Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions

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Volume 2009, Article ID 728612,13pages doi:10.1155/2009/728612

Research Article

Monotonic and Logarithmically Convex Properties

of a Function Involving Gamma Functions

Tie-Hong Zhao,

1

_{Yu-Ming Chu,}

2

_{and Yue-Ping Jiang}

3

1_{Institut de Math´ematiques, Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France} 2_{Department of Mathematics, Huzhou Teachers College, Huzhou 313000, Zhejiang, China} 3_{College of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, China}

Correspondence should be addressed to Yu-Ming Chu,[email protected]

Received 14 October 2008; Accepted 27 February 2009 Recommended by Sever Dragomir

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially aﬃrmative answer to an open problem posed by B.-N. Guo and F. Qi. Several inequalities for the geometric means of natural numbers are established.

Copyrightq2009 Tie-Hong Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

For real and positive values ofxthe Euler gamma functionΓand its logarithmic derivative

ψ, the so-called digamma function, are defined as

Γx

_∞

0

tx−1e−tdt, ψx Γ

_x

Γx. 1.1

For extension of these functions to complex variables and for basic properties see1 . In recent years, many monotonicity results and inequalities involving the Gamma and incomplete Gamma functions have been established. This article is stimulated by an open problem posed by Guo and Qi in2 . The extensions and generalizations of this problem can be found in3–5 and some references therein.

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in4 as follows:

_n_k

ik1i 1/n

_n_m_k

ik1 i

1/nm ≤

nk

nmk, 1.2

and the following lower bound was appeared in2,5 :

nk1

nmk1 <

n

nk!/k!

nm_n_m_k_!_/k_!, 1.3

SinceΓn1 n!,as a generalization of inequality1.3, the following monotonicity result was obtained by Guo and Qi in2 . The function

Γxy1/Γy1 1/x

xy1 1.4

is decreasing with respect toxon1,∞for fixedy ≥0.Hence, for positive real numbersx

andy, we have

xy1

xy2 ≤

Γxy1/Γy1 1/x

Γxy2/Γy1 1/x1. 1.5

Recently, in6 , Qi and Sun proved that the function

Γxy1/Γy1 1/x

√_x_y 1.6

is strictly increasing with respect tox∈y1,∞for ally≥y0.

Now, we generalize the function in1.4as follows: for positive real numbersxandy,

α≥0, let

Fαx, y

Γxy1/Γy1 1/x

xy1α . 1.7

The aim of this paper is to discuss the monotonicity and logarithmical convexity of the functionFαx, ywith respect to parameterα.

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Definition 1.1. LetD⊂R2_{be a convex set,}_f_:_D _→ _R_{is called a convex function on}_D_if

f

_x_y

2

≤ fx fy

2 1.8

for allx,y∈D, andfis called concave if−fis convex.

Definition 1.2. Let D ⊂ R2 _{be a convex set,} _f _: _D _→ ₀_,_∞ _{is called a logarithmically}

convex function onD if lnf is convex onD, andf is called logarithmically concave if lnf

is concave.

The following criterion for convexity of function was established by Fichtenholz in7 .

Proposition 1.3. Let D ⊂ R2 _{be a convex set, if} _f _: _D _→ _R _{have continuous second partial} derivatives, thenfis a convex (or concave) function onDif and only ifLxis a positive (or negative) semidefinite matrix for allx∈D, where

Lx

f₁₁ f₁₂ f₂₁ f₂₂

1.9

andf_ij ∂2_f_x

1, x2/∂xi∂xjforx x1, x2,i, j1,2.

Notation 1. In Definitions1.1,1.2andProposition 1.3, we denotex,yby the pointsor vectors ofR2_{, and denote}_{x, y}_{by the real variables in the later.}

Our main results are Theorems1.4and1.5.

Theorem 1.4. 1 For any fixed y ≥ 0,Fαx, yis strictly increasing (or decreasing, resp.) with

respect toxon0,∞if and only if0≤α≤1/2 (orα≥1, resp.);

2 For any fixedx >0,Fαx, yis strictly increasing with respect toyon0,∞if and only

if0≤α≤1.

Theorem 1.5. 1 If0≤α≤1/4, thenFαx, yis logarithmically concave with respect tox, y∈

0,∞×0,∞;

2 IfE⊂0,∞×0,∞is a convex set with nonempty interior andα≥1, thenFαx, yis

neither logarithmically convex nor logarithmically concave with respect tox, yonE.

The following two corollaries can be derived from Theorems1.4and1.5immediately.

