volume 7, issue 2, article 45, 2006.
Received 28 October, 2005;
accepted 27 November, 2005.
Communicated by:F. Qi
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Journal of Inequalities in Pure and Applied Mathematics
MONOTONICITY AND CONCAVITY PROPERTIES OF SOME FUNCTIONS INVOLVING THE GAMMA FUNCTION
WITH APPLICATIONS
SENLIN GUO
Department of Mathematics University of Manitoba Winnipeg, MB, R3T 2N2 Canada.
EMail:umguos@cc.umanitoba.ca
c
2000Victoria University ISSN (electronic): 1443-5756 322-05
Monotonicity and Concavity Properties of Some Functions Involving the Gamma Function
with Applications Senlin Guo
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Abstract
In this article, we give the monotonicity and concavity properties of some func- tions involving the gamma function and some equivalence sequences to the sequencen!with exact equivalence constants.
2000 Mathematics Subject Classification: Primary 33B15; Secondary 26D07, 26D20.
Key words: Gamma function, Monotonicity, Concavity, Equivalence
Contents
1 Introduction and Main Results. . . 3 2 Lemmas . . . 8 3 Proof of Main Results. . . 9
References
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1. Introduction and Main Results
Throughout the paper, let N denote the set of all positive integers and N0 = N∪ {0}.
We say an ' bn (n ≥ n0) if there exist two constantsc1 > 0and c2 > 0 such that
(1.1) c1bn≤an ≤c2bn
hold for alln ≥n0. The fixed numbersc1andc2in (1.1) are called equivalence constants.
The incomplete gamma function is defined forRez >0by Γ(z, x) =
Z ∞
x
tz−1e−tdt, γ(z, x) = Z x
0
tz−1e−tdt, (1.2)
and Γ(z,0) = Γ(z) is called the gamma function. The logarithmic derivative ofΓ(z), denoted byψ(z) = Γ0(z)/Γ(z), is called the psi or digamma function, andψ(k)fork ∈Nare called the polygamma functions. One of the elementary properties of the gamma function isΓ(x+1) =xΓ(x). In particular,Γ(n+1) = n!.
In [13], it was proved by F. Qi that the functions f(s, r) =
Γ(s) Γ(r)
s−r1 , (1.3)
f(s, r, x) =
Γ(s, x) Γ(r, x)
s−r1 (1.4)
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and
(1.5) g(s, r, x) =
γ(s, x) γ(r, x)
s−r1
are increasing with respect tor >0,s >0, orx >0.
E. A. Karatsuba [9] proved that the function
(1.6) f1(x) = [g(x)]6−(8x3+ 4x2 +x), where
(1.7) g(x) =e
x
x Γ(1 +x)
√π ,
is strictly increasing from[1,∞)onto[f1(1), f1(∞))with f1(1) = e6
π3 −13 and f1(∞) = 1 30. In 2003, in [1], H. Alzer proved that
α≤f1(x)< 1
30, x∈(0,∞), where
α = min
x>0 f1(x) = 0.0100450· · ·=f1(x0) for somex0 ∈[0.6,0.7]. Sincef1(x0)< f1(1)and
f1(x0)< lim
x→0+f1(x) = 1
√π ,
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his result shows thatf1(x)is not still monotonic on(0,1].
In [3], it was shown in 1997, by G. Anderson and S. Qiu, that the function
(1.8) f2(x) = ln Γ(x+ 1)
xlnx
is strictly increasing from(1,∞)onto(1−γ,1), whereγis the Euler-Mascheroni constant. H. Alzer, in 1998 in [2], proved thatf2(x), with
(1.9) f2(1) = lim
x→1f2(x) = 1−γ,
is strictly increasing on(0,∞). Also note that the functionf2(x)was proved to be concave on(1,∞)in [6] in 2000 by A. Elbert and A. Laforgia.
In [5, 8, 10, 12, 14, 17], monotonicity properties of other functions related to the (di)gamma function were obtained.
In this article, we shall give some monotonicity and concavity properties of several functions involving the gamma function and, as applications, deduce some equivalence sequences to the sequencen!with best equivalence constants.
Our main results are as follows.
Theorem 1.1. The functions
(1.10) f(x) = xx+12
exΓ(x+ 1) and
(1.11) F(x) = exΓ(x+ 1)
xx
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are strictly logarithmically concave and strictly increasing from(0,∞), respec- tively, onto 0,1/√
2π
and onto(1,∞).
