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On the monotonicity and convexity of the remainder of the Stirling formula
Applied Mathematics Letters 24 (2011) 869–871
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On the monotonicity and convexity of the remainder of the Stirling formula Cristinel Mortici Valahia University of Târgovişte, Department of Mathematics, Bd. Unirii 18, 130082 Târgovişte, Romania
article
abstract
info
Article history: Received 25 May 2009 Accepted 28 December 2010
Shi, Liu and Hu [X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. √ Math. 195 (2006) 134–154] proved that the function θ (x) defined by Γ (x + 1) = 2π (x/e)x eθ (x)/12x is strictly increasing for x ≥ 1. The aim of our work is to prove that −x−1 θ ′′′ (x) is strictly completely monotonic on (0, ∞) . As direct consequences, we show that θ is strictly convex on (0, ∞) , and then we prove that θ is strictly decreasing on (0, β) , and strictly increasing on (β, ∞) , where β = 0.34142 . . . . © 2010 Elsevier Ltd. All rights reserved.
Keywords: Stirling formula Gamma and polygamma functions Complete monotonicity
1. Introduction and motivation Maybe one of the most well known formulas for approximation of the large factorials is Stirling’s formula
√
n! ≈
2π nn+1/2 e−n .
The gamma function
Γ (x + 1) =
∞
∫
t x e−t dt ,
x > 0,
0
is the natural extension of the factorial function, since Γ (n + 1) = n!, for every n = 1, 2, 3, . . .. Shi et al. [1, Rel. 6] introduced the following generalized Stirling formula in a new approach (see [2, p. 253]):
Γ (x + 1) =
√
2π (x/e)x eθ(x)/12x
(1.1)
and proved [1, Theorem 3] that θ (x) is a strictly increasing function for real numbers x ≥ 1. In this work we give a complete characterization of the monotonicity of the function θ on its domain (0, ∞). To be precise, we prove that θ is strictly decreasing on (0, β) and strictly increasing on (β, ∞), where β = 0.34142 . . . is the unique solution of the equation ln Γ (x + 1) + xψ (x + 1) − ln
√
2π − 2x ln x + x = 0
(ψ is the digamma function). Moreover, we show that the function θ is strictly convex on (0, ∞) and −x−1 θ ′′′ is strictly completely monotonic on (0, ∞). 2. The results The digamma function ψ is defined as the logarithmic derivative of the gamma function,
ψ (x) =
d dx
(ln Γ (x)) =
Γ ′ (x) Γ (x)
E-mail addresses: [email protected], [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.12.034
870
C. Mortici / Applied Mathematics Letters 24 (2011) 869–871
for all x > 0, and their derivatives ψ ′ , ψ ′′ , . . . , known as polygamma functions, have the following integral representations:
ψ (n) (x) = (−1)n−1
∞
∫
t n e−xt 1 − e− t
0
dt
(2.1)
for n = 1, 2, 3, . . . . For proofs, details, and other related results, see, for example, [3, p. 260], or [4,5]. We also use the following integral representations: 1 xn
=
∞
∫
1
(n − 1)!
t n−1 e−xt dt ,
n ≥ 1.
(2.2)
0
There exists a very broad literature on these functions. In particular, many inequalities, sharp bounds for these functions, and accurate approximations have been published. Please refer to the papers [6–13] and the references therein. Recall that a function f is completely monotonic in an interval I if f has derivatives of all orders in I such that
(−1)n f (n) (x) ≥ 0, for all x ∈ I and n = 0, 1, 2, 3, . . . . If this inequality is strict for all x ∈ I and all non-negative integers n, then f is said to be strictly completely monotonic. Completely monotonic functions have many applications in different branches of science. For instance, they play a role in potential theory, probability theory, physics, numerical and asymptotic analysis, and combinatorics. Completely monotonic functions involving ln Γ (x) are important because they produce sharp bounds for the polygamma functions. A consequence of the famous Hausdorff–Bernstein–Widder theorem states that f is completely monotonic on (0, ∞) if and only if f (x) =
∞
∫
e−xt ϕ (t ) dt , 0
where ϕ is a non-negative function on (0, ∞) such that the integral converges for all x > 0; see [14, p. 161]. For the sake of simplicity, we define the function w (x) = θ (x) /12, and from (1.1), we have
w (x) = x ln Γ (x + 1) + x2 − x ln
√
2π − x2 ln x.
