函数和硕士论文范文 与Gamma函数和Psi函数的单调性和凹凸性方面在职研究生论文范文

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Gamma函数和Psi函数的单调性和凹凸性

QIU Songliang, CAI Chuanyu

(School of Sciences, Zhejiang SciTech University, Hangzhou 310018, China)

Abstract： The authors present several monotonicity and logconvexity properties of the gamma function Γ(x), and some monotonicity and convexity properties of certain combinations defined in terms of Γ(x), the psi function ψ(x), ψ′(x) and ψ″(x), by which several known results are improved.

Key words： gamma function; psi function; monotonicity; convexity; inequality

CLC number： O174.6

Document code： A

文章编号： 1673\|3851 (2018) 01\|0108\|05

0Introduction

For n∈N等于{n|n is a positive integer}, let γ等于limn→∞∑nk等于1k－1－logn等于0.57721… and ζ(s)等于∑∞k等于1k－s denote the EulerMascheroni constant and the Riemann zeta function, respectively. Throughout this paper, we let α等于75［28ζ(3)+π3］/64－75等于0.77797…, β等于2ζ(3)/3等于0.80137… and δ等于18(3－γ－logπ－π2/8)等于0.79837… As usual, for x,y>0, the gamma and psi functions are defined as

Γ(x)等于∫∞0tx－1e－tdt and ψ(x)等于ddxlogΓ(x)等于Γ′(x)Γ(x),

respectively (cf. ［14］).

During the past decades, many authors he obtained various properties for the functions Γ(x), ψ(x)and its derivatives. (Cf.［3］,［5］［18］ and bibliographies there.) For example, in ［5, Theorem 1.1 and Lemma 2.1］, some inequalities were obtained for the function f(x)≡xψ(x+1)－logΓ(x+1), and it was proved that g(x)≡x2［ψ′(x+1)+xψ″(x+1)］ is strictly increasing from ［0,∞) onto ［0,1/2), while the function x→g(x)/x is not monotone on (0,∞). In ［67］ and ［18］, several monotonicity properties and inequalities were obtained for the gamma function Γ(x). In ［18］, it was proved that the function F(x)≡(x+1)－1［Γ(x+1)］1/x(G(x)≡(x+1)－1/2［Γ(x+1)］1/x) is strictly decreasing (increasing, respectively) on ［1,∞). In ［7］, it was shown that the function F(G) is strictly decreasing and logconvex(increasing and logconce, respectively) on (0,∞) by using complicated methods, and some other properties of Γ(x) were derived. Such kind of studies usually rely on analytic properties of Γ(x), ψ(x), ψ(n)(x), and those of certain combinations defined in terms of these functions.

The main purpose of this paper is to improve the abovementioned known conclusions for the functions f,g,F and G, and some other main results proved in ［7］, by applying recent results for Γ(x), ψ(x) and ψ(n)(x).

1Preliminaries

In the sequel, we shall frequently apply the following formulas ［1, 6.1.40, 6.4.2 & 6.4.11］:

logΓ(x)~x－12logx－x+12log(2π)+∑∞k等于1B2k2k(2k－1)x2k－1(x→∞)

(1)

ψ(n)(1)等于(－1)n+1n！ζ(n+1)(2)

ψ(n)(x)~(－1)n+1(n－1)！xn+n！2xn+1+(n+1)！12xn+2+…(x→∞)

(3)

where B2k for k∈N are the Bernoulli numbers (see ［1, 23.1］).

First, we record the following theorem proved in ［19, Theorems 1.11.2］ and needed in the proofs of our results stated in Section 2.

Theorem A.a) For each n∈N, the function Gn(x)≡(－1)n+1［nψ(n)(x+1)+xψ(n+1)(x+1)］ is completely monotonic on ［0,∞), with Gn(0)等于n！nζ(n+1) and Gn(∞)等于0.

b) Let f1(x)等于xGn+1(x)/Gn(x) for each n∈N and for x∈［0,∞). Then for each n∈N and for all x∈［0,∞),

f1(0)等于0等于inf0≤x<∞f1(x)≤f1(x)<sup0≤x<∞f1(x)等于n+1等于f1(∞)

(4)

c) For x∈(0,∞), let f2(x)≡x－2(x+1)f(x), where f(x)等于xψ(x+1)－logΓ(x+1). Then f2 is strictly increasing from (0,∞) onto (π2/12,1). In particular,

π2x212(x+1)≤xψ(x+1)－logΓ(x+1)≤x2x+1(5)

for all x∈［0,∞), with equality in each instance if and only if x等于0.

d) For x∈(0,∞), let f3(x)等于x－3(x+1)2［2f(x)－x2ψ′(x+1)］, f4(x)等于(x+1)－2f3(x), and put c0等于infx∈(0,∞)f3(x). Then f3(0+)等于β, f3(1/2)等于δ, f3 is not monotone on (0,∞), and f4 is strictly decreasing and convex from (0,∞) onto (0,β). Furthermore,

α≤c0<δ(6)

c0≤f3(x)<supx∈(0,∞)f3(x)等于f3(∞)等于1(7)

and

αx3(x+1)2≤c0x3(x+1)2≤2f(x)－x2ψ′(x+1)≤x3(x+1)2min{1,β(x+1)2}

(8)

for x∈(0,∞). Each of the equalities in (8) holds if and only if x等于0.

