Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 4, Article 100, 2005
MONOTONICITY AND CONVEXITY FOR THE GAMMA FUNCTION
CHAO-PING CHEN
C OLLEGE OF M ATHEMATICS AND I NFORMATICS
H ENAN P OLYTECHNIC U NIVERSITY
J IAOZUO C ITY, H ENAN 454010, C HINA
chenchaoping@hpu.edu.cn
Received 02 July, 2005; accepted 08 September, 2005
Communicated by A. Laforgia
A BSTRACT. Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function
θa,b (x) = [Γ(x + b)/Γ(x + a)]1/(b−a) − x is strictly convex and decreasing on (−a, ∞) with
θa,b (∞) = a+b−1
and θa,b (−a) = a, where Γ denotes the Euler’s gamma function.
2
Key words and phrases: Gamma function; monotonicity; convexity.
2000 Mathematics Subject Classification. 33B15, 26A48.
1. I NTRODUCTION
Kazarinoff [10] proved that the function θ(n),
1 · 3 · 5 · · · (2n − 1)
1
,
=p
2 · 4 · 6 · · · (2n)
π(n + θ(n))
satisfies
(1.1)
1
1
< θ(n) < ,
4
2
n ∈ N.
More generally, set
Γ(x + 1)
θ(x) =
Γ(x + 12 )
where
Z
Γ(x) =
2
− x,
1
x>− ,
2
∞
tx−1 e−t dt (x > 0)
0
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
The author was supported in part by the Science Foundation of the Project for Fostering Innovation Talents at Universities of Henan
Province, China.
201-05
2
C H .-P. C HEN
is the Euler’s gamma function. Watson [15] proved that the function θ is strictly decreasing on
(−1/2, ∞). Applying this result, together with the observation that θ(∞) = 1/4, θ(−1/2) =
1/2 and θ(1) = 4π −1 − 1, we obviously imply sharper inequalities:
1
1
1
(1.2)
< θ(x) <
for x > − ,
4
2
2
1
−1
(1.3)
< θ(x) ≤ 4π − 1 for x ≥ 1.
4
In particular, take in (1.3) x = n, we get
(1.4)
1
p
π(n + 4π −1 − 1)
≤
1 · 3 · 5 · · · (2n − 1)
1
<p
,
2 · 4 · 6 · · · (2n)
π(n + 1/4)
and the constants 4π −1 − 1 and 1/4 are the best possible.
The inequality (1.4) is called Wallis’ inequality. For more information on Wallis’ inequality,
please refer to the paper [6] and the references therein.
H. Alzer [2] proved that the function θ is strictly decreasing on [0, ∞). Applying this result,
he showed that for all integers n ≥ 1,
r
r
n+A
Ωn−1
n+B
(1.5)
<
≤
2π
Ωn
2π
with the best possible constants
1
π
A=
and B = = 0.57079 . . . ,
2
2
n/2
where Ωn = π /Γ(1+n/2) denotes the volume of the unit ball in Rn . (1.5) is an improvement
of the following result given by Borewardt [5, p. 253]
r
r
n
Ωn−1
n+1
(1.6)
≤
≤
.
2π
Ωn
2π
If we denote by
1
Γ(x + b) b−a
θa,b (x) =
− x,
Γ(x + a)
then we conclude from the representations [1, p. 257]
Γ(x + a)
(a − b)(a + b − 1)
(1.7)
xb−a
=1+
+ O x−2
(x → ∞),
Γ(x + b)
2x
that
(1.8)
( )
1/(b−a)
1 Γ(x + b)
a+b−1
θa,b (x) = x
−1 →
x Γ(x + a)
2
as x → ∞.
Hence it is of interest to investigate the possible monotonic character of the function x 7→
θa,b (x).
Theorem 1.1. Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function
θa,b (x) = [Γ(x + b)/Γ(x + a)]1/(b−a) − x is strictly convex and decreasing on (−a, ∞).
Since θa,b (x) = θb,a (x), it is clear that x 7→ θa,b (x) is strictly convex and decreasing on
(−b, ∞) for 0 ≤ b < a < b + 1.
