Journal of Inequalities in Pure and
Applied Mathematics
MONOTONICITY AND CONVEXITY FOR THE GAMMA FUNCTION
CHAO-PING CHEN
College of Mathematics and Informatics
Henan Polytechnic University
Jiaozuo City, Henan 454010, China.
volume 6, issue 4, article 100,
2005.
Received 02 July, 2005;
accepted 08 September, 2005.
Communicated by: A. Laforgia
EMail: chenchaoping@hpu.edu.cn
Abstract
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Abstract
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
2
gamma function.
Monotonicity and Convexity for
the Gamma Function
2000 Mathematics Subject Classification: 33B15, 26A48.
Key words: Gamma function; monotonicity; convexity.
Chao-Ping Chen
The author was supported in part by the Science Foundation of the Project for Fostering Innovation Talents at Universities of Henan Province, China
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Contents
3
9
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1.
Introduction
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
Chao-Ping Chen
Γ(x + 1)
θ(x) =
Γ(x + 12 )
2
− x,
1
x>− ,
2
Z
Γ(x) =
∞
tx−1 e−t dt (x > 0)
0
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:
(1.3)
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where
(1.2)
Monotonicity and Convexity for
the Gamma Function
1
1
1
< θ(x) <
for x > − ,
4
2
2
1
< θ(x) ≤ 4π −1 − 1 for x ≥ 1.
4
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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
Monotonicity and Convexity for
the Gamma Function
Chao-Ping Chen
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with the best possible constants
1
A=
2
Contents
π
and B = = 0.57079 . . . ,
2
where Ωn = π n/2 /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
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If we denote by
Γ(x + b)
θa,b (x) =
Γ(x + a)
1
b−a
− x,
J. Ineq. Pure and Appl. Math. 6(4) Art. 100, 2005
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then we conclude from the representations [1, p. 257]
(1.7)
xb−a
Γ(x + a)
(a − b)(a + b − 1)
=1+
+ O x−2
Γ(x + b)
2x
(x → ∞),
that
)
( 1/(b−a)
a+b−1
1 Γ(x + b)
−1 →
(1.8) θa,b (x) = x
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.
From θa,b (∞) = a+b−1
, θa,b (−a) = a and the monotonicity of x 7→ θa,b (x),
2
we obtain the following
Corollary 1.2. Let a and b be given real numbers with 0 ≤ a < b < a + 1, then
for x > −a,
(1.9)
a+b−1
x+
2
b−a
Γ(x + b)
<
< (x + a)b−a .
Γ(x + a)
Monotonicity and Convexity for
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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
(1.11)
s 1−s Γ(x + 1)
x+
<
<
2
Γ(x + s)
1
x− +
2
r
1
x+
4
!1−s
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
k=0
(−1)k
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
(1.12)
21−λ λ−1
n ,
(sin θ)λ Pn(λ) cos θ <
Γ(λ)
Monotonicity and Convexity for
the Gamma Function
Chao-Ping Chen
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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
(1/2)
for the Legendre polynomials Pn = Pn :
(1.13)
1/2
(sin θ)
1/2
2
n−1/2 .
|Pn cos θ| <
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 lefthand inequality of (1.11) was also considered in 1984 by L. Lorch [13]. He
obtained the following results for integer x > 0:
(1.14)
(1.15)
Γ(x + 1) > x+
Γ(x + s)
Γ(x + 1) > x+
Γ(x + s)
s 1−s
2
s 1−s
2
for
0 < s < 1 or
for
1 < s < 2.
s > 2,
Monotonicity and Convexity for
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Chao-Ping Chen
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Lorch [13] used (1.14) to prove a sharpened inequality for ultraspherical polynomials:
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1−λ
(1.16)
(sin θ)λ |Pn(λ) cos θ| <
2
(n + λ)λ−1 .
Γ(λ)
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λ−1
In 1992, A. Laforgia [12] proved in (1.12) that the term n
can be replaced
λ−1
by Γ(n + λ)/Γ(n + 1). Since Γ(n + λ)/Γ(n + 1) < n , see [9], this provides
another refinement of Bernstein’s inequality. In 1994, Y. Chow et al. [7] showed
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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
Γ n + 32 λ
21−λ
λ
(λ)
.
