Sharp inequalities and complete monotonicity for the Wallis ratio Cristinel Mortici Abstract The aim of this paper is to prove the complete monotonicity of a class of functions arising from Kazarinoff s inequality [Edinburgh Math. Notes 40 956) 9 ]. As applications, new sharp inequalities for the gamma and digamma functions are established. Introduction and motivation In this paper we study the complete monotonicity of the functions f a : 0, ) R, f a x) = ln Γx+ ) ln Γ x+ ) lnx+a), a 0,.) related to the Kazarinoff s inequality: π ) < 3 5... n ) n+ 4 6... n) < π ), n..) n+ 4 For proof and other details, see [5, 3, 4, 6, 8]. As for the Euler s gamma function Γ see [, 8, 9]), we have Γn+) = n!, Γ n+ ) = 3... n ) π, n Received by the editors May 009 - In revised form in October 009. Communicated by A. Bultheel. 000 Mathematics Subject Classification : 6D5; 33B5; 6D07. Key words and phrases : Gamma function; digamma function; polygamma functions; completely monotonic functions; Kazarinoff s inequality. Bull. Belg. Math. Soc. Simon Stevin 7 00), 99 936
930 C. Mortici for every positive integer n, the inequality.) can be extended in the form x+ Γx+ ) < ) < x+ 4 Γ, x > 0. x+ In the papers [3, 7, 3, 30, 3] the inequality.) is proved mainly using the Γx+ ) variation of the function ). Inequalities for the ratio Γx+) or more Γ x+ Γx+ ) general, for ratio Γx+), with s > 0) have been studied extensively by many Γx+s) authors; for results and useful references, see, e.g., [, 4, 6,,, 5, 7, 9, 3, 33]. In the last section of this work, we prove the following sharp inequalities for x, x+ Γx+ ) < ) ω x+ 4 Γ 4, and x+ µ x+ Γx+) ) < x+ Γ, x+ where ω = 4 5π =.00953... and µ = 3π = 0.937... are the best possible. Then we establish some sharp inequalities for the digamma function ψ, that is the logarithmic derivative of the gamma function, ψx) = d dx ln Γx) = Γ x) Γx). More precisely, we prove that for every x, ) ρ ψx+) ψ x+ 4 x+ ) < ) x+ 4 and ) < ψx+) ψ x+ ) x+ x+ ) + σ, where the constants ρ = 7 5 ln = 0.03706... and σ = ln 4 3 = 0.0596... are the best possible. A monotonicity result The derivatives ψ, ψ, ψ,... are known as polygamma functions. In what follows, we use the following integral representations, for every positive integer n, ψ n) x) = ) n 0 t n e xt dt.) e t
Sharp inequalities and complete monotonicity for the Wallis ratio 93 and for every r > 0, x r = t r e xt dt..) Γr) 0 See, e.g., [, 8]. Recall that a function g is completely monotonic in an interval I if g has derivatives of all orders in I such that ) n g n) x) 0,.3) for all x I and n = 0,,, 3.... Dubourdieu [0] proved that if a non constant function g is completely monotonic, then strict inequalities hold in.3). Completely monotonic functions involving ln Γx) are important because they produce sharp bounds for the polygamma functions, see, e.g., [, 4, 7, 0-9]. The famous Hausdorff-Bernstein-Widder theorem [34, p. 6] states that g is completely monotonic on [0, ) if and only if gx) = 0 e xt dµt), where µ is a non-negative measure on [0, ) such that the integral converges for all x > 0. Lemma.. Letw k ) k be the sequence defined by w k = a k a+ ) k +, k. [ ] i) If a 0, 4, then w k 0, for every k. [ ii) If a, ), then w k 0, for every k. Proof. Regarded as a function of a, w k = w k a) is strictly decreasing, since d a k a+ ) ) k + = k a k a+ ) ) k < 0. da For a 4 For a and k, we have ) w k = w k a) w k = ) 3 k 4 4 k + 4 0. and k, we have ) w k = w k a) w k = k < 0. Now we are in position to give the following Theorem [ ].. i) The function f a given by.) is completely monotonic, for every a 0, 4.
