Chen, Chao-Ping

Qi, Feng

English

Victoria University Institutional Repository (VUIR)

The Best Bounds in Wallis' Inequality

Name not available

A New Proof of the Best Bounds in Wallis' Inequality

Victoria University Eprints Repository

THE BEST BOUNDS IN WALLIS’ INEQUALITYCHAO-PING CHEN AND FENG QIAbstract. For all natural number n, we have1√pi(n+ 4/pi − 1) ≤(2n− 1)!!(2n)!!<1√pi(n+ 1/4).The constants 4pi− 1 and 14are the best possible. From this, the well-knownWallis’ inequality is improved.1. IntroductionLetPn =(2n− 1)!!(2n)!!, (1)then we have12√n<√2√(2n+ 1)pi< Pn <2√(4n+ 1)pi<1√3n+ 1<1√2n+ 1<1√2n(2)for n > 1. The inequality (2) is called Wallis’ inequality in [8, p. 103].The lower and upper bounds of Pn in (2) are always cited and applied by math-ematicians. The smallest upper bound 2√(4n+1)piand the largest lower bound√2√(2n+1)piin (2), that is, the following inequalities√2√(2n+ 1)pi< Pn <2√(4n+ 1)pi(3)are obtained by N. D. Kazarinoff. See [7, pp. 47–48 and pp. 65–67].We can rewrite inequality (3) as1√pi(n+ 12 )< Pn <1√pi(n+ 14 )(4)2000 Mathematics Subject Classification. Primary 26D15; Secondary 26A48.Key words and phrases. Wallis’ inequality, best bound, gamma function, monotonicity.The authors were supported in part by NNSF (#10001016) of China, SF for the ProminentYouth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, NSFof Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China.This paper was typeset using AMS-LATEX.12 CH.-P. CHEN AND F. QIfor n ∈ N. See [4, p. 259].It is well-known that factorials and their ‘continuous’ extension play an eminentrole, for instance, in Combinatorics, Graph Theory, and Special Functions.In this article, we will refine inequality (4). More precisely, we will ask for twobest possible constants A and B such that the following double inequality1√pi(n+A)≤ Pn ≤ 1√pi(n+B)(5)holds for all natural number n. In other words, the constants A = 4pi −1 and B = 14can not be replaced by smaller and larger numbers in (5) respectively.2. LemmasLemma 1. For x > 0, we have2x+ 1x(4x+ 1)<Γ′(x+ 12)Γ(x+ 12) − Γ′(x)Γ(x), (6)xb−aΓ(x+ a)Γ(x+ b)= 1 +(a− b)(a+ b− 1)2x+O(1x2), x→∞. (7)The proof of inequality (6) is given in [2, 3, 9], and the proof of the asymptoticexpansion (7) can be found in [6] and [10, p. 378]. See also [1, p. 257].Remark 1. Replacing x by x+ 12 in (6) yields4x+ 4(2x+ 1)(4x+ 3)<Γ′(x+ 1)Γ(x+ 1)− Γ′(x+ 12 )Γ(x+ 12 ). (8)Lemma 2. For x > 0, we haveΓ(x+ 1)Γ(x+ 12 )<2x+ 1√4x+ 3. (9)Proof. Define for positive real number xf(x) = ln(2x+ 1)− 12ln(4x+ 3)− ln Γ(x+ 1) + lnΓ(x+12).Differentiation f(x) gives usf ′(x) =22x+ 1− 24x+ 3−[Γ′(x+ 1)Γ(x+ 1)− Γ′(x+ 12)Γ(x+ 12 )],utilizing (8), we obtainf ′(x) >22x+ 1− 24x+ 3− 4x+ 4(2x+ 1)(4x+ 3)= 0.THE BEST BOUNDS IN WALLIS’ INEQUALITY 3Therefore, f(x) is strictly increasing in (0,+∞), andf(x) > f(0) =12lnpi3> 0,which leads to the inequality (9). Corollary 1. For all natural number n, we haveΓ(n+ 1)Γ(n+ 12 )<2n+ 1√4n+ 3. (10)Corollary 2. The sequence{Qn}∞n=1 ,{[Γ(n+ 1)Γ(n+ 12 )]2− n}∞n=1(11)is strictly decreasing.Proof. By standard argument, we can rewrite Qn+1 < Qn for any natural numbern as inequality (10). Therefore, the monotonicity follows. 3. Main resultsNow we give the main results of this paper.Theorem 1. For all natural number n, we have1√pi(n+ 4pi − 1) ≤ (2n− 1)!!(2n)!! < 1√pi(n+ 14) . (12)The constants 4pi − 1 and 14 are the best possible.Proof. SinceΓ(n+ 1) = n!, Γ(n+12)=(2n− 1)!!2n√pi, 2nn! = (2n)!!,the double inequality (12) is equivalent to14< Qn =[Γ(n+ 1)Γ(n+ 12 )]2− n ≤ 4pi− 1. (13)From the monotonicity of the sequence Qn provided in Corollary 2, it followsthatlimn→∞Qn < Qn ≤ Q1 =4pi− 1.Using the asymptotic formula (7), we conclude fromQn = n[n−12Γ(n+ 1)Γ(n+ 12 )− 1] [n−12Γ(n+ 1)Γ(n+ 12 )+ 1]4 CH.-P. CHEN AND F. QIthatlimn→∞Qn =14.Thus, inequality (13) follows. The proof is complete. References[1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tebles, National Bureau of Standards, Applied Mathematics Series55, 4th printing, Washington, 1965.[2] H. Alzer, On some inequalities for the gamma and psi function, Math. Comp. 66 (1997),373–389.[3] G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen,Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (1995),1713–1723.[4] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure andApplied Mathematics 97, Addison Wesley Longman Limited, 1998.[5] Ch.-P. Chen and F. Qi, Improvement of lower bound in Wallis’ inequality, RGMIA Res. Rep.Coll. 5 (2002), supplement, Art. 23. Available online at http://rgmia.vu.edu.au/v5(E).html.[6] C. L. Frenzer, Error bounds for asymptotic expansions of the ratio of two gamma functions,SIAM J. Math. Anal. 18 (1987), 890–896.[7] N. D. Kazarrinoff, Analytic Inequalities, Holt, Rhinehart and Winston, New York, 1961.[8] J.-Ch. Kuang, Cha´ngyo`ng Bu`deˇngshi` (Applied Inequalities), 2nd edition, Hunan EducationPress, Changsha, China, 1993. (Chinese)[9] Y. L. Luke, Inequalities for the gamma function and its logarithmic derivative, Math. Balkan-ica (N. S.) 2 (1972), 118–123.[10] A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhauser,Basel, 1988.(Chen and Qi) Department of Applied Mathematics and Informatics, Jiaozuo Instituteof Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, CHINAE-mail address, Qi: qifeng@jzit.edu.cnURL, Qi: http://rgmia.vu.edu.au/qi.html

A Dictionary of Inequalities, Pitman Monographs and Surveys

Analytic Inequalities,

Ch´ angy` ong B` udˇ engsh` i (Applied Inequalities), 2nd edition,

Error bounds for asymptotic expansions of the ratio of two gamma functions,

Improvement of lower bound in Wallis’ inequality,

Inequalities for the gamma function and its logarithmic derivative,

Inequalities for zero-balanced hypergeometric functions,

On some inequalities for the gamma and psi function,

Special Functions of Mathematical Physics,

A NEW PROOF OF THE BEST BOUNDS IN WALLIS’INEQUALITYCHAO-PING CHEN AND FENG QIAbstract. By using some properties of gamma function and psi function andthe convolution theorem, a new proof of the following double inequality isgiven: For all natural number n, we have1√pi(n+ 4pi− 1) ≤ (2n− 1)!!(2n)!! < 1√pi(n+ 14) ,and the constants 4pi− 1 and 14are the best possible.1. IntroductionDefine (2m)!! =∏mi=1(2i) and (2m − 1)!! =∏mi=1(2i − 1) for any given positiveinteger m. Then we have1√pi(n+ 12) < (2n− 1)!!(2n)!! < 1√pi(n+ 14) . (1)The inequality (1) is called Wallis’ inequality in [7, p. 103] and can be improvedto the followingTheorem 1. For all natural number n, we have1√pi(n+ 4pi − 1) ≤ (2n− 1)!!(2n)!! < 1√pi(n+ 14) . (2)The constants 4pi − 1 and 14 are the best possible.In [2, pp. 358–359] and [9], it was twice proved that the function[Γ(x+1)Γ(x+ 12 )]2− xis decreasing for x > 0. This implies that the constants 4pi − 1 and 14 in the lowerand upper bounds of inequality (2) are the best possible.Recently, inequality (2) in Theorem 1 was obtained using different approachesby the authors in [3, 4, 5].In this short note, we will give a new proof of Theorem 1 by using some propertiesof gamma and psi functions and the convolution theorem.2000 Mathematics Subject Classification. Primary 05A10, 26D20; Secondary 33B15.Key words and phrases. Wallis’ inequality, best bound, gamma function, psi function, convo-lution theorem, monotonicity.The authors were supported in part by NNSF (#10001016) of CHINA, SF for the ProminentYouth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, DoctorFund of Jiaozuo Institute of Technology, CHINA.This paper was typeset using AMS-LATEX.12 CH.-P. CHEN AND F. QI2. LemmasThe following lemmas regarding to gamma function Γ(x) and psi function ψ = Γ′Γare necessary.Lemma 1 ([6]). For x > 0, we havexb−aΓ(x+ a)Γ(x+ b)= 1 +(a− b)(a+ b− 1)2x+O(x−2). (3)Lemma 2 ([1, 8]). For x > 0, we haveψ(x) = −γ +∫ ∞0e−t − e−xt1− e−t dt, (4)ψ(x) = lnx− 12x−n∑r=1(−1)r−1Br2rx−2r +O(x−2n−2), (5)where γ = 0.57721566490153286060651 · · · is the Euler’s constant. In particular,ψ(x) = lnx− 12x+O(x−2). (6)Lemma 3. Let f1(t) and f2(t) be piecewise continuous for t ≥ 0 on any given finiteinterval and there exist two constants M > 0 and c ≥ 0 such that |f(t)| ≤ Mect,then we have∫ ∞0[∫ s0f1(u)f2(t− u) du]e−st dt =∫ ∞0f1(u)e−su du∫ ∞0f2(v)e−sv dv. (7)Remark 1. Lemma 3 is a convolution theorem of Laplace transform, which can befound in standard textbooks, for example, [1, 10].3. A new proof of Theorem 1SinceΓ(n+ 1) = n!, Γ(n+12)=(2n− 1)!!2n√pi , 2nn! = (2n)!!, (8)the double inequality (2) can be rewritten as14<[Γ(n+ 1)Γ(n+ 12)]2 − n ≤ 4pi− 1. (9)Letf(x) =[Γ(x+ 1)Γ(x+ 12)]2 − x, x > 0. (10)Direct computation givesf ′(x) = 2[Γ(x+ 1)Γ(x+ 12)]2[ψ(x+ 1)− ψ(x+ 12)]− 1 (11)andψ(x+ 1)− ψ(x+ 12)1 + f ′(x)f ′′(x)= ψ′(x+ 1)− ψ′(x+12)+ 2[ψ(x+ 1)− ψ(x+12)]2, g(x).(12)A NEW PROOF OF THE BEST BOUNDS IN WALLIS’ INEQUALITY 3Differentiating (4) yieldsψ′(x) =∫ ∞0te−xt1− e−t dt. (13)From (4) and (13), it follows thatg(x) = −∫ ∞0te−xth(t) dt+ 2(∫ ∞0e−xth(t) dt)2, (14)whereh(x) =(et/2 + 1)−1. (15)By using the convolution theorem, Lemma 3, we haveg(x) = −∫ ∞0te−xth(t) dt+ 2∫ ∞0[∫ t0h(s)h(t− s) ds]dt=∫ ∞0e−xtI(t) dt,(16)whereI(t) =∫ ∞0[2h(s)h(t− s)− h(t)]ds. (17)We claim that for 0 < s < t the following inequality holds:2h(s)h(t− s)− h(t) > 0, (18)which is equivalent to (1 + es/2)(1 + e(t−s)/2)< 2(1 + et/2). (19)LetJ(t) = ln(1 + es/2)+ ln(1 + e(t−s)/2)− ln[2(1 + et/2)], 0 < s < t.Calculating straightforwardly yieldsJ ′(t) =et/2[1− es/2]2es/2(1 + et/2)(1 + e(t−s)/2) < 0.Therefore we have J(t) < J(s) = 0, which means that inequality (18) is valid.Combining (16), (17) and (18) leads to g(x) > 0. From (13), it follows thatψ′(x) > 0, and ψ(x) is increasing in (0,∞). Since 1+ f ′(x) ≥ 0 by (11), f ′′(x) andg(x) have the same sign by (12), thus f ′′(x) > 0 and f ′(x) is increasing in (0,∞).From (3), we havelimx→∞x− 12 Γ(x+ 1)Γ(x+ 12) = 1, (20)From (6), it follows thatlimx→∞x[ψ(x+ 1)− ψ(x+12)]=12. (21)Combination of (11), (20) and (21) yieldsf ′(x) < limx→∞ f′(x) = 0,which implies that f(x) is decreasing in (0,∞). Hencelimn→∞{[Γ(n+ 1)Γ(n+ 12 )]2− n}<[Γ(n+ 1)Γ(n+ 12 )]2− n ≤[Γ(1 + 1)Γ(1 + 12 )]2− 1 = 4pi− 1. (22)4 CH.-P. CHEN AND F. QIWe can rewrite f(x) asf(x) = x[x−1/2Γ(x+ 1)Γ(x+ 12) − 1][x−1/2 Γ(x+ 1)Γ(x+ 12) + 1]. (23)Using (3) yieldslimn→∞{[Γ(n+ 1)Γ(n+ 12 )]2− n}= limx→∞ f(x) =14. (24)The double inequality (2) follows from (22) and (24), and the constants 4pi − 1 and14 are the best possible. The proof is complete.References[1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formu-las, Graphs, and Mathematical Tables, National Bureau of Standards, Applied MathematicsSeries 55, 4th printing, Washington, 1965.[2] H. Alzer, Inequalities for the volume of the unit ball in Rn, J. Math. Anal. Appl. 252 (2000),353–363.[3] Ch.-P. Chen and F. Qi, Improvement of lower bound in Wallis’ inequality, RGMIA Res. Rep.Coll. 5 (2002), suppl., Art. 23. Available online at http://rgmia.vu.edu.au/v5(E).html.[4] Ch.-P. Chen and F. Qi, The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc.(2003), accepted. RGMIA Res. Rep. Coll. 5 (2002), no. 4, Art 13. Available online at http://rgmia.vu.edu.au/v5n4.html.[5] Ch.-P. Chen and F. Qi, The best bounds to(2n)!22n(n!)2, Math. Gaz., to appear in November,2004.[6] C. L. Frenzer, Error bounds for asymptotic expansions of the ratio of two gamma functions,SIAM J. Math. Anal. 18 (1987), 890–896.[7] J.-Ch. Kuang, Cha´ngyo`ng Bu`deˇngshi` (Applied Inequalities), 2nd ed., Hunan Education Press,Changsha, CHINA, 1993. (Chinese)[8] Zh.-X. Wang and D.-R. Guo, Te`shu¯ Ha´nshu` Ga`ilu`n (Introduction to Special Function), TheSeries of Advanced Physics of Peking University, Peking University Press, Beijing, CHINA,2000. (Chinese)[9] G. N. Watson, A note on Gamma functions, Proc. Edinburgh Math. Soc. (2) 11 1958/1959Edinburgh Math. Notes No. 42 (misprinted 41) (1959), 7–9.[10] Zh.-Zh. Zhou and J.-F. Zheng, Fu`bia`n Ha´nshu` yuˇ Ji¯fe¯n Bia`nhua`n (Complex Functions andIntegral Transforms), Higher Education Press, Beijing, CHINA, 1995. (Chinese)(Ch.-P. Chen) Department of Applied Mathematics and Informatics, Jiaozuo Instituteof Technology, Jiaozuo City, Henan 454000, CHINA(F. Qi) Department of Applied Mathematics and Informatics, Jiaozuo Institute ofTechnology, Jiaozuo City, Henan 454000, CHINAE-mail address: qifeng@jzit.edu.cnURL: http://rgmia.vu.edu.au/qi.html

A note on Gamma functions,

Ch´ angy` ong B` udˇ engsh` i (Applied Inequalities), 2nd ed.,

F` ubi` an H´ ansh` u yˇ u J¯ if¯ en Bi` anhu` an (Complex Functions and Integral Transforms),

Inequalities for the volume of the unit ball in

T` esh¯ u H´ ansh` u G` ail` un (Introduction to Special Function),