The main result of our work is the following. Theorem 2.1. The function −x−1 w ′′′ (x) is strictly completely monotonic on (0, ∞). Proof. We successively have
w′ (x) = ln Γ (x + 1) + xψ (x + 1) − ln
√
2π − 2x ln x + x,
then
w′′ (x) = 2ψ (x) +
1
+ xψ ′ (x) − 2 ln x − 1
x
and
w′′′ (x) = 3ψ ′ (x) −
1 x2
2
+ xψ ′′ (x) − . x
Now, using (2.1)–(2.2) and the convolution theorem for Laplace transforms we get, for x > 0, x−1 w ′′′ (x) =
3 x
1
2
ψ ′ (x) − 3 + ψ ′′ (x) − 2 x x ∫ ∞ ∫ ∞ −tx te
e−tx dt
=3 0
∞
∫
1−e
0 t
∫
=3 0
0
s 1−e
dt − −t
1 2
ds e−tx dt − −s
∞
∫
t 2 e−tx dt − 0
1 2
∞
∫
1−e
0
∫ 0
∞
t 2 e−tx dt −
t 2 e−tx ∞
∫ 0
t 2 e−tx 1−e
∞
∫
dt − 2 −t
te−tx dt 0
dt − 2 −t
∫
∞
te−tx dt ,
0
or x−1 w ′′′ (x) =
∞
∫
ϕ (t ) e−tx dt ,
(2.3)
0
where
ϕ (t ) = 3
t
∫ 0
s 1−
e− s
ds −
1 2
t2 −
t2 1 − e− t
− 2t .
C. Mortici / Applied Mathematics Letters 24 (2011) 869–871
871
Differentiation yields
ϕ ′ (t ) =
t 2 et + tet − t − 2e2t + 4et − 2
(et − 1)2
=
∞ n +1 tn −1 − 2 2 − n − 4 < 0, n! (et − 1)2 n=3
so ϕ is strictly decreasing. For t > 0, we have ϕ (t ) < ϕ (0) = 0, so from (2.3) it results that the function −x−1 w ′′′ (x) is strictly completely monotonic. As a direct consequence of Theorem 2.1, w ′′′ < 0, so w ′′ is strictly decreasing. The function w ′′ can be equivalently written as
1 1 w ′′ (x) = 2 (ψ (x) − ln x) + x ψ ′ (x) − + . x
x
Using the asymptotic formulas [3, Rel. 6.3.18, p. 259, and Rel. 6.4.12, p. 260]
ψ (x) ∼ ln x −
1 2x
−
1 12x2
+ ···,
ψ ′ (x) ∼
1 x
+
1 2x2
+
1 6x3
+ ···,
we have limx→∞ w ′′ (x) = 0; thus w ′′ > 0. In consequence, w is strictly convex. In particular, w ′ is strictly increasing with lim w ′ (x) = − ln
x→ 0
√
2π < 0
and
w′ (1) = 2 − γ − ln
√
2π = 1.4228 . . . > 0
(γ = 0.577215 . . . is the Euler–Mascheroni constant). Now, if β is the unique positive real number such that w ′ (β) = 0, i.e., ln Γ (β + 1) + βψ (β + 1) − ln
√
2π − 2β ln β + β = 0,
then w < 0 on (0, β), and w > 0 on (β, ∞). The numerical value β = 0.34142 . . . can be easily found using computer software such as MAPLE. Finally, w is strictly decreasing on (0, β), and strictly increasing on (β, ∞). ′
′
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. Math. 195 (2006) 134–154. E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, fourth ed., Cambridge University Press, Cambridge, England, 1990. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. G. Andrews, R. Askey, R. Roy, Special functions, in: Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, 1999. P.J. Davis, Leonhard Euler’s integral: a historical profile of the gamma function, Amer. Math. Monthly 66 (1959) 849–869. H. Alzer, Some gamma function inequalities, Math. Comp. 60 (201) (1993) 337–346. H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (217) (1997) 373–389. H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg 68 (1998) 363–372. C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. 93 (1) (2009) 37–45. C. Mortici, Monotonicity properties of the volume of the unit ball in Rn, Optimization Lett. 4 (3) (2010) 457–464. C. Mortici, Very accurate estimates of the polygamma functions, Asymptot. Anal. 68 (3) (2010) 125–134. C. Mortici, The proof of Muqattash-Yahdi conjecture, Math. Comp. Modelling 51 (9–10) (2010) 1154–1159. C. Mortici, A class of integral approximations for the factorial function, Comput. Math. Appl. 59 (6) (2010) 2053–2058. D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1981.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On the monotonicity and convexity of the remainder of the Stirling formula Cristinel Mortici Valahia University of Târgovişte, Department of Mathematics, Bd. Unirii 18, 130082 Târgovişte, Romania
article
abstract
info
Article history: Received 25 May 2009 Accepted 28 December 2010
Shi, Liu and Hu [X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. √ Math. 195 (2006) 134–154] proved that the function θ (x) defined by Γ (x + 1) = 2π (x/e)x eθ (x)/12x is strictly increasing for x ≥ 1. The aim of our work is to prove that −x−1 θ ′′′ (x) is strictly completely monotonic on (0, ∞) . As direct consequences, we show that θ is strictly convex on (0, ∞) , and then we prove that θ is strictly decreasing on (0, β) , and strictly increasing on (β, ∞) , where β = 0.34142 . . . . © 2010 Elsevier Ltd. All rights reserved.