93 C. Mortici ii) The function f b is completely monotonic, for every b [, ). Proof. We have f ax) = ψx+) ψ x+ ) x+a) and Using.)-.), we get or where i) If a monotonic, that is f a x) = ψ x+ ) ψ x+ ) + x+a). f a te x) = x+)t te x+ )t 0 e t dt 0 e t dt+ te x+a)t dt, 0 f a te x++a)t x) = 0 e t ϕ a t) dt, ϕ a t) = e at e a+ )t + e t ) = w k t k. k= [ ] 0, 4, then w k 0 and then ϕ a > 0. In consequence, f a is completely ) n f n) a x) > 0,.4) for every x 0, ) and n. Further, f a > 0, so f a is strictly increasing. As lim x f ax) = 0, we have f ax) < 0, for every x > 0, so f a is strictly decreasing. As lim x f a x) = 0, it results that f a > 0. Now.4) holds also for n = and n = 0, meaning [ that f a is completely monotonic. ii) If b, ), then w k 0 and then ϕ b < 0. In consequence, f b is completely monotonic, that is ) n f n) b x) < 0,.5) for every x 0, ) and n. Further, f b < 0, so f b is strictly decreasing. As lim x f b x) = 0, we have f b x) > 0, for every x > 0, so f b is strictly increasing. As lim x f b x) = 0, it results that f b < 0. Now.5) holds also for n = and n = 0, meaning that f b is completely monotonic. 3 Applications In view of their importance, the gamma and polygamma functions have incited the work of many researches, so that numerous remarkable estimates were discovered. We refer here to [4, 5, 7]. We establish in this section some new sharp inequalities for the gamma and digamma functions, using the monotonicity results stated in Theorem..
Sharp inequalities and complete monotonicity for the Wallis ratio 933 More precisely, for a = 4, the function f /4 x) = ln Γx+) ln Γ x+ ) ln x+ ) 4 is completely monotonic, in particular it is strictly decreasing. In consequence, we have, for every x, 0 = lim x f /4 x) < f /4 x) f /4 ). By exponentiating, we obtain the sharp inequalities for x, x+ 4 < Γx+) ) ω x+ Γ 4, x+ where the constant ω = exp f /4 ) = 4 5π =.00953... is the best possible. The function f /4 x) = ψx+ ) ψ x+ ) x+ 4 is strictly increasing. In consequence, for every x, we have ) thus f /4 ) f /4 x) < lim x f /4 x) = 0, ) ρ ψx+) ψ x+ ) < x+ 4 x+ 4 ), where the constant ρ = f /4 ) = 5 7 ln = 0.03706... is the best possible. For b =, the function f / is completely monotonic, in particular, the function gx) = ln Γx+) ln Γ x+ ) ln x+ ) is strictly increasing. In consequence, for every x, we have g) gx) lim x gx) = 0. By exponentiating, we obtain the sharp inequalities for x, µ x+ Γx+ ) ) < x+ Γ, x+ where the constant µ = exp g) = 3π = 0.9 3... is the best possible. The function f / x) = ψx+ ) ψ x+ ) x+ )
934 C. Mortici is strictly decreasing. In consequence, for every x, we have thus 0 = lim x f / x) < f / x) f / ), ) < ψx+) ψ x+ ) x+ x+ ) + σ, where the constant σ = f / ) = ln 4 3 = 0.0596... is the best possible. Acknowledgements: The author thanks the anonymous referees for useful comments and corrections that improved the initial form of this paper. References [] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 965. [] H. Alzer, Some gamma function inequalities, Math. Comp., 60 993), no. 0, 337 346. [3] H. Alzer, Inequalities for the volume of the unit ball in R n, J. Math. Anal. Appl., 5 000), 353-363. [4] N. Batir, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math., 64), 005), Article 03. [5] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics, 97, Addison Wesley Longman Limited, 998. [6] J. Bustoz and M.E.H. Ismail, On the gamma function inequalities, Math. Comp., 47 986), no. 76, 659 667. [7] Ch.-P. Chen and F. Qi, The best bounds in Wallis inequality, Proc. Amer. Math. Soc., 33 005), 397-40. [8] P. J. Davis, Leonhard Euler s integral: A historical profile of the gamma function, Amer. Math. Monthly 66 959), 849 869. [9] S. S. Dragomir, R. P. Agarwal and N. S. Barnett, Inequalities for beta and gamma functions via some classical and new integral inequalities, J. of Inequal. Appl., 5 000), 03-65. [0] J. Dubourdieu, Sur un théorème de M. S. Bernstein relatif á la transformation de Laplace-Stieltjes, Compositio Math., 7 939), 96-4. [] N. Elezović, C. Giordano and J. Pečarić, The best bounds in Gautschi s inequality, Math. Inequal. Appl., 3) 000), 39 5. [] C. Giordano and A. Laforgia, Inequalities and monotonicity properties for the gamma function, J. Comput. Appl. Math., 33 ) 00), 387 396.