Next, we prove the following theorem, which improves ［5,Theorem 1.1 & Lemma 2.1］.

Theorem 1.Let Gn and f be as in Theorem A. For real numbers a and b, define the functions g1,a and g2,b on (0,∞) by

g1,a(x)等于x－af(x) and g2,b(x)等于xbG1(x),

respectively. Then we he the following conclusions:

a) The function g1,a is strictly increasing(decreasing) on (0,∞) if and only if a≤1(a≥2, respectively), with g1,1((0,∞))等于(0,1) and g1,2((0,∞))等于(0,π2/12). In particular, for x∈［0,∞),

π2x212(x+1)≤xψ(x+1)－logΓ(x+1)≤minπ212x2,x2x+1

(9)

with equality in each instance if and only if x等于0.

b) The function g2,b is strictly increasing (decreasing) on (0,∞) if and only if b≥2(b≤0, respectively), with g2,2(［0,∞))等于［0,1/2) and g2,0(［0,∞))等于(0,π2/6］.

Proof: a) Let g1(x)等于x2ψ′(x+1)/f(x) for x∈(0,∞). Since f′(x)等于xψ′(x+1)>0, f is strictly increasing on (0,∞) and f(x)>f(0)等于0 for x∈(0,∞). By differentiation,

xa+1g′1,a(x)/f(x)等于g1(x)－a(10)

By ［5, Theorem 1.1(4)］, we he

inf0<x<∞g1(x)等于g1(∞)等于1 and sup0<x<∞g1(x)等于g1(0+)等于2

(11)

Hence it follows from (10) that

g′1,a(x)≤0a≥sup0<x<∞g1(x)等于2

and

g′1,a(x)≥0a≤inf0<x<∞g1(x)等于1.

This yields the assertion on the monotonicity of g1,a.

By lHpitals rule, (2) and (3), we obtain

g1,1(0+)等于limx→0f(x)x等于0, g1,1(∞)等于limx→∞f(x)x等于1,

g1,2(0+)等于limx→0f(x)x2等于12ψ′(1)等于π212,

g1,2(∞)等于limx→∞f(x)x等于0.

The first inequality and the second upper bound in (9) follow from (5), and the first upper bound in (9) follows from the monotonicity property of g1,2. The equality case in (9) is clear.

b) Clearly, g2,2(0)等于0, g2,0(0)等于ψ′(1)等于ζ(2)等于π2/6 and g2,0(∞)等于G1(∞)等于0. By (3), we obtain the limiting value g2,2(∞)等于limx→∞x2G1(x)等于1/2.

Let G1 and f1, with n等于1, be as in Theorem A. Then by differentiation,

x1－bg′2,b(x)/G1(x)等于b－f1(x)(12)

which yields the assertion on the monotonicity of g2,b by Theorem A(2).

2Some Properties of the Gamma Function

In ［7, Theorem 1］ (［7, Theorem 2］), it was proved that the function

F(x)≡(x+1)－1［Γ(x+1)］1/x(G(x)

≡(x+1)－1/2［Γ(x+1)］1/x)

is strictly decreasing and strictly logconvex(increasing and logconce, respectively) on (0,∞). Our following theorem improves these known conclusions.

Theorem 2.Let c0 and α be as in Theorem A, and for each c∈R, define the function F on (0,∞) by F(x)等于(x+1)－c［Γ(x+1)］1/x. Then we he the following conclusions:

a) F is strictly decreasing on (0,∞) if and only if c≥1, with F((0,∞))等于(e－1,e－γ) if c等于1, and F((0,∞))等于(0,e－γ) if c>1. Moreover, F is logconvex on (0,∞) if and only if c≥1.

b) F is strictly increasing on (0,∞) if and only if c≤π2/12. If c≤π2/12, then F((0,∞))等于(e－γ,∞).

c) F is logconce on (0,∞) if and only if c≤c0. In particular, F is logconce on (0,∞) if c≤α.

Proof: Let g1,2, f2 and f3 be as in Theorem 1, Theorem A(c) and Theorem A(d), respectively. Then by differentiation, we obtain

F′(x)F(x)等于h1(x)≡g1,2(x)－cx+1(13)

F′(x)F(x)等于f2(x)－cx+1(14)

h′1(x)等于c－f3(x)(x+1)2(15)

a) By (14) and Theorem A(3), for x∈(0,∞),

F′(x)<0c>f2(x)c≥sup0<x<∞f2(x)等于1,

which shows that F is strictly decreasing on (0,∞) if and only if c≥1.

It follows from (7) and (13) that

F is logconvex on (0,∞)h1 is strictly increasing on (0,∞)c≥sup0<x<∞f3(x)等于1.

