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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:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 322-05

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Monotonicity and Concavity Properties of Some Functions Involving the Gamma Function

with Applications Senlin Guo

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J. Ineq. Pure and Appl. Math. 7(2) Art. 45, 2006

http://jipam.vu.edu.au

Abstract

In this article, we give the monotonicity and concavity properties of some functions 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

1. Introduction and Main Results

Throughout the paper, let N denote the set of all positive integers and N0 = N∪ {0}.

We say a_n ' b_n (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

t^z−1e^−tdt, γ(z, x) = Z x

0

t^z−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) = x^x+¹²

e^xΓ(x+ 1) and

(1.11) F(x) = e^xΓ(x+ 1)

x^x

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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) = e^xΓ(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) = e^xΓ(x+ 1)√

x−1 x^x+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⁻ⁿn^n+1/2.

Moreover, for alln ∈N,

(1.15) √

2π ·e⁻ⁿn^n+1/2 < n!≤e·e⁻ⁿn^n+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⁻ⁿn^n+1/2. Furthermore, for alln≥2,

(1.19) e

2 2r

n

n−1e⁻ⁿn^n+1/2 ≤n!<√ 2π

r n

n−1e⁻ⁿn^n+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⁻ⁿn^n+1/2 < n!<

n n−1

¹₂ √

2π e⁻ⁿn^n+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)(e^2πt−1), (2.2)

ψ

x+1 2

= lnx+ 2 Z ∞

0

tdt

(t²+ 4x²)(e^πt+ 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 differentiating 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)(e^2πt−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 thatlim_x→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 implieslim_x→∞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

(e^πt+ 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 differentiating 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](e^πt+ 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 differentiating 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 monotonicity 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 concave.

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) = n^n+1/2 eⁿn! < 1

√2π

forn∈N, and

(3.19) lim

n→∞

n^n+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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