Corollary 1.6. Ifx, y∈0,∞×0,∞, then

xy1

xy2 <

Γxy1/Γy1 1/x

Γxy2/Γy1 1/x1 <

xy1

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Remark 1.7. Inequality1.3can be derived fromCorollary 1.6if we takex, y∈N. Although we cannot get the inequality1.2exactly fromCorollary 1.6, but we can get the following inequality which is close to inequality1.2:

_n_k

ik1i 1/n

_n_m_k

ik1 i

1/nm ≤

nk1

nmk1. 1.11

Corollary 1.8. Ifx1, y1,x2, y2∈0,∞×0,∞, then

Γx1y11

/Γy11 1/x1·

Γx2y21

/Γy21 1/x2

Γx1x2y1y2/21

/Γy1y2/21 4/x1x2

≤ √

2x1y11

x2y21 1/4

x1y1x2y22

.

1.12

Remark 1.9. We conjecture that the inequality 1.2can be improved if we can choose two pairs of integersx1, y1andx2, y2properly.

2. Lemmas

It is well known that the Bernoulli numbersBnis defined8 in general by

1

et−₁

1 2 −

1

t

∞

n1

−1n−1 t

2n

2n!Bn. 2.1

In particular, we have

B1

1

6, B2 1

30, B3 1

42, B4 1

30. 2.2

In9 , the following summation formula is given:

∞

n0

−1n

2n12k1

π2k1_Ek

22k2₂_k_! 2.3

for nonnegative integerk, whereEkdenotes the Euler number, which implies

Bn

22n!

2π2n

∞

m1

1

m2n, n∈N. 2.4

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Lemma 2.1see3 . For real numberx >0and natural numberm, one has

lnΓx 1

2ln2π

x− 1

2

lnx−x m

n1

−1n−1 Bn 22n−1n·

1

x2n−1

−1mθ1 Bm1

2m12m2· 1

x2m1, 0< θ1<1;

2.5

ψx lnx− 1

2x m

n1

−1nBn 2n·

1

x2n −1 m1_θ

2

Bm1 2m2·

1

x2m2, 0< θ2<1; 2.6

ψx 1 x

1 2x2

m

n1

−1n−1 Bn

x2n1 −1

m_θ

3·Bm1

x2m3, 0< θ3<1; 2.7

ψx −1 x2 −

1

x3

m

n1

−1n2n1 Bn

x2n2 −1

m1₂_m₃_θ 4·Bm1

x2m4, 0< θ4<1. 2.8

Lemma 2.2see14 . Inequalities

lnx− 1

x ≤ψx≤lnx−

1

2x, 2.9

k−1!

xk k!

2xk1 ≤−1

k1_ψk_x_≤ k−1!

xk k!

xk1 2.10

hold in0,∞fork∈N.

Lemma 2.3. Letrx, y ψxy1−ψy1−αx/xy1, then the following statements are true:

1if0≤α≤1, thenrx, y≥0forx, y∈0,∞×0,∞; 2ifα >1, thenrα, y<0fory∈2/α−1,∞.

Proof. 1Making use of2.6we get

lim

y→ ∞rx, y ylim→ ∞

lnxy1−lny1 0 2.11

for any fixedx >0.

Sinceψx1 1/xψxand 0≤α≤1, we have

rx, y−rx, y1 x

1−αyx2−α

y1xy1xy2 >0 2.12

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Therefore,Lemma 2.31follows from2.11and2.12.

2Ifα >1, then2.12leads to

rα, y−rα, y1<0 2.13

fory∈2/α−1,∞.

Therefore,Lemma 2.32follows from2.11and2.13.

Lemma 2.4. Ifgx, y 2xψy1−2lnΓxy1−lnΓy1 x2_ψ_y₁_{, then}_g_{x, y}_>₀ forx, y∈0,∞×0,∞.

Proof. It is easy to see that

g0, y 0 2.14

for ally∈0,∞.

Letg1x, y ∂gx, y/∂x, then

g1x, y 2

xψy1−ψxy1 ψy1 , 2.15

g10, y 0, 2.16

∂g1x, y

∂x 2

ψy1−ψxy1 >0 2.17

forx >0. On the other hand, from2.10we know thatψxis strictly decreasing on0,∞. Therefore,Lemma 2.4follows from2.14–2.17.

Remark 2.5. Let

ax, y 2 x3

lnΓxy1−lnΓy1 − 2

x2ψxy1,

bx, y − 1 x2

ψxy1−ψy1 ,

cx, y −1 xψ

_y₁_.

2.18

Then simple computation shows that

gx, y x32bx, y−ax, y−cx, y . 2.19

Lemma 2.6. Letdx, y 1/xψxy1 α/xy12,then the following statements are true:

1if0≤α≤1/4, then

ax, y dx, y cx, y dx, y >bx, y dx, y 2 2.20

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2ifα≥1,then

ax, y dx, y cx, y dx, y <bx, y dx, y 2 2.21

forx, y∈0,∞×0,∞. Proof. Let

fx, y 2ψy1xψxy1−lnΓxy1 lnΓy1 −ψxy1−ψy1 2,

px, y fx, y−gx, y

ψxy1 αx

xy12

.