Keywords: Stirling formula Gamma and polygamma functions Complete monotonicity
1. Introduction and motivation Maybe one of the most well known formulas for approximation of the large factorials is Stirling’s formula
√
n! ≈
2π nn+1/2 e−n .
The gamma function
Γ (x + 1) =
∞
∫
t x e−t dt ,
x > 0,
0
is the natural extension of the factorial function, since Γ (n + 1) = n!, for every n = 1, 2, 3, . . .. Shi et al. [1, Rel. 6] introduced the following generalized Stirling formula in a new approach (see [2, p. 253]):
Γ (x + 1) =
√
2π (x/e)x eθ(x)/12x
(1.1)
and proved [1, Theorem 3] that θ (x) is a strictly increasing function for real numbers x ≥ 1. In this work we give a complete characterization of the monotonicity of the function θ on its domain (0, ∞). To be precise, we prove that θ is strictly decreasing on (0, β) and strictly increasing on (β, ∞), where β = 0.34142 . . . is the unique solution of the equation ln Γ (x + 1) + xψ (x + 1) − ln
√
2π − 2x ln x + x = 0
(ψ is the digamma function). Moreover, we show that the function θ is strictly convex on (0, ∞) and −x−1 θ ′′′ is strictly completely monotonic on (0, ∞). 2. The results The digamma function ψ is defined as the logarithmic derivative of the gamma function,
ψ (x) =
d dx
(ln Γ (x)) =
Γ ′ (x) Γ (x)
E-mail addresses: [email protected], [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.12.034
870
C. Mortici / Applied Mathematics Letters 24 (2011) 869–871
for all x > 0, and their derivatives ψ ′ , ψ ′′ , . . . , known as polygamma functions, have the following integral representations:
ψ (n) (x) = (−1)n−1
∞
∫
t n e−xt 1 − e− t
0
dt
(2.1)
for n = 1, 2, 3, . . . . For proofs, details, and other related results, see, for example, [3, p. 260], or [4,5]. We also use the following integral representations: 1 xn
=
∞
∫
1
(n − 1)!
t n−1 e−xt dt ,
n ≥ 1.
(2.2)
0
There exists a very broad literature on these functions. In particular, many inequalities, sharp bounds for these functions, and accurate approximations have been published. Please refer to the papers [6–13] and the references therein. Recall that a function f is completely monotonic in an interval I if f has derivatives of all orders in I such that
(−1)n f (n) (x) ≥ 0, for all x ∈ I and n = 0, 1, 2, 3, . . . . If this inequality is strict for all x ∈ I and all non-negative integers n, then f is said to be strictly completely monotonic. Completely monotonic functions have many applications in different branches of science. For instance, they play a role in potential theory, probability theory, physics, numerical and asymptotic analysis, and combinatorics. Completely monotonic functions involving ln Γ (x) are important because they produce sharp bounds for the polygamma functions. A consequence of the famous Hausdorff–Bernstein–Widder theorem states that f is completely monotonic on (0, ∞) if and only if f (x) =
∞
∫
e−xt ϕ (t ) dt , 0
where ϕ is a non-negative function on (0, ∞) such that the integral converges for all x > 0; see [14, p. 161]. For the sake of simplicity, we define the function w (x) = θ (x) /12, and from (1.1), we have
w (x) = x ln Γ (x + 1) + x2 − x ln
√
2π − x2 ln x.