Sharp inequalities and complete monotonicity for the Wallis ratio 935 [3] D. K. Kazarinoff, On Wallis formula, Edinburgh Math. Notes, 40 956), 9. [4] N. D. Kazarinoff, Analytic Inequalities, Holt, Rhinehart and Winston, NewYork, 96. [5] D. Kershaw, Some extensions of of W. Gautschi inequalities for the gamma function, Math. Comp., 4 983), no. 64, 607-6. [6] J.-Ch. Kuang, Chángyòng Bùdĕngshì Applied Inequalities), nd edition, Hunan Education Press, Changsha, China, 993. Chinese) [7] A. Laforgia, Further inequalities for the gamma function, Math. Comp., 4 984), no. 66, 597-600. [8] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Springer, Berlin, 966. [9] M. Merkle, Convexity, Schur-convexity and bounds for the gamma function involving the digamma function, Rocky Mountain J. Math., 83) 998), 053 066. [0] C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. Basel), 93 009), no., 37 45. [] C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett., 3 00), no., 97-00. [] C. Mortici, The proof of Muqattash-Yahdi conjecture Math. Comp. Modelling, 5 00), no. 9-0, 54-59. [3] C. Mortici, Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 5 009), no., 86-9. [4] C. Mortici, Optimizing the rate of convergence in some new classes of sequences convergent to Euler s constant, Anal. Appl. Singap.), 8 00), no., 99-07. [5] C. Mortici, Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput., 5 00), 3443-3448. [6] C. Mortici, Best estimates of the generalized Stirling formula, Appl. Math. Comput., 00), 5 00), no., 4044-4048. [7] C. Mortici, A class of integral approximations for the factorial function, Comput. Math. Appl., 00), 59 00), no. 6, 053-058. [8] C. Mortici, Sharp inequalities related to Gosper s formula, C. R. Acad. Sci. Paris, 48 00), no. 3-4, 37-40. [9] C. Mortici, Product approximations via asymptotic integration, Amer. Math. Monthly, 7 00), no. 5, 434-44.
936 C. Mortici [30] F. Qi, D.-W. Niu, J. Cao and S.-X. Chen, Four logarithmically completely monotonic functions involving gamma function, J. Korean Math. Soc., 45 008), no., 559-573. [3] F. Qi, X.-A. Li and S.-X. Chen, Refinements, extensions and generalizations of the second Kershaw s double inequality, Math. Inequal. Appl., 008), no. 3, 457-465. [3] G. N. Watson, A note on gamma function, Proc. Edinburgh Math. Soc., ) 959), 7-9. [33] J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 948), no. 9, 563 564. [34] D. V. Widder, The Laplace Transform, 98. Valahia University of Târgovişte, Department of Mathematics, Bd. Unirii 8, 3008 Târgovişte, Romania, E-mail: cmortici@valahia.ro Website: www.cristinelmortici.ro