General Mathematics Vol. 13, No. 3 (2005), 65–72
Inequalities for the Polygamma Functions
with Application
Chao-Ping Chen
Dedicated to Professor Dumitru Acu on his 60th anniversary
Abstract
We present some inequalities for the polygamma functions. As an
application, we give the upper and lower bounds for the expression
n
P
1 − ln n − γ, where γ = 0.57721... is the Euler’s constant.
k
k=1
2000 Mathematics Subject Classification: 26D15, 33B15.
Keywords: Inequality, polygamma function, harmonic sequence, Euler’s
constant.
1
Inequalities for the polygamma functions
The gamma function is usually defined for Re z > 0 by
Z∞
tz−1 e−t dt.
Γ(z) =
0
65
66
Chao-Ping Chen
The psi or digamma function, the logarithmic derivative of the gamma
function, and the polygamma functions can be expressed as
∞ X
Γ0 (z)
1
1
ψ(z) =
= −γ +
−
,
Γ(z)
1
+
k
z
+
k
k=0
ψ
(n)
n+1
(z) = (−1)
n!
∞
X
k=0
1
(z + k)n+1
for Re z > 0 and n = 1, 2, ..., where γ = 0.57721... is the Euler’s constant.
M. Merkle [2] established the inequality
∞
2N
X B2k
X
1
1
1
<
+ 2+
2 <
2k+1
x 2x
(x
+
k)
x
k=0
k=1
2N
+1
X
B2k
1
1
< + 2+
x 2x
x2k+1
k=1
for all real x > 0 and all integers N ≥ 1, where Bk denotes Bernoulli
numbers, defined by
∞
X Bj
t
=
tj .
et − 1
j!
j=0
The first five Bernoulli numbers with even indices are
1
1
1
1
5
B2 = , B4 = − , B6 = , B8 = − , B10 = .
6
30
42
30
66
The following Theorem establishes a more general result.
Theorem 1. let m ≥ 0 and n ≥ 1 be integers, then we have for x > 0,
(1)
2m+1
X B2j 1
1
−
ln x −
< ψ(x) <
2x
2j x2j
j=1
2m
X B2j 1
1
−
< ln x −
2x j=1 2j x2j
Inequalities for the Polygamma Functions with Application
67
and
2m
X B2j Γ(n + 2j)
n!
(n − 1)!
+
<
+
xn
2xn+1 j=1 (2j)! xn+2j
(2)
n+1
< (−1)
ψ
(n)
2m+1
X B2j Γ(n + 2j)
n!
(n − 1)!
+
.
(x) <
+
xn
(2j)! xn+2j
2xn+1
j=1
Proof. From Binet’s formula [6, p. 103]
ln Γ(x) =
1
x−
2
ln x − x + ln
√
Z∞ 2π +
0
t
t
−1+
t
2
e −1
e−xt
dt,
t2
we conclude that
1
−
ψ(x) = ln(x) −
2x
(3)
Z∞ 0
t
t
−1+
t
2
e −1
e−xt
dt
t
and therefore
n+1
(4) (−1)
ψ
(n)
n!
(n − 1)!
+ n+1 +
(x) =
n
x
2x
Z∞ 0
t
t
−1+
t
2
e −1
tn−1 e−xt dt.
It follows from Problem 154 in Part I, Chapter 4, of [3] that
(5)
2m
2m+1
X
X B2j
B2j 2j
t
t
t < t
−1+ <
(2j)!
2
(2j)!
e −1
j=1
j=1
for all integers m ≥ 0. the inequality (5) can be also found in [4].
From (3) and (5) we conclude (1), and we obtain (2) from (4) and (5).
The proof of Theorem 1 is complete.
1 (see [1, pag. 258]), (1) can be written as
Note that ψ(x + 1) = ψ(x) + x
(6)
2m+1
2m+1
X B2j 1
X B2j 1
1
1
<
ψ(x
+
1)
−
ln
x
<
,
−
−
2x
2j x2j
2x
2j x2j
j=1
j=1
68
Chao-Ping Chen
and (2) can be written as
(7)
2m+1
X B2j Γ(n + 2j)
n!
(n − 1)!
−
+
<
xn
2j xn+2j
2xn+1
j=1
(−1)n+1 ψ (n) (x + 1) <
2m+1
X B2j Γ(n + 2j)
(n − 1)!
n!
+
.
−
n+2j
xn
(2j)!
2xn+1
x
j=1
In particular, taking in (6) m = 0 we obtain for x > 0,
(8)
1
1
1
−
,
2 < ψ(x + 1) − ln x <
2x 12x
2x
and taking in (7) m = 1 and n = 1 we obtain for x > 0,
(9)
1
1
1
1
1
− ψ 0 (x + 1) <
2 −
3 +
5 −
7 <
x
2x
6x
30x
42x
<
1
1
1
.
