Journal of Inequalities in Pure and
Applied Mathematics
GOOD LOWER AND UPPER BOUNDS ON BINOMIAL
COEFFICIENTS
volume 2, issue 3, article 30,
2001.
PANTELIMON STANICA
Auburn University Montgomery,
Department of Mathematics,
Montgomery, Al 36124-4023, USA
Received 6 November, 2000;
accepted 26 March, 2001.
Communicated by: J. Sandor
and
Institute of Mathematics of Romanian Academy,
Bucharest-Romania
EMail: stanica@strudel.aum.edu
URL: http://sciences.aum.edu/ stanpan
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
We provide good bounds on binomial coefficients, generalizing known ones,
using some results of H. Robbins and of Sasvári.
2000 Mathematics Subject Classification: 05A20, 11B65, 26D15.
Key words: Binomial Coefficients, Stirling’s Formula, Inequalities.
Good Lower and Upper Bounds
on Binomial Coefficients
Contents
1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
Pantelimon Stănică
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1.
Motivation
Analytic techniques can be often used to obtain asymptotics for simply-indexed
sequences. Asymptotic estimates for doubly(multiply)-indexed sequences are
considerably more difficult to obtain (cf. [4], p. 204). Very little is known about
how to obtain asymptotic estimates of these sequences. The estimates that are
known are based on summing over one index at a time. For instance, according
to the same source, the formula
(n−2k)2
n
2n e− 2n
p nπ
∼
k
2
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
3
4
is valid only when |2n − k| ∈ o(n ).
We raise the question of getting good bounds for the binomial coefficient,
which should be valid for any n, k.
In the August-September 2000 issue of American Mathematical Monthly, O.
Krafft proposed the following problem (P10819):
For m ≥ 2, n ≥ 1, we have
mn
mm(n−1)+1
− 12
≥
n
.
n
(m − 1)(m−1)(n−1)
In this note, we are able to improve this inequality (by replacing 1 in the
right-hand
side by a better absolute constant) and also generalize the inequality
mn
to pn .
We also employ a method of Sasvári [5] (see also [2]), to derive better lower
and upper bounds, with the absolute constants replaced by appropriate functions
of m, n, p.
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2.
The Results
The following double inequality for the factorial was shown
n √by H. Robbins in
[3] (1955), a step in a proof of Stirling’s formula n! ∼ ne
2πn.
Lemma 2.1 (Robbins). For n ≥ 1,
√
1
(2.1)
n! = 2π nn+ 2 e−n+r(n) ,
where r(n) satisfies
1
12n+1
< r(n) <
1
.
12n
One approach to get approximations for the binomial coefficient mn
,m≥
pn
p, would be to use Stirling’s approximation for the factorial of Lemma 2.1,
namely
√
(2.2)
1
1
2π nn+ 2 e−n+ 12n+1 < n! <
√
1
1
2π nn+ 2 e−n+ 12n .
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mn
pn
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(mn)!
=
(pn)!((m − p)n)!
>√
Pantelimon Stănică
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Thus
(2.3)
Good Lower and Upper Bounds
on Binomial Coefficients
1
2π (pn)pn+ 2
Close
√
1
1
2π (mn)mn+ 2 e−mn+ 12mn+1
√
1
1
1
e−pn+ 12pn 2π ((m − p)n)(m−p)n+ 2 e−(m−p)n+ 12n(m−p)
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Page 4 of 12
1
1
1
1
1
1
mmn+ 2
− 12pn
− 12n(m−p)
12nm+1
= √ n− 2
1
1 e
2π
(m − p)(m−p)n+ 2 ppn+ 2
J. Ineq. Pure and Appl. Math. 2(3) Art. 30, 2001
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and
(2.4)
mn
pn
<√
√
1
2π (pn)pn+ 2
1
1
2π (mn)mn+ 2 e−mn+ 12mn
√
1
1
1
e−pn+ 12pn+1 2π ((m − p)n)(m−p)n+ 2 e−(m−p)n+ 12n(m−p)+1
1
1
1
1
1
1
mmn+ 2
− 12pn+1
− 12n(m−p)+1
12nm
= √ n− 2
.