, θa,b (−a) = a and the monotonicity of x 7→ θa,b (x), we obtain the
From θa,b (∞) = a+b−1
2
following
J. Inequal. Pure and Appl. Math., 6(4) Art. 100, 2005
http://jipam.vu.edu.au/
M ONOTONICITY AND C ONVEXITY FOR
THE
G AMMA F UNCTION
3
Corollary 1.2. Let a and b be given real numbers with 0 ≤ a < b < a + 1, then for x > −a,
b−a
a+b−1
Γ(x + b)
(1.9)
x+
<
< (x + a)b−a .
2
Γ(x + a)
A proof of the theorem above has been shown in [8], here we provide another proof. The
ratio of two gamma functions has been investigated intensively by many authors. For example,
Gautschi [9] proved the following inequalities
(1.10)
x1−x <
Γ(x + 1)
< (x + 1)1−s ,
Γ(x + s)
0 < s < 1, x = 1, 2, . . . .
Kershaw [11] has given some improvements of these inequalities such as
!1−s
r
s 1−s Γ(x + 1)
1
1
<
< x− + x+
(1.11)
x+
4
2
Γ(x + s)
2
for real x > 0 and 0 < s < 1.
Inequalities for the ratio Γ(x + 1)/Γ(1 + λ) (x > 0; λ ∈ (0, 1)) have a remarkable application, they can be used to obtain estimates for ultraspherical polynomials. The ultraspherical
polynomials are defined by
[n/2]
Pn(λ) (x)
=
X
(−1)k
k=0
Γ(n − k + λ)
(2x)n−2k ,
Γ(λ)Γ(k + 1)Γ(n − 2k + 1)
where n ≥ 0 is an integer and λ > 0 is a real number.
In 1931, S. Bernstein [4] proved the following inequality for ultraspherical polynomials: If
0 < λ < 1, n ≥ 1, and 0 < θ < π, then
21−λ λ−1
(1.12)
(sin θ)λ Pn(λ) cos θ <
n ,
Γ(λ)
where the constant 21−λ /Γ(λ) cannot be replaced by a smaller one.
We note that inequality (1.12) with λ = 1/2 leads to a well-known inequality for the Legendre
(1/2)
polynomials Pn = Pn :
1/2
2
1/2
(1.13)
(sin θ) |Pn cos θ| <
n−1/2 .
π
Several authors presented remarkable refinements of (1.12). They proved that in (1.12) the
factor nλ−1 can be replaced by smaller expressions. The left-hand inequality of (1.11) was also
considered in 1984 by L. Lorch [13]. He obtained the following results for integer x > 0:
Γ(x + 1) s 1−s
(1.14)
> x+
for 0 < s < 1 or s > 2,
Γ(x + s)
2
Γ(x + 1) s 1−s
(1.15)
> x+
for 1 < s < 2.
Γ(x + s)
2
Lorch [13] used (1.14) to prove a sharpened inequality for ultraspherical polynomials:
(1.16)
(sin θ)λ |Pn(λ) cos θ| <
21−λ
(n + λ)λ−1 .
Γ(λ)
In 1992, A. Laforgia [12] proved in (1.12) that the term nλ−1 can be replaced by Γ(n+λ)/Γ(n+
1). Since Γ(n + λ)/Γ(n + 1) < nλ−1 , see [9], this provides another refinement of Bernstein’s
J. Inequal. Pure and Appl. Math., 6(4) Art. 100, 2005
http://jipam.vu.edu.au/
4
C H .-P. C HEN
inequality. In 1994, Y. Chow et al. [7] showed that (1.12) holds with Γ(n+2λ)(Γ(n+1))−1 (n+
λ)−λ instead of nλ−1 . This sharpens Lorch’s result for all λ ∈ (0, 1/2), since the inequality
Γ(n + 2λ)
< (n + λ)λ−1
Γ(n + 1)(n + λ)λ
is valid for all λ ∈ (0, 1/2). In 1997, H. Alzer [3] showed the following inequality
3
1−λ
Γ
n
+
λ
2
2
.
