Complete monotonicity of a function involving the p-psi function and alternative proofs

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1 Global Journal of Mathematical Analysis, 2 (3) (24) c Science Publishing Corporation doi:.449/gjma.v2i3.396 Research Paper Complete monotonicity of a function involving the p-psi function and alternative proofs Valmir Krasniqi & Feng Qi 2,3,4, Department of Mathematics, University of Prishtina, Prishtinë, Republic of Kosova 2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 2843, China 3 Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 3387, China 4 Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454, China *Corresponding author s qifeng68@gmail.com, qifeng68@hotmail.com, qifeng68@qq.com *Corresponding author s URL: http: // qifeng68. wordpress. com Copyright c 24 Valmir Krasniqi& Feng Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, providehe original work is properly cited. Abstract In the paper, the authors prove that the function x α x+p+ ψ p(x) is completely monotonic on (, ) if and only if α, where p N and ψ p (x) is the p-analogue of the classical psi function ψ(x). Keywords: completely monotonic function; necessary and sufficient condition; p-gamma function; p-psi function; inequality MSC : Primary 33D5; Secondary 26A48, 33B5, 33E5. Introduction Recall from 2, Chapter XIII, 6, Chapter and 7, Chapter IV that a function f is saio be completely monotonic on an interval I if f has derivatives of all orders on I and satisfies ( ) n f (n) (x) < (.) for x I and n. The celebrated Bernstein-Widder s Theorem (see 6, p. 3, Theorem.4 or 7, p. 6, Theorem 2b) characterizes that a necessary and sufficient condition that f(x) should be completely monotonic for < x < is that f(x) e xt d α(t), where α(t) is non-decreasing anhe integral converges for < x <. This expresses that a completely monotonic function f on, ) is a Laplace transform of the measure α. (.2)

2 Global Journal of Mathematical Analysis 25 It is common knowledge that the classical Euler s gamma function Γ(x) may be defined for x > by Γ(x) t x e t. The logarithmic derivative of Γ(x), denoted by ψ(x) Γ (x) Γ(x), is called psi function or digamma function. An alternative definition of the gamma function Γ(x) is Γ(x) lim p Γ p(x), where Γ p (x) p!p x x(x + ) (x + p) p x x( + x/) ( + x/p) (.3) (.4) for x > and p N, the set of all positive integers. See 3, p. 25. The p-analogue of the psi function ψ(x) is defined as the logarithmic derivative of the Γ p function, that is, ψ p (x) d d x Γ p(x) Γ p(x) Γ p (x). (.5) The function ψ p has the following properties:. It has the following representations ψ p (x) p p k x + k p (p+)t t e xt. (.6) 2. It is increasing on (, ) and ψ p is completely monotonic on (, ). The very right hand side of the formula (.6) corrects errors appeared in 8, p. 374, Lemma 5 and, p. 29, Lemma 2.3. In 2, pp , Theorem, it was provehat the function θ α (x) x α x ψ(x) (.7) is completely monotonic on (, ) if and only if α. For the history, background, applications and alternative proofs of this conclusion, please refer to 4, 3, p. 8, Section.6.6 and closely related references therein. The aim of this paper is to generalize 2, pp , Theorem and 4, p. 5, Theorem to the case of the p-analogue ψ p (x) of the psi function ψ(x) as follows. Theorem.. The function θ p,α (x) x α x + p + ψ p(x) (.8) for p N is completely monotonic on (, ) if and only if α. Remark.. Letting p in Theorem., we obtain 2, pp , Theorem and 4, p. 5, Theorem. 2. Proofs of Theorem. First Proof. From the identity (.6) anhe integral expression b a e at e bt t in, p. 23, 5..32, we obtain (2.) θ p, (x) x (p+)t ϕ(t)e xt, (2.2)

