] GUO, B.-N. QI, F.: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48 (2), ; [4] GUO, B.-N. QI, F.: A completely monotonic function involving the tri-gamma function andwithdegreeone, Appl. Math. Comput. 28 (22), ; [5] GUO, B.-N. QI, F. SRIVASTAVA, H. M.: Some uniqueness results for the nontrivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 2 (2), ; [6] MITRINOVIĆ, D. S. PEČARIĆ, J. E. FINK, A. M.: Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 993. [7] QI, F.: Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2 (2), Article ID 49358, 84 pages; [8] QI, F.: Completely monotonic degree of a function involving the tri- and tetra-gamma functions, Download Date /25/7 5: PM

10 BAI-NI GUO JIAO-LIAN ZHAO FENG QI [9] QI, F.: Complete monotonicity of a family of functions involving the tri- and tetra-gamma functions, [] QI, F.: Some completely monotonic functions involving the q-tri- and -tetra-gamma functions and applications, [] QI, F. CERONE, P. DRAGOMIR, S. S.: Complete monotonicity of a function involving the divided difference of psi functions, Bull. Austral. Math. Soc. (23) (To appear) [2] QI, F. GUO, B.-N.: A completely monotonic function involving the tri- and tetragamma functions, [3] QI, F. GUO, B.-N.: Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications, Commun. Pure Appl. Anal. 8 (29), ; [4] QI, F. GUO, B.-N.: Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic, [5] QI, F. GUO, B.-N.: Necessary and sufficient conditions for functions involving the triand tetra-gamma functions to be completely monotonic, Adv. in Appl. Math. 44 (2), 7 83; [6] QI, F. LUO, Q.-M.: Bounds for the ratio of two gamma functions From Wendel s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (22), [7] QI, F. LUO, Q.-M. GUO, B.-N.: Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math. (23) (To appear); [8] WIDDER, D. V.: The Laplace Transform, Princeton University Press, Princeton, 946. [9] ZHAO, J.-L. GUO, B.-N. QI, F.: Complete monotonicity of two functions involving the tri- and tetra-gamma functions, Period. Math. Hungar. 65 (22), no., 47 55; Received Accepted * School of Mathematics and Informatics Henan Polytechnic University Jiaozuo City Henan Province, 454 CHINA bai.ni.guo@gmail.com bai.ni.guo@hotmail.com qifeng68@gmail.com qifeng68@hotmail.com qifeng68@qq.com URL: ** Department of Mathematics and Informatics Weinan Teachers University Weinan City Shaanxi Province, 74 CHINA zhaojl24@gmail.com darren24@26.com 478 Download Date /25/7 5: PM