2 −
3 +
2x
6x
30x5
The inequalities (8) and (9) play an important role in the proof of Theorem 2 in Section 2.
2
Inequalities for Euler’s constant
Euler’s constant γ = 0.57721... is defined by
1
1 1
γ = lim 1 + + + ... + − ln n .
n→∞
2 3
n
It is of interest to investigate the bounds for the expression
n
P
1 − ln n − γ. The inequality
k
k=1
n
X1
1
1
1
− 2 <
− ln n − γ <
2n 8n
k
2n
k=1
Inequalities for the Polygamma Functions with Application
69
is called in literature Franel’s inequality [3, Ex. 18].
n−1
P 1
It is given [1, p. 258] that ψ(n) =
− γ, and then we get
k
k=1
n−1
X
1
(10)
k=1
k
− γ = ψ(n + 1) − ln n.
Taking in (6) x = n we obtain that
(11)
2m+1
n
2m+1
X B2j 1
X
X B2j 1
1
1
1
−
<
−
ln
n
−
γ
<
−
2j
2j .
2n
2j
k
2n
2j
n
n
j=1
j=1
k=1
n
P
1 − ln n − γ.
k
k=1
L. Toth [5, pag. 264] proposef the following problems:
The inequality (11) provides closer bounds for
(i) Prove that for every positive integers n we have
1
2
2n +
5
<
n
X
1
k=1
k
1
− ln n − γ <
2n +
1
3
.
(ii) Show that 52 can be replaced by a slightly smaller number, but that
1 cannot be replaced by a slightly larger number.
3
The following Theorem 2 answers the problem due to Tóth.
Theorem 2. For every positive integers n,
n
(12)
X1
1
1
≤
− ln n − γ <
,
2n + a
i
2n + b
i=1
with the posible constants
a=
1
1
− 2 and b = .
1−γ
3
70
Chao-Ping Chen
Proof. By (10), the inequality (12) can be rearranged as
b<
1
− 2n ≤ a.
ψ(n + 1) − ln n
Define for x > 0,
φ(x) =
1
− 2x.
ψ(x + 1) − ln x
Differentiating φ and utilizing (8) and (9) reveals that for x > 12
5,
1
− φ0 (x + 1) − 2(ψ(x + 1) − ln x)2 <
x
2
1
1
1
1
12 − 5x
1
−
=
< 0,
< 2− 3+
5 −2
2
2x 12x
2x
6x
30x
360x5
(φ(x + 1) − ln x)2 ψ 0 (x) =
and then the function φ strictly decreases with x > 12
5.
φ(1) =
1
− 2 = 0.3652721186544155...,
1−γ
1
φ(2) =
− 4 = 0.35469600731465752...,
3
− γ − ln 2
2
1
φ(3) =
− 6 = 0.34898948531361115... .
11
− γ − ln 3
6
Therefore, the sequence
φ(n) =
1
− 2n, n ∈ N
ψ(n + 1) − ln n
is strictly decreasing. This leads to
lim φ(n) < φ(n) ≤ φ(1) =
n→∞
1
− 2.
1−γ
Inequalities for the Polygamma Functions with Application
71
Making use of asymptotic formula of ψ (see [1, pag. 259])
ψ(x) = ln x −
1
1
−
+ O(x−4 )(x → ∞),
2x 12x2
we conclude that
1
+ O(x−2 )
lim φ(n) = lim φ(x) = lim 3
.
n→∞
x→∞
x→∞ 1 + O(x−1 )
References
[1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 4th printing, with
corrections, Applied Mathematics Series 55, National Bureau of Standards, Washington, 1965.
[2] M. Merkle, Logarithmic convexity inequalities for the gamma function,
J. Math. Anal. Appl. 203, 1996, 369-380.
[3] G. Pólya, C. Szegö, Problems and Theorems in Analysis, Vol. I and II,
Springer-Verlag, Berlin, Heidelberg.
[4] Z. Sasvári, Inequalities for binomial coefficients, J. Math. Anal. Appl.
236, 1999, 223-226.
[5] L. Tòth, E 3432, Amer. Math. Monthly 98, 1991, no. 3, 264, 99, 1992,
684-685.
72
Chao-Ping Chen
[6] Zh. - X. Wang, D. R. Guo, Introduction to Special Function, The Series of Advanced Physics of Pekin University, Pekin University Press,
Beijing, China, 2000 (in Chinese).
Department of Applied Mathematics and Informatics
Henan Polytechnic University
Jiaozuo City,
Henan 454010 - China
E-mail: chenchaoping@hpu.edu.cn ; chenchaoping@sohu.com