R E S E A R C H
Open Access
An inequality for the gamma function via
statistics and applications
Pingping Zhang and Mingjin Wang
**Correspondence:
[email protected] Department of Mathematics, Changzhou University, Changzhou, Jiangsu 213164, P.R. China
Abstract
The aim of this paper is to establish an inequality for the gamma function, using a statistical method. Applications of the inequality are also given, including some estimates of
π
.MSC: Primary 62F11; 33B15; 33B10
Keywords: inequality; variance; unbiased estimate; UMVUE
1 Introduction and main result
Recently, there have been many papers about the ratio of gamma functions in the liter-ature; see [–]. Some of the papers use statistical methods. Gurland [] has given an inequality satisfied by the gamma function, using the so-called Cramér-Rao lower bound for the variance of unbiased estimators. Olkin [] has given an extension of Gurland’s in-equality. Gokhale [] has given another inequality, which used an analogue of the Cramér-Rao lower bound derived by Cramér-Rao []. Cramér-Rao gave a stronger version of Wallis’ formula []. We, inspired by the above papers, give an inequality concerning the gamma function. Ap-plications of the inequality are also given. We first recall some definitions, notation, and well-known results in statistical theory, which will be used in this paper.
A normal distributionN(μ,σ) is described by the probability density function,
p(x) =√ π σe
–(x–μ)
σ , x∈R. (.)
When a random variableXis distributed normally with meanμand varianceσ, we write
X∼N(μ,σ).
Suppose thatx,x, . . . ,xnis a sample from a population with a distribution function
Fθ(x) (θ∈). Letgˆ=gˆ(x,x, . . . ,xn) be an estimator of a parametric functiong(θ). IfE(gˆ) =
g(θ) for all values of parameterθ∈, we callgˆan unbiased estimator ofg(θ).
Consider an estimation ofg(θ) based on a samplex,x, . . . ,xnfrom some member of a
family of distribution functionsFθ(x),θ∈, whereis the parameter space. An unbiased estimatorgˆ(x,x, . . . ,xn) ofg(θ) is UMVUE, if∀θ∈,
varθ(gˆ(x,x, . . . ,xn)≤varθ
˜
g(x,x, . . . ,xn)
, (.)
for any other unbiased estimatorg˜.
Euler’s gamma functionis defined forx> by
(x) =
∞
tx–e–tdt. (.)
Ifx,x, . . . ,xnis a sample from a population with distributionN(μ,σ), then
ˆ σ= (
n ) √
(n+ )
n
i=
x
i (.)
is the UMVUE ofσ.
The main result of this paper is the following theorem.
Theorem . Suppose that nk(k= , , . . . ,m)are nonnegative integers andλk∈R,such
that≤λk≤, mk=λk= .Then we have
n(n) (n+ )
– ≤
m
k= λk
nk(nk)
(nk+ )
–
, (.)
where n= mk=nk.
2 Proof of the main result
In this section, we use statistical methods to prove the theorem.
Proof Letx,x, . . . ,xn,x,x, . . . ,xn,xm,xm, . . . ,xmnm be a random sample from a
normal distributionX∼N(μ,σ). From (.), it is known that
ˆ σ= (
n ) √
(n+ )
m
i= ni
j=
x
ij (.)
is the UMVUE ofσ, wheren= mi=ni.
For anyxk,xk, . . . ,xknk, ≤k≤m
(nk)
√
(nk+ )
nk
k=
x
ki (.)
is an unbiased estimate ofσ.
Using (.), we construct a new unbiased estimate ofσ,i.e.,
ˆ σ=
m
k=
λk(nk)
√
(nk+)
nk
k=
x
ki, (.)
where ≤λk≤, mk=λk= .
Due to the definition of the UMVUE, the following inequality holds:
After some simple computations, we can obtain
D(σˆ) =
n(n )
(n+ )
–
σ (.)
and
D(σˆ) = m
k= λk
nk(nk)
(nk+ )
–
σ. (.)
Substituting (.) and (.) into (.) gives (.). Thus, we complete the proof.
3 Some applications of Theorem 1.1
In this section, we show some applications of the main result of this paper. First, we give the following inequalities, which include the gamma function and a trigonometric function.
Theorem . Suppose n is any positive integer,andθ∈R,then
(n+ ) (n+
)
≤nsinθ+ –sinθ (n+ ) (n+
)
. (.)