It is well know that ψ(1)等于－γ (see ［1, 6.3.2］). If c等于1, then by lHpitals rule and (1),

F(0+)等于limx→0［Γ(x+1)］1/x等于limx→0explogΓ(x+1)x

等于limx→0exp(ψ(x+1))等于eψ(1)等于e－γ,

F(∞)等于limx→∞explogx+logΓ(x)x－log(x+1)

等于limx→∞exp(x+1/2)logx－x－xlog(x+1)x

等于limx→∞explogx2x－1等于e－1.

Similarly, if c>1, then F(0+)等于e－γ and

F(∞)等于limx→∞explogx+logΓ(x)x－clog(x+1)

等于limx→∞exp(x+1/2)logx－xx－clog(x+1)

等于limx→∞exp(c－1)log1x+logx2x－1等于0.

b) It follows from (14) and Theorem A(3) that for all x∈(0,∞),

F′(x)>0c<f2(x)c≤f2(0+)等于π212,

that is, F is strictly increasing on (0,∞) if and only if c≤π2/12.

Clearly, if c≤π2/12, then F(0+)等于e－γ. Since F(∞)等于e－1 when c等于1,

F(∞)等于limx→∞(x+1)1－c·Γ(x+1)1/xx+1等于∞.

c) It follows from (13), (15) and Theorem A(d) that on (0,∞),

F is logconceh1 is strictly decreasingc≤f3(x)c≤inf0<x<∞f3(x)等于c0.

The remaining conclusion is clear.

The following corollary improves ［7, Corollaries 12］.

Corollary 3.For x,y∈(0,∞) with y≥x,

x+1y+1≤Γ(x+1)1/xΓ(y+1)1/y≤x+1y+1π2/12(16)

with equality in each instance if and only if y等于x. Moreover, for x∈(0,∞),

h2(x)x<Γ(x+1)<［e－γ(x+1)］x(17)

where h2(x)等于max{e－1(x+1), e－γ(x+1)π2/12}.

Proof: It follows from Theorem 2(a)(b) that

Γ(x+1)1/xx+1≥Γ(y+1)1/yy+1 and Γ(x+1)1/x(x+1)π2/12≤Γ(y+1)1/y(y+1)π2/12,

with equality in each instance if and only if y等于x. This yields the double inequality (16) and its equality case.

The double inequality (17) follows from Theorem 2(a)(b).

Remark.Let h2 be as in Corollary 3, and h3(x)等于eγ－1(x+1)1－π2/12 for x∈(0,∞). Then it is clear that h3 is strictly increasing from (0,∞) onto (eγ－1,∞), and

e－1(x+1)/［e－γ(x+1)π2/12］等于h3(x).

Hence there exists a unique number x1∈(0,∞) such that the function

h2(x)等于e－γ(x+1)π2/12,if x∈(0,x1)

e－1(x+1),if x∈［x1,∞).

In ［7, Theorems 45］, it was proved that the function G(x)≡Γ(x+1)1/x is strictly increasing on (0,∞), and H(x)≡xηΓ(x+1)1/x is strictly increasing (decreasing) on (0,∞) if η≥0(η≤－1, respectively). The following theorem strengthens these results.

Theorem 4.a) The function G(x)≡Γ(x+1)1/x is strictly increasing and logconce from (0,∞) onto (e－γ,∞).

b) For each η∈R, define the function H on (0,∞) by H(x)≡xηΓ(x+1)1/x. Then H is strictly increasing(decreasing) on (0,∞) if and only if η≥0(η≤－1, respectively), with H((0,∞))等于(e－γ,∞) if η等于0, and H((0,∞))等于(1/e,∞) if η等于－1.

Proof: a) By logarithmic differentiation, G′(x)/G(x)等于g1,2(x), where g1,2 is as in Theorem 1. This yields the monotonicity and logconcity properties of G by Theorem 1(a).

Clearly, G(0+)等于e－γ. Applying (1), we can obtain the limiting value G(∞)等于∞.

b) Let g1,1 be as in Theorem 1. Then by logarithmic differentiation, xH′(x)/H(x)等于η+g1,1(x), and hence the assertion on the monotonicity of H follows from Theorem 1(a).

It is clear that H(x)等于G(x) if η等于0. Hence H((0,∞))等于(e－γ,∞) if η等于0.

If η等于－1, then H(x)等于G(x)/x, so that H(0+)等于∞. By (1.1), H(∞)等于e－1.

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Gamma函数和Psi函数的单调性与凹凸性

裘松良,蔡传宇

(浙江理工大学理学院,杭州 310018)

摘 要： 给出了gamma函数Γ(x)的几个单调性和对数凹凸性,以及由Γ(x)、psi函数ψ(x)及其导数ψ′(x)和ψ″(x)定义的某种组合的单调性与凹凸性,并运用这些结果实质性地改进了关于Γ(x)和ψ(x)的几个已知结果.

关键词： gamma函数;psi函数；单调性；凹凸性；不等式

(责任编辑： 康锋)

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