1 e
1
2π
(m − p)(m−p)n+ 2 ppn+ 2
Good Lower and Upper Bounds
on Binomial Coefficients
However, we can improve the lower bound, by employing a method of Sasvári
[5] (see also [2]). Let
Pantelimon Stănică
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DN (n, m, p) =
N
X
j=1
B2j
2j(2j − 1)
1
1
1
−
−
2j−1
2j−1
(mn)
(np)
((m − p)n)2j−1
,
with B2j , the Bernoulli numbers defined by
∞
t X B2j 2j
t
=
1
−
+
t
et − 1
2 j=1 (2j)!
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and
∆(n, m, p) = r(mn) − r(pn) − r((m − p)n).
We show that ∆(n, m, p) − DN (n, m, p) is an increasing (decreasing) function
of n if N is even (respectively, odd). We proceed to the proof of the above fact.
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By the Binet formula (see [2]), we get
Z ∞ t
t
1
−1+
e−tx d x, x ∈ (0, ∞),
r(x) =
2
t−1
t
e
2
0
R ∞ j −t
and using j! = 0 t e d t, we get
Z ∞
1
∆(n, m, p) − DN (n, m, p) =
PN (t)Qn (t)d t,
t2
0
where
N
t
t X B2j 2j
PN (t) = t
−1+ −
t
e −1
2 j=1 (2j)!
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
and
Qn (t) = e−mnt − e(m−p)nt − e−pnt .
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Sasvári proved that PN (t) is positive (negative) if N is even (respectively, odd).
So we need to show that Qn (t) is increasing with respect to n, if t > 0 and
m > p ≥ 1. Since Qn (t) = f (e−nt ), for f (u) = um − um−p − up , it suffices to
show that f is decreasing on (0, 1), that is f 0 (u) < 0 on (0, 1). Now, f 0 (u) < 0
is equivalent to mum−1 − (m − p)um−p−1 − pup−1 < 0, which is equivalent to
g(u) = um−2p (mup − m + p) < p. If m ≥ 2p, then g(u) ≤ mup − m + p < p.
If 1 < m < 2p, then
Contents
g 0 (u) = (m − 2p)um−2p−1 (mup − m + p) + mpum−p−1
= um−2p−1 (m − 2p)(mup − m + 2p) > 0.
Therefore, for 0 < u < 1, we have g(u) < g(1) = p and the claim is proved.
Thus, we have
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Theorem 2.2.
1
1
1
mmn+ 2
(2.5) √ eD2N +1 (n,m,p) n− 2
1
1
2π
(m − p)(m−p)n+ 2 ppn+ 2
1
mn
1 D2N (n,m,p) − 1
mmn+ 2
<
<√ e
n 2
1
1 .
pn
2π
(m − p)(m−p)n+ 2 ppn+ 2
Taking N = 0 and observing that B2 = 16 , we get
Corollary 2.3.
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
mn+ 12
1
1
1
1
1
m
1
(2.6) √ e 12n ( m − p − m−p ) n− 2
1
1
2π
(m − p)(m−p)n+ 2 ppn+ 2
1
mmn+ 2
mn
1
− 12
<
<√ n
1 .
1
pn
2π
(m − p)(m−p)n+ 2 ppn+ 2
By using (2.4), the upper bound can be improved and we get
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Corollary 2.4.
1
1
1
1
1
1
1
mmn+ 2
(2.7) √ e 12n ( m − p − m−p ) n− 2
1
1
2π
(m − p)(m−p)n+ 2 ppn+ 2
1
1
1
1
mn
1
mmn+ 2
− 12pn+1
− 12n(m−p)+1
− 12
12nm
<
<√ e
n
1
1
pn
2π
(m − p)(m−p)n+ 2 ppn+ 2
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To show that the upper bound of Corollary 2.4 improves upon the one of
Corollary 2.3 we use (2.4) and prove that
1
1
1
−
−
<0
12nm 12pn + 1 12n(m − p) + 1
(2.8)
by rewriting as
1
1
1
−
−
12nm 12pn + 1 12n(m − p) + 1
144mnp(m − p) + 12n(m − p) + 12pm + 1
=
−144mn2 (m − p) − 12mn − 144m2 np − 12mn
12mn(12pn + 1)(12n(m − p) + 1)
2
−144mnp − 12np + 12pm + 1 − 144mn2 (m − p) − 12mn
=
< 0.