2.22

Then it is not diﬃcult to verify

p0, y 0, 2.23

px, y x4ax, y dx, y cx, y dx, y −bx, y dx, y 2, 2.24

∂px, y ∂x −

αx

xy12

∂gx, y

∂x −gx, y

ψxy1 α

xy12 −

2αx

xy13

.

2.25

1If 0≤α≤1/4, then making use of Lemmas2.2,2.4and2.25we get

∂px, y ∂x >−

αx

xy12

∂gx, y ∂x

gx, y

1

xy12 1

xy13 −

α

xy12

2αx

xy13

> 1

xy12

1−αgx, y−αx∂gx, y ∂x

, 2.26

forx, y∈0,∞×0,∞.

Letgix, y ∂igx, y/∂xi, i 1,2,3,4,qx, y 1−αgx, y−αx∂gx, y/∂x,

andqjx, y ∂jqx, y/∂xj, j1,2. Then simple computation leads to

g3x, y −2ψxy1, 2.27

g4x, y −2ψxy1, 2.28

∂q2x, y

∂x 1−4αg3x, y−αxg4x, y, 2.29 q20, y q10, y q0, y 0 2.30

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It is well known that lnΓx −cx ∞_k₁x/k − ln1 x/k − lnx, where c

0.577215· · · is the Euler’s constant. From this we get

ψn −1n1n!

∞

k0

1

kxn1. 2.31

FromLemma 2.2,2.27–2.29,2.31and the assumption 0 ≤ α≤ 1/4, we conclude that

∂q2x, y

∂x >0. 2.32

Therefore,Lemma 2.61follows from2.23–2.26,2.30, and2.32.

2Ifα≥1, then making use of2.8,Lemma 2.4and2.25we obtain

∂px, y ∂x <−

αx

xy12

∂gx, y

∂x gx, y

1

xy13 1

2xy14

2αx

xy13

<− αx

xy12

∂gx, y

∂x gx, y

2αx1

xy13

< αx1

xy13

2gx, y−x∂gx, y ∂x

.

2.33

Let

vx, y 2gx, y−x∂gx, y

∂x , vix, y

∂ivx, y

∂xi , i1,2. 2.34

Then

v2x, y 2xψxy1<0 2.35

forx, y∈0,∞×0,∞byLemma 2.2, and

v0, y v10, y 0 2.36

fory∈0,∞.

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3. Proofs of Theorems

1.4 and

1.5

Proof ofTheorem 1.4. 1LetGx, y lnFαx, yandG1x, y x2∂Gx, y/∂x, then

G1x, y −

lnΓxy1−lnΓy1 xψxy1− αx

2

xy1. 3.1

The following three cases will complete the proof ofTheorem 1.41.

Case 1. If 0≤α≤1/2, then3.1andLemma 2.2imply

∂G1x, y

∂x x

ψxy1−αx2y2

xy12

> x

1

xy1 1

2xy12 −

αx2y2

xy12

x

2xy12

2−2αx 2−4αy3−4α

>0

3.2

forx, y∈0,∞×0,∞.

From3.2and the fact thatG10, y 0 for ally ∈ 0,∞we know thatFαx, yis

strictly increasing with respect toxon0,∞for any fixedy∈0,∞.

Case 2. Ifα≥1, then3.1and2.7imply

∂G1x, y

∂x < x

1

xy1 1

2xy12 1

6xy13 −

αx2y2

xy12

x

6xy13

6−6αx2λ1yxλ2y

<0

3.3

forx, y∈0,∞×0,∞, whereλ1y 12−18αy15−18α <0 andλ2y 61−2αy2 15−24αy10−12α <0.

From3.3and the fact thatG10, y 0 for ally ∈ 0,∞we know thatFαx, yis

strictly decreasing with respect toxon0,∞for any fixedy∈0,∞.

Case 3. If 1/2< α <1, let

G2x, y ψxy1−

αx2y2

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Then

∂G1x, y

∂x xG2x, y, 3.5

G20, y< 1

y1 1 2y12

1 6y13 −

2α y1

1

6y13

61−2αy2 15−24αy10−12α <0

3.6

fory≥15−24α√48α−15/24α−12.

It is obvious that3.6implies

G2

0,15

√

48α−15 24α−12

<0. 3.7

The continuity ofG2x, ywith respect tox∈0,∞for any fixedy∈0,∞and3.7imply

that there existsδδα>0 such that

G2

x,15

√

48α−15 24α−12

<0 3.8

forx∈0, δ.

From3.5,3.8andG10,15

√

48α−15/24α−12 0 we know thatFαx, yis strictly decreasing with respect toxon0, δfory 15√48α−15/24α−12.