Proof Lettingm= andλ=sinθ,λ=cosθin (.) gives
(n+n)(n+n )
(n+n+
)
–
≤sinθ
n(n)
(n+ )
–
+cosθ
n(n)
(n+ )
–
. (.)
Lettingn=n=ain (.), one obtains (a+ )
(a+ )
– sinθ+cosθ (a+ ) (a+
)
≤a –sinθ–cosθ. (.)
Employing the following trigonometric formula:
sinθ+cosθ= –sin
θ
, (.)
(.) becomes
(a+ ) (a+
)
≤asin(θ) + –sin(θ)
(a+ ) (a+
)
. (.)
Replacing θbyθ andabyn, we obtain (.). Thus, we finish the proof. In [], Gurland gave the following estimator ofπ:
n+ (n+ )
(n)!! (n– )!!
Mortici [] gave the refinements of Gurland’s formula forπ:
n+
n+ n+
+
,n –
,n
(n)!! (n– )!!
<π. (.)
Using (.), we can get the following similar result.
Corollary . Suppose n is any nonnegative integer,then
n
(n)!! (n– )!!
–
(n)!! (n– )!!
<π. (.)
Proof Lettingθ=π in (.) gives
(n+ )
(n+ )
≤n+
(n+ ) (n+
)
. (.)
We have
(q)!! (q– )!!=
√
π(q+ )
(q+). (.)
See,e.g., []. So
n
(n)!! (n– )!!
–
(n)!! (n– )!!
≤π. (.)
Because the equality in (.) cannot hold, we get (.).
Now we give an inequality involving combinational coefficientsmn, defined by
n m
= n!
m!(n–m)!.
Theorem . Suppose m,w are any positive integers,then
m
(mw)!! (mw– )!!
+
m
wπ–
(w)!! (w– )!!
m m
–
≤wπ. (.)
Proof Since
m
k=
m k
m = . (.)
Lettingnk= w,λk=
m k
m in (.), thenn= mk=nk= mw. We get
mw
(mw) (mw+
)
– ≤
m
k=
m k
m
w
(w) (w+
)
–
Using the inequality of (.) and after some simple derivations, we have
m
(mw)!! (mw– )!!
–
m
k=
m k
(w)!! (w– )!!
m
π
≤w
–
m
k=
m k
m
. (.)
Substituting
m
k=
m k
=
m m
(.)
into (.) one obtains (.).
The special casew= of (.) results in
m
(m)!! (m– )!!
+π– m
m m
–
≤π. (.)
Finally, we give the following double inequality forπ.
Theorem . Let p,d are positive integers,then
pd
(pd+ p)!! (pd+ p– )!!
–
(p)!! (p– )!!
<π< d
(p+ )[(d+ )((pd+p+d–)!! (pd+p+d)!! )– (
(p–)!! (p)!! )]
. (.)
Proof Lettingm= d+ andn=n=· · ·=nm= p+ in (.) gives
(p+ )(d+ )
(pd+p+d+ ) (pd+p+d+ ) – ≤
d+
k= λk
p+
(p+ )
(p+ )
–
. (.)
Using (.), the inequality (.) can be written in the equivalent form
π< ( –
d+ k= λk)
(p+ )[(d+ )((pd+p+d–)!!(pd+p+d)!! )– d+ k= λk(
(p–)!! (p)!! )]
. (.)
Lettingλ=λ=· · ·=λm=d+ in (.) gives
π< d
(p+ )[(d+ )((pd+p+d–)!! (pd+p+d)!! )– (
(p–)!! (p)!! )]
Similarly, lettingm= d+ ,n=n=· · ·=nm= p, andλ=λ=· · ·=λm=d+ in (.),
one can obtain
pd
(pd+ p)!! (pd+ p– )!!
–
(p)!! (p– )!!
<π. (.)
Then the inequality of (.) is the combination of the inequality (.) and the inequality
(.).
The special casep= of (.) results in
d
(d+ )!! (d+ )!!
–
<π< d [(d+ )((d+)!!
(d+)!!)– ]
. (.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank two anonymous referees for many helpful comments and suggestions. We acknowledge support by the National Natural Science Foundation (grant 11271057) of China.
Received: 23 December 2014 Accepted: 20 May 2015 References
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