Theorem 1.2. The function
(1.12) g(x) = exΓ(x+ 1)
x+12x+12
is strictly logarithmically concave and strictly increasing from −12,∞ onto p
π/e,p 2π/e
.
Theorem 1.3. The function
(1.13) h(x) = exΓ(x+ 1)√
x−1 xx+1
is strictly logarithmically concave and strictly increasing from(1,∞)onto 0,√ 2π
. As applications of these theorems, we have the following corollaries.
Corollary 1.4. Forn ∈N,
(1.14) n!'e−nnn+1/2.
Moreover, for alln ∈N,
(1.15) √
2π ·e−nnn+1/2 < n!≤e·e−nnn+1/2. The equivalence constants√
2πandein (1.15) are best possible.
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Corollary 1.5. Forn ∈N0,
(1.16) n!'e−n
n+1
2 n+12
.
Moreover, for alln ∈N0,
(1.17) √
2e−n
n+1 2
n+12
≤n!<
r2π e e−n
n+1
2 n+12
. The equivalence constants√
2 andp
2π/e in (1.17) are best possible.
Corollary 1.6. Forn ≥2,
(1.18) n!'
r n
n−1e−nnn+1/2. Furthermore, for alln≥2,
(1.19) e
2 2r
n
n−1e−nnn+1/2 ≤n!<√ 2π
r n
n−1e−nnn+1/2. The equivalence constants(e/2)2 and√
2π in (1.19) are best possible.
Remark 1. In [16, Theorem 5], it was proved that forn ≥2,
(1.20) √
2π e−nnn+1/2 < n!<
n n−1
12 √
2π e−nnn+1/2, which can be directly deduced from (1.15) and (1.19).
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2. Lemmas
We need the following lemmas to prove our results.
Lemma 2.1 ([4, p. 20]). Asx→ ∞, (2.1) ln Γ(x) =
x− 1
2
lnx−x+ ln√
2π +O 1
x
. Lemma 2.2 ([7, p. 892] and [11, p. 17]). Forx >0,
ψ(x) = lnx− 1 2x −2
Z ∞
0
tdt
(t2+x2)(e2πt−1), (2.2)
ψ
x+1 2
= lnx+ 2 Z ∞
0
tdt
(t2+ 4x2)(eπt+ 1). (2.3)
Lemma 2.3. The function
(2.4) ϕ(x) = ln x+ 1
x+ 12 − 1 2x is strictly increasing from(0,∞)onto(−∞,0).
Proof. We omit the proof of this lemma due to its simplicity.
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3. Proof of Main Results
Proof of Theorem1.1. Taking the logarithm of f(x)defined by (1.10) and dif- ferentiating directly yields
lnf(x) =
x−1 2
lnx−x−ln Γ(x), (3.1)
[lnf(x)]0 = lnx− 1
2x−ψ(x).
(3.2)
Then by formula (2.2) of Lemma2.2, (3.3) [lnf(x)]0 = 2
Z ∞
0
tdt
(t2 +x2)(e2πt−1), x >0.
Hence,[lnf(x)]0 >0forx∈(0,∞), which means thatlnf(x), and thenf(x), is strictly increasing on(0,∞).
It is easy to see thatlimx→0+f(x) = 0. By (3.1) and Lemma2.1, we have (3.4) lnf(x) =−ln√
2π+O 1
x
→ln 1
√2π, x→ ∞, which implieslimx→∞f(x) = 1/√
2π.
Taking the logarithm of F(x) defined by (1.11) and differentiating easily gives
lnF(x) =x+ ln Γ(x+ 1)−xlnx, (3.5)
[lnF(x)]0 =ψ(x+ 1)−lnx.
(3.6)
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Then by (2.3) of Lemma2.2, for allx >0, (3.7) [lnF(x)]0 = ln
1 + 1
2x
+ 2 Z ∞
0
tdt h
t2+ 4 x+122i
(eπt+ 1)
>0.
Hence,lnF(x), and thenF(x), is strictly increasing on(0,∞).
It is easy to see thatlimx→0+F(x) = 1. By using Lemma2.1, from (3.5), (3.8) lnF(x) = 1
2lnx+ ln√
2π+O 1
x
, x→ ∞.