On the other hand, making use of2.5and2.6we have

lim

x→∞G1x, y xlim→∞x

1−

y 1

2

lnxy1

x − αx xy1

Cy, θ1

lim

x→∞1−αxC

y, θ1

∞,

3.9

where

Cy, θ1

y1

2

lny1 1 12y1−

1 2−

θ1

360y13 3.10

fory∈0,∞and 0< θ1 <1.

Equation3.9implies that there existsMMα> δαsuch that

G1

x,15

√

48α−15 24α−12

>0 3.11

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Hence, from3.11we know thatFαx, yis strictly increasing with respect to xon

M,∞fory 15√48α−15/24α−12.

2Since

x∂Gx, y

∂y ψxy1−ψy1− αx

xy1 rx, y, 3.12

then,Theorem 1.42follows from3.12andLemma 2.3.

Proof ofTheorem 1.5. LetGx, y lnFαx, y, G₁₁x, y ∂2Gx, y/∂x2, G₁₂ ∂2Gx, y/ ∂x∂yandG₂₂x, y ∂2_G_{x, y}_/∂y2_{, then simple calculation yields}

G₁₁x, y 2 x3

lnΓxy1−lnΓy1 − 2

x2ψxy1 1

xψ

_x_y₁ α

xy12

ax, y dx, y,

3.13

G₁₂x, y − 1 x2

ψxy1−ψy1 1

xψ

_x_y₁ α

xy12

bx, y dx, y,

3.14

G₂₂x, y 1 x

ψxy1−ψy1 α

xy12

cx, y dx, y,

3.15

whereax, y, bx, y, cx, y, anddx, yare defined inRemark 2.5andLemma 2.6.

According to the Definition 1.2 and Proposition 1.3, to prove Theorem 1.5 we need only to show that

G₁₁x, y≤0, 3.16

G₁₁x, yG₂₂x, y−G₁₂x, y 2≥0 3.17

for 0≤α≤1/4 andx, y∈0,∞×0,∞, and

G₁₁x, yG₂₂x, y−G₁₂x, y 2<0 3.18

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Next, let wx, y x3_G

11x, y, then

wx, y 2lnΓxy1−lnΓy1 −2xψxy1 x2_ψ_x_y₁ αx3 xy12,

w0, y 0,

3.19

∂wx, y ∂x x

2

ψxy1 αx3y3

xy13

< x2

αx3y3

xy13 − 1

xy12 − 1

xy13

x2 xy13

α−1x 3α−1y3α−2

<0

3.20

forx, y∈0,∞×0,∞byLemma 2.2and 0≤α≤1/4.

Therefore, 3.16 follows from 3.19 and 3.20, and 3.17 and 3.18 follow from Lemma 2.6. The proof ofTheorem 1.5is completed.

Acknowledgments

This research is partly supported by 973 Project of China under grant 2006CB708304, N S Foundation of China under Grant 10771195, and N S Foundation Zhejiang Province under Grant Y607128.

References

1 E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1996.

2 B.-N. Guo and F. Qi, “Inequalities and monotonicity for the ratio of gamma functions,”Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 239–247, 2003.

3 F. Qi and B.-N. Guo, “Monotonicity and convexity of ratio between gamma functions to diﬀerent powers,”Journal of the Indonesian Mathematical Society, vol. 11, no. 1, pp. 39–49, 2005.

4 F. Qi, “Inequalities and monotonicity of sequences involving n_n₁_!_/k_!,” _{Soochow Journal of}

Mathematics, vol. 29, no. 4, pp. 353–361, 2003.

5 F. Qi and Q.-M. Luo, “Generalization of H. Minc and L. Sathre’s inequality,”Tamkang Journal of Mathematics, vol. 31, no. 2, pp. 145–148, 2000.

6 F. Qi and J.-S. Sun, “A mononotonicity result of a function involving the gamma function,”Analysis Mathematica, vol. 32, no. 4, pp. 279–282, 2006.

7 G. M. Fichtenholz,Diﬀerential- und Integralrechnung. II, VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1966.

8 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 ofNational Bureau of Standards Applied Mathematics Series, U.S.Government Printing Oﬃce, Washington, DC, USA, 1964.

9 Zh.-X. Wang and D.-R. Guo,Introduction to Special Function, The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000.

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11 Q.-M. Luo, B.-N. Guo, F. Qi, and L. Debnath, “Generalizations of Bernoulli numbers and polynomials,”International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 59, pp. 3769– 3776, 2003.

12 Q.-M. Luo and F. Qi, “Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 7, no. 1, pp. 11–18, 2003.

13 Q.-M. Luo, F. Qi, and L. Debnath, “Generalizations of Euler numbers and polynomials,”International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 61, pp. 3893–3901, 2003.

14 F. Qi and B.-N. Guo, “A new proof of complete monotonicity of a function involving psi function,”