The main result of our work is the following. Theorem 2.1. The function −x−1 w ′′′ (x) is strictly completely monotonic on (0, ∞). Proof. We successively have
w′ (x) = ln Γ (x + 1) + xψ (x + 1) − ln
√
2π − 2x ln x + x,
then
w′′ (x) = 2ψ (x) +
1
+ xψ ′ (x) − 2 ln x − 1
x
and
w′′′ (x) = 3ψ ′ (x) −
1 x2
2
+ xψ ′′ (x) − . x
Now, using (2.1)–(2.2) and the convolution theorem for Laplace transforms we get, for x > 0, x−1 w ′′′ (x) =
3 x
1
2
ψ ′ (x) − 3 + ψ ′′ (x) − 2 x x ∫ ∞ ∫ ∞ −tx te
e−tx dt
=3 0
∞
∫
1−e
0 t
∫
=3 0
0
s 1−e
dt − −t
1 2
ds e−tx dt − −s
∞
∫
t 2 e−tx dt − 0
1 2
∞
∫
1−e
0
∫ 0
∞
t 2 e−tx dt −
t 2 e−tx ∞
∫ 0
t 2 e−tx 1−e
∞
∫
dt − 2 −t
te−tx dt 0
dt − 2 −t
∫
∞
te−tx dt ,
0
or x−1 w ′′′ (x) =
∞
∫
ϕ (t ) e−tx dt ,
(2.3)
0
where
ϕ (t ) = 3
t
∫ 0
s 1−
e− s
ds −
1 2
t2 −
t2 1 − e− t
− 2t .
C. Mortici / Applied Mathematics Letters 24 (2011) 869–871
871
Differentiation yields
ϕ ′ (t ) =
t 2 et + tet − t − 2e2t + 4et − 2
(et − 1)2
=
∞ n +1 tn −1 − 2 2 − n − 4 < 0, n! (et − 1)2 n=3
so ϕ is strictly decreasing. For t > 0, we have ϕ (t ) < ϕ (0) = 0, so from (2.3) it results that the function −x−1 w ′′′ (x) is strictly completely monotonic. As a direct consequence of Theorem 2.1, w ′′′ < 0, so w ′′ is strictly decreasing. The function w ′′ can be equivalently written as
1 1 w ′′ (x) = 2 (ψ (x) − ln x) + x ψ ′ (x) − + . x
x
Using the asymptotic formulas [3, Rel. 6.3.18, p. 259, and Rel. 6.4.12, p. 260]
ψ (x) ∼ ln x −
1 2x
−
1 12x2
+ ···,
ψ ′ (x) ∼
1 x
+
1 2x2
+
1 6x3
+ ···,
we have limx→∞ w ′′ (x) = 0; thus w ′′ > 0. In consequence, w is strictly convex. In particular, w ′ is strictly increasing with lim w ′ (x) = − ln
x→ 0
√
2π < 0
and
w′ (1) = 2 − γ − ln
√
2π = 1.4228 . . . > 0
(γ = 0.577215 . . . is the Euler–Mascheroni constant). Now, if β is the unique positive real number such that w ′ (β) = 0, i.e., ln Γ (β + 1) + βψ (β + 1) − ln
√
2π − 2β ln β + β = 0,
then w < 0 on (0, β), and w > 0 on (β, ∞). The numerical value β = 0.34142 . . . can be easily found using computer software such as MAPLE. Finally, w is strictly decreasing on (0, β), and strictly increasing on (β, ∞). ′
′
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. Math. 195 (2006) 134–154. E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, fourth ed., Cambridge University Press, Cambridge, England, 1990. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. G. Andrews, R. Askey, R. Roy, Special functions, in: Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, 1999. P.J. Davis, Leonhard Euler’s integral: a historical profile of the gamma function, Amer. Math. Monthly 66 (1959) 849–869. H. Alzer, Some gamma function inequalities, Math. Comp. 60 (201) (1993) 337–346. H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (217) (1997) 373–389. H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg 68 (1998) 363–372. C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. 93 (1) (2009) 37–45. C. Mortici, Monotonicity properties of the volume of the unit ball in Rn, Optimization Lett. 4 (3) (2010) 457–464. C. Mortici, Very accurate estimates of the polygamma functions, Asymptot. Anal. 68 (3) (2010) 125–134. C. Mortici, The proof of Muqattash-Yahdi conjecture, Math. Comp. Modelling 51 (9–10) (2010) 1154–1159. C. Mortici, A class of integral approximations for the factorial function, Comput. Math. Appl. 59 (6) (2010) 2053–2058. D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1981.