3 26 Global Journal of Mathematical Analysis where ϕ(t) t t. The function ϕ(t) is increasing on (, ) with lim t + ϕ(t) 2 and lim ϕ(t). t See 5, 6, 7,, 4, 5, 8 and related references therein. Therefore, for x > and n N, we have (2.3) (2.4) ( ) n θ (n) dn p, (x) x( )n d x n x n/x (p+)t ϕ(t)e xt ( ) n n dn d x n t n ϕ(t) (p+)t e xt n t n (p+)t ϕ(t)(tx n)e xt + t n ϕ(t) (p+)t e xt n/x (p+)t ϕ(t)e xt t n (p+)t ϕ(t)(tx n)e xt ( ) n n/x > ϕ t n (p+)t ( ) n (tx n)e xt + ϕ x x ( ) n ϕ t n (p+)t (tx n)e xt x ( ) n ϕ x t n (p+)t e xt n t n (p+)t e xt x ( ) n ϕ x t n e xt x t n e (x+p+)t n t n e xt + n x ( ) n ϕ x n! x x n+ x n! )! (n )! n(n (x + p + ) n+ x n + n (x + p + ) ( ) n n ϕ n! x x n x (x + p + ) n+ x n + (x + p + ) ( ) ( ) n n n! x ϕ x (x + p + ) n x + p + ( ) n n!(p + ) ϕ x (x + p + ) n+ >, where we usehe formula x ω Γ(ω) t ω e xt t n (p+)t (tx n)e xt n/x t n e (x+p+)t for real numbers x > and ω >, see, p. 255, 6... So we obtain that the function θ p, (x) is completely monotonic on (, ). Since n ( ) n ( ) ( ) n u(x)v(x) (n) i u (i) (x) ( ) n i v (n i) (x), i i the product of any two completely monotonic function is also completely monotonic on their common domain. On the other hand, the function x α for α < is clearly completely monotonic on (, ). Consequently the function θ p,α (x) x α θ p, (x) for α is completely monotonic on (, ). Conversely, if θ p,α (x) is completely monotonic on (, ), then d θ p,α (x) x {α α d x x + p + ψ p(x) + p + } x + p + xψ p(x) (2.5)

4 Global Journal of Mathematical Analysis 27 for x >, equivalently, α xψ p(x) p+ x+p+ x+p+ ψ p(x). Employing L Hôspital s rule and (.6) results in lim xψ p(x) p+ x+p+ x+p+ ψ p(x) lim xψ p (x) + ψ p(x) + p+ (x+p+) 2 x x+p+ ψ p(x) so it is necessary that α. The proof is complete. Second Proof. From (2.2) and by integration by part leao θ p, (x) lim p+ (x+p+) 2 (p+)t ϕ(t) d e xt { (p+)t ϕ(t) } e xt { (p+)t ϕ(t)e xt} t t { (p+)t ϕ (t) + (p + )e (p+)t ϕ(t) } e xt. x p k 2 (x+k) + p 3 x x+p+ p k k (x+k) 2, (x+k) 2 Therefore, for showing that the function θ p, (x) is completely monotonic on (, ) for all p N, it suffices to prove that the function (p+)t ϕ (t) + (p + )e (p+)t ϕ(t) (2.6) is positive. Since the function ϕ(t) is increasing on (, ), the derivative ϕ (t) is positive on (, ). Further considering the limits in (2.4), the positivity of ϕ(t) follows. As a result, the function (2.6) is positive. The rest of the proof is the same as the first proof. Remark 2.. This paper is a slightly modified version of the preprint 9. Acknowledgements The second author was partially supported by the National Natural Science Foundation of China under Grant No and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY492, China. References 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, 9th printing, Washington, H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (997), no. 27, ; Available online at 3 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2), no., 3 ; Available online at 5 B.-N. Guo, A.-Q. Liu, and F. Qi, Monotonicity and logarithmic convexity of three functions involving exponential function, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 5 (28), no. 4, B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms 52 (29), no., 89 92; Available online at

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