(1.17)
(sin θ) |Pn cos θ| <
·
Γ(λ) Γ n + 1 + 12 λ
Inequality (1.17) refines the results given by S. Bernstein, L. Lorch and A.
Laforgia.
Monotonicity and Convexity for
the Gamma Function
Chao-Ping Chen
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2.
Proof of Theorem 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
0
θa,b
(x) =
Monotonicity and Convexity for
the Gamma Function
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) =
1 − e−t
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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
Monotonicity and Convexity for
the Gamma Function
where
Z t
(2.2)
ω(t) =
0
1
δ(s)δ(t − s) − δ(t) ds.
b−a
Now we are in a position to prove that
(2.3)
1
δ(s)δ(t − s) − δ(t) > 0 for
b−a
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t > s > 0.
Define for t > s > 0,
ϕ(t) = ln
Chao-Ping Chen
δ(s)
+ ln δ(t − s) − ln δ(t).
b−a
Elementary calculations reveal that
δ 0 (t − s) δ 0 (t)
−
,
ϕ0 (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]
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Defined for r ∈ (0, 1),
r2 ert
1
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
0 0
(t)
decreasing with r ∈ (0, 1). This implies that δδ(t)
< 0 for t > 0 and 0 <
0
b − a < 1, and then, ϕ (t) > 0 and ϕ(t) > ϕ(s) = 0. This means (2.3) holds,
00
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,
(2.4)
lim xa−b
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
θa,b
(x)
< lim
x→∞
0
θa,b
(x)
1
b−a
1
a−b Γ(x + b)
x
x [ψ(x + b) − ψ(x + a)] − 1
= lim
x→∞ b − a
Γ(x + a)
= 0.
The proof is complete.
Monotonicity and Convexity for
the Gamma Function
Chao-Ping Chen
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References
[1] M. ABRAMOWITZ AND I.A. STEGUN (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National
Bureau of Standards, Applied Mathematics Series 55, 4th printing, with
corrections, Washington, 1965.
[2] H. ALZER, Inequalities for the volume of unit ball in Rn , J. Math. Anal.
Appl., 252 (2000), 353–363.
[3] H. ALZER, On Bernstein’s inequality for ultraspherical polynomials, J.
Math. Anal. Appl., 252 (2000), 353–363.
[4] S. BERNSTEIN, Sur les polynômes orthogonaux relatifs à un segment
fini, J. Math., 10 (1931), 219–286.
[5] K.H. BORGWARDT, The Simplex Method, Springer-Verlag, Berlin,
1987.
[6] CHAO-PING CHEN AND FENG QI, The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc., 133(2) (2005), 397–401.
[7] Y. CHOW, L. GATTESCHI AND R. WONG, A Bernstein-type for the Jacobi polynomial, Proc. Amer. Math. Soc., 121(2) (1994), 703–709.
[8] N. ELEZOVIĆ, C. GIORDANO AND J. PEČARIĆ, The best bounds in
Gautschi’s inequality, Math. Inequal. Appl., 3(2) (2000), 239–252.
[9] W. GAUTSCHI, Some elementary inequalities relating to the gamme and
incomplete gamma function, J. Math. Phys., 38 (1959), 77–81.
Monotonicity and Convexity for
the Gamma Function
Chao-Ping Chen
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[10] D.K. KAZARINOFF, On Wallis’ formula, Edinburgh. Math. Soc. Notes,
No. 40 (1956), 19–21.
[11] D. KERSHAW, Some extensions of W. Gautschi’s inequalities for gamma
function, Math. Comp., 41 (164) (1983), 607–611.
[12] A. LAFORGIA, A simple proof of the Bernstein inequality for ultraspherical polynomials, Boll. Un. Mat. Ital., 6A (1992), 267–269.
[13] L. LORCH, Inequalities for ultraspherical polynomials and the gamma
function, J. Approx. Theory, 40 (1984), 115–120.
Monotonicity and Convexity for
the Gamma Function
[14] W. MAGNUS, F. OBERHETTINGER AND R.P. SONI, Formulas and
Theorems for the Special Functions of Mathematical Physics, Springer,
Berlin, 1966.
Chao-Ping Chen
[15] G.N. WATSON, A note on Gamma functions, Proc. Edinburgh Math. Soc.,
(2) 11 1958/1959 Edinburgh Math. Notes No. 42 (misprinted 41) (1959),
7–9.
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