Sharp inequalities and complete monotonicity for the Wallis ratio

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1 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 ) 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 ) < n ) n 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 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),

2 930 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 = and µ = 3π = 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 = and σ = ln 4 3 = 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

3 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,,, 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 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.

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..

5 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 = 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 = 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 = is the best possible. The function f / x) = ψx+ ) ψ x+ ) x+ )

6 934 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 = 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., ), no. 0, [3] H. Alzer, Inequalities for the volume of the unit ball in R n, J. Math. Anal. Appl., 5 000), [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., ), no. 76, [7] Ch.-P. Chen and F. Qi, The best bounds in Wallis inequality, Proc. Amer. Math. Soc., ), [8] P. J. Davis, Leonhard Euler s integral: A historical profile of the gamma function, Amer. Math. Monthly ), [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), [0] J. Dubourdieu, Sur un théorème de M. S. Bernstein relatif á la transformation de Laplace-Stieltjes, Compositio Math., 7 939), [] N. Elezović, C. Giordano and J. Pečarić, The best bounds in Gautschi s inequality, Math. Inequal. Appl., 3) 000), [] C. Giordano and A. Laforgia, Inequalities and monotonicity properties for the gamma function, J. Comput. Appl. Math., 33 ) 00),

7 Sharp inequalities and complete monotonicity for the Wallis ratio 935 [3] D. K. Kazarinoff, On Wallis formula, Edinburgh Math. Notes, ), 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, [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, [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), [0] C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. Basel), ), no., [] C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett., 3 00), no., [] C. Mortici, The proof of Muqattash-Yahdi conjecture Math. Comp. Modelling, 5 00), no. 9-0, [3] C. Mortici, Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 5 009), no., [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., [5] C. Mortici, Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput., 5 00), [6] C. Mortici, Best estimates of the generalized Stirling formula, Appl. Math. Comput., 00), 5 00), no., [7] C. Mortici, A class of integral approximations for the factorial function, Comput. Math. Appl., 00), 59 00), no. 6, [8] C. Mortici, Sharp inequalities related to Gosper s formula, C. R. Acad. Sci. Paris, 48 00), no. 3-4, [9] C. Mortici, Product approximations via asymptotic integration, Amer. Math. Monthly, 7 00), no. 5,

8 936 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., ), no., [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, [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 ), no. 9, [34] D. V. Widder, The Laplace Transform, 98. Valahia University of Târgovişte, Department of Mathematics, Bd. Unirii 8, 3008 Târgovişte, Romania, cmortici@valahia.ro Website: