Sharp Smith’s bounds for the gamma function

R E S E A R C H

Open Access

Sharp Smith’s bounds for the gamma

function

Xi-Qiao Li

1

_{, Zhi-Ming Liu}

1

_{, Zhen-Hang Yang}

2

_{and Shen-Zhou Zheng}

3*

*_{Correspondence:}

[email protected]

3_{College of Science, Beijing}

Jiaotong University, Beijing, China Full list of author information is available at the end of the article

Abstract

Among various approximation formulas for the gamma function, Smith showed that

x+1

2

∼S(x) =√2

π

_x e

x

2xtanh 1 2x

x/2

, x→ ∞,

which is a little-known but accurate and simple one. In this note, we prove that the functionx→ln

(x+ 1/2) – lnS(x) is strictly increasing and concave on (0,∞), which shows that Smith’s approximation is just an upper one.

MSC: Primary 33B15; 26A48; secondary 26D15; 26A51

Keywords: Gamma function; Approximation formula; Inequality

1 Introduction

The Stirling formula

n!∼√2πnnne–n (1.1)

has many important applications in statistical physics, probability theory and number the-ory. Due to its practical importance, it has attracted much interest of many mathemati-cians and has also motivated a large number of research papers concerning various gener-alizations and improvements; see for example, Burnside’s [1], Gosper [2], Batir [3], Mortici [4].

The gamma function(x) =₀∞tx–1_e–t_{dtforx> 0 is closely related to the Stirling}

for-mula, since(n+ 1) =n! for alln∈N. This inspired some authors to also pay attention to find various better approximations for the gamma function; see, for instance, Ramanujan [5, p. 339], Windschitl (see Nemes [6, Corollary 4.1]), Yang and Chu [7], Chen [8].

More results involving the approximation formulas for the factorial or gamma function can be found in [9–23] and the references cited therein.

In this note, we are interested in Smith’s approximation formula (see [24, equation (42)]):

x+1

2

∼√2π

x e

x

2xtanh 1

2x x/2

:=S(x), asx→ ∞. (1.2)

It is easy to check that

x+1

2

=√2π

x e

x

2xtanh 1

2x x/2

1 +O

1

x5

,

which shows that the rate ofS(x) converging to(x+ 1/2) asx→ ∞is likex–5_.

Accord-ing to the comment in [8, (3.5)–(3.10)], it is well known that Smith’s approximation is an accurate but simple one for gamma function.

The aim of this short note is to further prove the Smith approximationS(x) is an upper one. Our main result is stated as follows.

Theorem 1 The function

f(x) =ln

x+1

2

–ln√2π–xlnx+x–x 2ln

2xtanh 1

2x

is strictly increasing and concave from(0,∞)onto(–ln√2, 0).

2 Proof of Theorem 1

To prove Theorem 1 we need the following two lemmas.

Lemma 1 The inequality

ψ

x+1

2

<4 3

15x2_{+ 4}

x(20x2_{+ 7)} (2.1)

holds for all x> 0.

Proof Let

f1(x) =ψ

x+1 2

–4 3

15x2_{+ 4}

x(20x2_{+ 7)}. (2.2)

Using the recurrence formula [25, pp. 258–260]:

ψ(n)(x+ 1) –ψ(n)(x) = (–1)

n_n! xn+1 ,

we have

f1(x+ 1) –f1(x) =ψ

x+3

2

– 4 3(x+ 1)

15x2+ 30x+ 19 20x2_{+ 40x+ 27}

–ψ

x+1

2

+4 3

15x2_{+ 4} x(20x2_{+ 7)}

– 1

(x+ 1/2)2–

4 3(x+ 1)

15x2+ 30x+ 19 20x2_{+ 40x+ 27}+

4 3

15x2+ 4

x(20x2_{+ 7)}

= 144

It then follows that

f1(x) <f1(x+ 1) <· · ·<_n→∞lim f1(x+n) = 0,

which proves the desired inequality (2.1).

Lemma 2 The inequality

sinh2t

cosht >

t2_(21t2_{+ 60)}

31t2_{+ 60} (2.3)

holds for all t> 0.

Proof It is obvious that the inequality what we consider is equivalent to

f2(t) =31t2+ 60(sinht)2–t221t2+ 60cosht> 0.

Simplifying and expanding it in power series lead us to

2f2(t) = 60cosh2t+ 31t2cosh2t– 120t2cosht– 42t4cosht– 31t2– 60

= 60

∞

n=0

22n (2n)!t

2n_{+ 31} ∞

n=1

22n–2 (2n– 2)!t

2n

– 120

∞

n=1

1 (2n– 2)!t

2n_{– 42} ∞

n=2

1 (2n– 4)!t

2n_{– 31t}2_{– 60}

:=

∞

n=2 an

(2n)!t

2n_,

where

an=

62n2– 31n+ 12022n–1– 24n(2n– 1)14n2– 35n+ 31.

It is easy to check thata2=a3= 0 anda4= 49 184 > 0. It remains to provean> 0 forn≥5. To this end, it suffices to provebn= 22n–1–6n(2n–1) > 0 forn≥5, because the inequality

62n2– 31n+ 120> 414n2– 35n+ 31

is clearly valid forn≥5. We easily obtain

bn+1– 4bn= 6

6n2– 7n– 1> 0

forn≥5, which in combination withb5= 242 > 0 yieldsbn> 0 forn≥5. This completes

the proof.

Theorem1 Differentiating and simplifying yields

f(x) =ψ

x+1

2

–lnx–1 2ln

2xtanh 1

2x

+ 1

2xsinh(1/x)– 1 2,

f(x) =ψ

x+1

2

+ 1 2x3

cosh(1/x)

sinh2(1/x)– 3 2x.

As an application of inequalities (2.1) and (2.3) it gives

f(x) <4 3

15x2_{+ 4} x(20x2_{+ 7)}+

1 2x3

cosh(1/x)

sinh2_(1/x)–

3 2x

= 1 2x3

cosh(1/x)

sinh2(1/x)– 1 6

60x2_{+ 31} x(20x2_{+ 7)}

x=1/t

= t

3

2

cosht

sinh2t–

31t2_{+ 60} t2_(21t2_{+ 60)}

< 0.

Then it is deduced that

f(x) > lim

x→∞f _{(x) = 0,}

which in turn implies that

–1

2ln2 =x→0lim+f(x) <f(x) <_x→∞limf(x) = 0.

This completes the proof.

3 Corollaries and remarks

Using the increasing property off(x+ 1/2) given in Theorem 1 and noting that

f

1 2

=ln

√

e

√

π(tanh1)1/4 and f

3 2

=ln

2e√e33/4

27√πtanh3/4(1/3)

,

we have the corollaries.

Corollary 1 The double inequality

α1<

ex+1/2(x+ 1)

√

2π(x+ 1/2)x+1/2_{[(2x+ 1)tanh(1/(2x+ 1))]}(2x+1)/4 < 1

holds for all x> 0with the best constants1andα1=

√

e/π/(tanh1)1/4_≈0.99573.

Corollary 2 The double inequality

α2<

n!

√

2π((n+ 1/2)/e)n+1/2_{[(2n+ 1)tanh(1/(2n+ 1))]}(2n+1)/4 < 1

holds for all n∈Nwith the best constants1and

α2=

2e√e33/4

By the decreasing property off(x+ 1/2) given in Theorem 1 and the facts that

f

1 2

= 1

sinh2– 1

2ln(tanh1) +ln2 – 1

2–γ≈0.027823,

f

3 2

= 1

3sinh(2/3)– 1 2ln

3tanh1

3

–ln3

2+ψ(1) + 1

2≈0.00016946,

the following corollaries are immediate.

Corollary 3 For x> 0,the inequalities

1 2+

1 2ln

(2x+ 1)tanh 1

2x+ 1

– 1

(2x+ 1)sinh(2/(2x+ 1))

<ψ(x+ 1) –ln

x+1

2

<β1+

1 2ln

(2x+ 1)tanh 1

2x+ 1

– 1

(2x+ 1)sinh(2/(2x+ 1))

hold,where the constants1/2and

β1=ln2 –

1

2ln(tanh1) + 1

sinh2–γ ≈0.52782

are the best possible.

Corollary 4 Let Hn=n_k=1for n∈N.The inequalities

1 2+γ

+1 2ln

(2n+ 1)tanh 1

2n+ 1

– 1

(2n+ 1)sinh(2/(2x+ 1))

<Hn–ln

n+1

2

<β2+

1 2ln

(2n+ 1)tanh 1

2n+ 1

– 1

(2n+ 1)sinh(2/(2n+ 1))

hold,where1/2 +γ ≈1.0772and

β2=

1 3sinh(2/3)–

1 2ln

3tanh1

3

–ln3

2+ 1≈1.0774

are the best possible constants.

Finally, as a by-product of Lemma 1, we draw the following conclusion.

Theorem 2 Let g be defined on(0,∞)by

g(x) =ln

x+1

2

–

1 2ln2π+

16 21xlnx+

5x

42ln

x2+ 7

20

–x–

√

35

42 arccot

20 7 x

.

Proof Differentiation yields

g(x) =ψ

x+1

2 – 5 42ln x2+ 7

20

+16 21lnx

,

g(x) =ψ

x+1

2

–4 3

15x2_{+ 4}

x(20x2_{+ 7)}=f1(x) < 0,

where the inequality holds due to Lemma 1. This completes the proof.

Remark1 Theorem 2 gives a new approximation for the gamma function

x+1

2

∼√2πx26x/21

x2+ 7 20

5x/42

exp

–x–

√ 35 42 arccot 20 7x ,

asx→ ∞, which satisfies

x+1

2

=√2πx26x/21

x2+ 7 20

5x/42

exp

–x–

√ 35 42 arccot 20 7 x

1 +Ox–5.

Remark2 Theorem 2 also offers an asymptotic formula for the psi function

ψ

x+1

2

∼ 5

42ln

x2+ 7

20

+16

21lnx asx→ ∞.

Furthermore, by replacingxwithx+ 1/2, we have the following sharp inequalities:

5 42ln

x2+x+3 5

+16 21ln

x+1

2

<ψ(x+ 1) <λ0+

5 42ln

x2+x+3 5

+16 21ln

x+1

2

(3.1)

forx> 0 with the best constant

λ0=

16 21ln2 –

5 42ln

3

5–γ ≈0.011709;

γ + 5 42ln

n2+n+3 5

+16 21ln

n+1

2

<Hn<λ0+γ+

5 42ln

n2+n+3 5

+16 21ln

n+1

2

forn∈Nwith the best constant

λ1= 1 –

16 21ln 3 2– 5 42ln 13

5 –γ≈0.00010718.

Inequalities (3.1) first appeared in [26, Corollary 3.4].

4 Conclusions

Acknowledgements

This paper is supported by the National Science Foundation of China grant No. 11371050.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this work equally, and they all read and approved the final manuscript.

Author details

1_{School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China.}2_{Department of}

Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, China.3_{College of}

Science, Beijing Jiaotong University, Beijing, China.

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Received: 2 October 2017 Accepted: 15 January 2018 References

1. Burnside, W.: A rapidly convergent series for logN!. Messenger Math.46, 157–159 (1917)

2. Gosper, R.W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA75, 40–42 (1978)

3. Batir, N.: Sharp inequalities for factorialn. Proyecciones27(1), 97–102 (2008)

4. Mortici, C.: On the generalized Stirling formula. Creative Math. Inform.19(1), 53–56 (2010) 5. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Springer, Berlin (1988)

6. Nemes, G.: New asymptotic expansion for the gamma function. Arch. Math. (Basel)95, 161–169 (2010)

7. Yang, Z.H., Chu, Y.M.: Asymptotic formulas for gamma function with applications. Appl. Math. Comput.270, 665–680 (2015)

8. Chen, C.P.: A more accurate approximation for the gamma function. J. Number Theory164, 417–428 (2016) 9. Shi, X., Liu, F., Hu, M.: A new asymptotic series for the gamma function. J. Comput. Appl. Math.195, 134–154 (2006) 10. Guo, B.N., Zhang, Y.J., Qi, F.: Refinements and sharpenings of some double inequalities for bounding the gamma

function. J. Inequal. Pure Appl. Math.9(1), Article ID 17 (2008)

11. Batir, N.: Inequalities for the gamma function. Arch. Math.91, 554–563 (2008)

12. Mortici, C.: An ultimate extremely accurate formula for approximation of the factorial function. Arch. Math.93(1), 37–45 (2009)

13. Mortici, C.: New sharp inequalities for approximating the factorial function and the digamma functions. Miskolc Math. Notes11(1), 79–86 (2010)

14. Mortici, C.: Improved asymptotic formulas for the gamma function. Comput. Math. Appl.61, 3364–3369 (2011) 15. Zhao, J.L., Guo, B.N., Qi, F.: A refinement of a double inequality for the gamma function. Publ. Math. (Debr.)80(3–4),

333–342 (2012)

16. Mortici, C.: Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Model.57, 1360–1363 (2013)

17. Feng, L., Wang, W.: Two families of approximations for the gamma function. Numer. Algorithms64, 403–416 (2013) 18. Qi, F.: Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the

gamma function. J. Comput. Appl. Math.268, 155–167 (2014)

19. Lu, D.: A new sharp approximation for the gamma function related to Burnside’s formula. Ramanujan J.35(1), 121–129 (2014)

20. Yang, Z.H., Tian, J.F.: Monotonicity and inequalities for the gamma function. J. Inequal. Appl.2017, Article ID 317 (2017)

21. Yang, Z.H., Tian, J.F.: Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal.12(1) (2018). https://doi.org/10.7153/jmi-2018-12-01

22. Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput.66(217), 373–389 (1997) 23. Yang, Z.H.: Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely

monotonic functions related to gamma function. J. Math. Anal. Appl.441, 549–564 (2016)

24. Smith, W.D.: The gamma function revisited. http://schule.bayernport.com/gamma/gamma05.pdf (2006)

25. Abramowttz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)