(1.17)
(sin θ)λ |Pn(λ) cos θ| <
·
Γ(λ) Γ n + 1 + 12 λ
Inequality (1.17) refines the results given by S. Bernstein, L. Lorch and A. Laforgia.
2. P ROOF OF T HEOREM 1.1
Easy computation yields
1
[ψ(x + b) − ψ(x + a)](θa,b (x) + x) − 1,
b−a
00
(b − a)θa,b
(x)
1
= ψ 0 (x + b) − ψ 0 (x + a) +
[ψ(x + b) − ψ(x + a)]2 .
θa,b (x) + x
b−a
Using the representations [14, p. 16]
Z ∞ −t
Γ0 (x)
e − e−xt
ψ(x) =
= −γ +
dt,
Γ(x)
1 − e−t
0
Z ∞
t
0
ψ (x) =
e−xt dt
−t
1
−
e
0
for x > 0, γ = 0.57721 . . . is the Euler-Mascheroni constant, it follows that
Z ∞
2
Z ∞
00
(b − a)θa,b
(x)
1
−(x+a)t
−(x+a)t
=−
tδ(t)e
dt +
δ(t)e
dt ,
θa,b (x) + x
b−a
0
0
where
1 − e−(b−a)t
δ(t) =
and δ(0) = b − a.
1 − e−t
By using the convolution theorem for Laplace transforms, we have
Z ∞
00
(b − a)θa,b
(x)
=−
tδ(t)e−(x+a)t dt
θa,b (x) + x
0
Z ∞ Z t
1
(2.1)
+
δ(s)δ(t − s) ds e−(x+a)t dt
b−a 0
0
Z ∞
=
e−(x+a)t ω(t) dt,
0
θa,b
(x) =
0
where
Z t
1
(2.2)
ω(t) =
δ(s)δ(t − s) − δ(t) ds.
b−a
0
Now we are in a position to prove that
1
(2.3)
δ(s)δ(t − s) − δ(t) > 0 for t > s > 0.
b−a
Define for t > s > 0,
δ(s)
ϕ(t) = ln
+ ln δ(t − s) − ln δ(t).
b−a
J. Inequal. Pure and Appl. Math., 6(4) Art. 100, 2005
http://jipam.vu.edu.au/
M ONOTONICITY AND C ONVEXITY FOR
THE
G AMMA F UNCTION
5
Elementary calculations reveal that
δ 0 (t − s) δ 0 (t)
0
ϕ (t) =
−
,
δ(t − s)
δ(t)
0 0
δ (t)
et
(b − a)2 e(b−a)t
= (ln δ(t))00 = t
−
2 .
δ(t)
(e − 1)2
[e(b−a)t − 1]
Defined for r ∈ (0, 1),
1
r2 ert
g(r) = rt
= 2
2
(e − 1)
t
Since x 7→
rt/2
sinh(rt/2)
2
.
sinh x
x
is strictly increasing with x ∈ (0, ∞), we get g is strictly decreasing with
0 0
(t)
< 0 for t > 0 and 0 < b − a < 1, and then, ϕ0 (t) > 0 and
r ∈ (0, 1). This implies that δδ(t)
00
ϕ(t) > ϕ(s) = 0. This means (2.3) holds, and thus, θa,b
(x) > 0 (x > −a) follows from (2.1),
(2.2) and (2.3).
From the representations (1.7) and
1
ψ(x) = ln x −
+ O x−2
(x → ∞),
2x
(see [1, p. 259]), we conclude that
Γ(x + b)
= 1,
lim xa−b
(2.4)
x→∞
Γ x+a
1
(2.5)
lim x ψ(x + b) − ψ x + a
=
.
x→∞
b−a
0
From (2.4), (2.5) and the monotonicity of the function x 7→ θa,b
(x), we imply
0
0
θa,b
(x) < lim θa,b
(x)
x→∞
1
b−a
1
Γ(x
+
b)
= lim
xa−b
x [ψ(x + b) − ψ(x + a)] − 1
x→∞ b − a
Γ(x + a)
= 0.
The proof is complete.
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J. Inequal. Pure and Appl. Math., 6(4) Art. 100, 2005
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