12mn(12pn + 1)(12n(m − p) + 1)
Remark 2.1. The left side of Corollary 2.3 differs slightly from (2.3), in that
12mn + 1 is replaced by 12mn. Therefore, the left side of (2.6) is an improvement of (2.3).
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
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Next, we prove another result, where the expressions given by exponential
powers are replaced by functions of n only. We prove
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Theorem 2.5. Let m, n, p be positive integers, with m > p ≥ 1 and n ≥ 1.
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Then
1
1
1
1
mmn+ 2
(2.9) √ e− 8n n− 2
1
1
2π
(m − p)(m−p)n+ 2 ppn+ 2
1
mn
1
mmn+ 2
− 12
<
<√ n
1
1
pn
2π
(m − p)(m−p)n+ 2 ppn+ 2
Proof. Using Corollary 2.3, we need to show that
(2.10)
1
1
1
1
−
−
≥− .
12nm 12np 12n(m − p)
8n
The inequality (2.10) is equivalent to
(2.11)
Good Lower and Upper Bounds
on Binomial Coefficients
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m
3
1
+
≤ .
m p(m − p)
2
Let x = m − p. Thus, x ≥ 1. We show first that the left side of (2.11),
2 +px+p2
g(x, p) = xpx(p+x)
is decreasing with respect to x, that is
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d g(x, p)
1
1
=− 2 +
< 0,
dx
x
(p + x)2
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which is certainly true. Therefore,
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2
g(x, p) ≤ g(1, p) =
p +p+1
(= h(p)).
p(p + 1)
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Since h0 (p) = − p22p+1
< 0, we get that h is decreasing with respect to p, so
(p+1)2
3
g(x, p) ≤ h(p) ≤ h(1) = .
2
Now we provide a further simplification of Theorem 2.5. The following
lemma proves to be very useful.
Lemma 2.6. Let p ≥ 1 be a fixed natural number and m ≥ p + 1. Then the
m− 12
m
function m−p
is decreasing (with respect to m) and
lim
m→∞
m
m−p
m− 12
= ep .
Proof. It suffices to prove that the function h(x) = log
is decreasing and its limit is ep . By differentiation
h0 (x) = log
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
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x
x−p
x− 12
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, x ≥ p + 1,
x
2xp − p
−
.
x − p 2x(x − p)
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Since
x
p
p
p2
= − log (1 − ) < + 2
log
x−p
x
x 2x
(by Taylor expansion), we get
h0 (x) <
p
p2
p
2p2 − p
x − px − p2
+ 2− −
=
< 0,
x 2x
x 2x(x − p)
x(x − p)
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since p ≥ 1, so h is decreasing. The lower bound of this function is its limit,
x
− 12
which is ep , since 1 − xp → e−p , and x−p
→ 1 as x → ∞.
x
Using Theorem 2.5 and Lemma 2.6, we get
Theorem 2.7. We have, for m > p ≥ 1 and n ≥ 2,
1
1
mn
1
mm(n−1)+1
> √ ep− 8n n− 2
(2.12)
1 .
pn
2π
(m − p)(m−p)(n−1)−p+1 ppn+ 2
Taking p = 1, we obtain a stronger version of the inequality P10819, namely
Corollary 2.8. We have, for m > 1 and n ≥ 2,
1
1
mn
mm(n−1)+1
.
(2.13)
> 1.08444 e− 8n n− 2
n
(m − 1)(m−1)(n−1)
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
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References
[1] O. KRAFFT, Problem P10819, Amer. Math. Monthly, 107 (2000), 652.
[2] E. RODNEY, Problem 10310, Amer. Math. Monthly, (1993), 499; with a
solution in Amer. Math. Monthly, (1996), 431–432, by MMRS.
[3] H. ROBBINS, A Remark on Stirling Formula, Amer. Math. Monthly, 62
(1955), 26–29.
[4] K. ROSEN (ed.), Handbook of Discrete Combinatorial Mathematics, CRC
Press, 2000.
Good Lower and Upper Bounds
on Binomial Coefficients
Pantelimon Stănică
[5] Z. SASVÁRI, Inequalities for Binomial Coefficients, J. Math. Anal. and
App., 236 (1999), 223–226.
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