Therefore,lnF(x), and thenF(x)tends to∞asx→ ∞.
Formulas (3.3) and (3.7) tell us that[lnf(x)]0and[lnF(x)]0 are both strictly decreasing. Therefore, lnf(x) and lnF(x) are strictly concave, that is, the functionf(x)andF(x)are both logarithmically concave.
Proof of Theorem1.2. Taking the logarithm ofg(x)defined by (1.12) and dif- ferentiating shows
lng(x) =x+ ln Γ(x+ 1)−
x+ 1 2
ln
x+ 1
2
, (3.9)
[lng(x)]0 =ψ(x+ 1)−ln
x+1 2
. (3.10)
Then, by formula (2.3) of Lemma2.2, we have (3.11) [lng(x)]0 = 2
Z ∞
0
tdt
[t2 + (2x+ 1)2](eπt+ 1), x >−1 2.
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So
(3.12) [lng(x)]0 >0, x∈
−1 2,∞
,
which means thatlng(x), theng(x), is strictly increasing on −12,∞ . SinceΓ(1/2) =√
π, it is easy to verify that lim
x→−1/2g(x) =p π/e. From (3.9) and Lemma2.1, it is obtained that
(3.13) lng(x) =
x+1 2
lnx+ 1
x+12 + ln√
2π −1 +O 1
x
, x→ ∞.
Hencelng(x)→lnp
2π/e asx→ ∞, and then lim
x→∞g(x) =p 2π/e.
Formula (3.11) shows that[lng(x)]0is strictly decreasing. Therefore,lng(x) is strictly concave, that is, the functiong(x)is logarithmically concave.
Proof of Theorem1.3. Taking the logarithm of h(x)defined by (1.13) and dif- ferentiating straightforwardly reveals
lnh(x) = ln Γ(x) +x+1
2ln(x−1)−xlnx, (3.14)
[lnh(x)]0 =ψ(x) + 1
2(x−1) −lnx.
(3.15)
By settingx=u+ 1withu >0, we have (3.16) [lnh(x)]0 =ψ(u+ 1) + 1
2u −ln(u+ 1) = [lng(u)]0−ϕ(u),
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whereg(u)andϕ(u)are respectively defined by (1.12) and (2.4). From (3.12) and Lemma2.3, it is deduced that[lnh(x)]0 >0forx >1. Therefore,lnh(x), and thenh(x), is strictly increasing on(1,∞).
It is obvious that lim
x→1+h(x) = 0. From (3.14) and Lemma2.1, we see (3.17) lnh(x) = 1
2lnx−1
x + ln√
2π+O 1
x
→ln√
2π, x→ ∞.
So lim
x→∞h(x) = √ 2π.
Considering the logarithmic concavity ofg(x)and the increasing monotonic- ity of ϕ(x) in (3.16) reveals that [lnh(x)]0 is strictly decreasing. Therefore, lnh(x) is strictly concave, that is, the function h(x) is logarithmically con- cave.
Proof of Corollary1.4. By Theorem 1.1, we know that the function f(x) is strictly increasing from(0,∞)onto
0,√1
2π
, hence
(3.18) 1
e =f(1) ≤f(n) = nn+1/2 enn! < 1
√2π
forn∈N, and
(3.19) lim
n→∞
nn+1/2 enn! = 1
√2π. From (3.18) and (3.19), we see that Corollary1.4is true.
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Proof of Corollary1.5. By Theorem1.2, we see that the functiong(x)is strictly increasing from −12,∞
onto p
π/e,p 2π/e
. So
(3.20) √
2 = g(0)≤g(n) = enn!
n+ 12n+1/2 <
r2π
e , n∈N0
and
(3.21) lim
n→∞
enn!
n+12n+1/2 = r2π
e .
Inequality (3.20) is equivalent to (1.17). Since the constants√
2 andp 2π/e are best possible in (3.20), they are also best possible in (1.17).
Proof of Corollary1.6. The monotonicity ofh(x)by Theorem1.3implies
(3.22) e
2 2
=h(2) ≤h(n) = enn!√ n−1 nn+1 <√
2π, n ≥2
and
(3.23) lim
n→∞
enn!√ n−1 nn+1 =√
2π.
From (3.22) and (3.23), we see that Corollary1.6is valid.
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