Some properties of log-convex function and applications for the

247742 Computers and Mathematics with Applications 63 (2012) 1111-1116

Contents lists available at

SciVerse ScienceDirect

Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa Some properties of log-convex function and applications for the exponential function

Xiaoming Zhanga,∗, Weidong Jiangb

a Zhejiang Broadcast & Radio University Haining College, Zhejiang 314400, PR China

bDepartment of Information Engineering, Weihai Vocational College, Weihai, Shandong province, 264200, PR Chinaa r t i c l e i n f o

Article history:

Received 21 September 2009

Received in revised form 27 December 2010

Accepted 5 December 2011Keywords:

Log-convex function

Integral inequality

Taylor series expansion

Estimation of remainder termsa b s t r a c tIn this paper, some properties of log-convex function are researched, and integral inequal-

ities of log-convex functions are proved. As an application, an estimation formula of re- mainder terms in Taylor series expansion is given. Crown Copyright©2011 Published by Elsevier Ltd. All rights reserved.1. Introduction

Throughout the paper we assume thatR,R++andNrespectively stands for real number set, positive real number set

and natural number set. Recall that the definition of a log-convex function. Definition 1.Letf: [a,b] ⊆R→R++. Thenfis called a log-convex(concave) function, if f(αx+(1 holds for anyx,y∈ [a,b],α∈ [0,1].

The authors of [

1 ] proved the following results on the log-convex functions. Theorem 1.Let f: [a,b] ⊆R→R++be log-convex (concave), denote

M=

(b-a)(f(b)-f(a))lnf(b)-lnf(a),if f(a)̸=f(b); (b-a)f(a),if f(a)=f(b).

Then

b a

Corresponding author. Tel.: +86 13957359895.

E-mail addresses:zjzxm79@126.com(X. Zhang), jackjwd@163.com (W. Jiang).

0898-1221/$ - see front matter Crown Copyright©2011 Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.camwa.2011.12.019brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by Elsevier - Publisher Connector

1112X. Zhang, W. Jiang / Computers and Mathematics with Applications 63 (2012) 1111-1116

The aim of this paper is to show some results on the log-convex functions. In Section 2 , we give some integral properties

of the log-convex function, including a lower bound of its integral inequality. As an application, in Section

3 , an estimation

formula of remainder terms in Taylor series expansion ofex(x>0)is given. This result is better than the result in [2,3]

or [ 4

2. Some properties of the log-convex function

Theorem 2.(i)Let f: [a,b] → [0,+∞)be a strictly decreasing and differentiable function, f(x) >0for x∈(a,b]. Define

F(x)=x

af(t)dt with x∈(a,b]. Then F is a log-concave function.

(ii)Let f: [a,b] → [0,+∞)be a twice differentiable log-concave function, f(x) >0and f′(x) >0for x∈(a,b]. Define

F(x)=x

af(t)dt with x∈(a,b]. Then F is a log-concave function.

(iii)Let f: [a,b] → [0,+∞)be a twice differentiable log-convex function, f(x) >0and f′(x) >0for x∈(a,b],

lim x→a+f2(x)/f′(x)=0. Define F(x)=x af(t)dt with x∈(a,b]. Then F is log-convex function. Proof.We only prove (ii), the other proof is similar. Let

G(x):=F′′(x)F(x)-(F′(x))2f

′(x),x∈(a,b]. Then

G(x)=

x af(t)dt′′·x af(t)dt-f2(x)f ′(x) x a f(t)dt-f2(x)f ′(x). and G ′(x)=f(x)-2f(x)(f′(x))2-f2(x)f′′(x)(f′(x))2

ThenGis decreasing. We have

2(x)f and F

The proof of (ii) is completed.

Theorem 3.Let f: [a,b] ⊆R→R++be log-convex (concave), c∈(a,b), f′-(c)̸=0and f′+(c)̸=0. Then

b a ′-(c) Computers and Mathematics with Applications 63 (2012) 1111-1116

Contents lists available at

SciVerse ScienceDirect

Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa Some properties of log-convex function and applications for the exponential function

Xiaoming Zhanga,∗, Weidong Jiangb

a Zhejiang Broadcast & Radio University Haining College, Zhejiang 314400, PR China

bDepartment of Information Engineering, Weihai Vocational College, Weihai, Shandong province, 264200, PR Chinaa r t i c l e i n f o

Article history:

Received 21 September 2009

Received in revised form 27 December 2010

Accepted 5 December 2011Keywords:

Log-convex function

Integral inequality

Taylor series expansion

Estimation of remainder termsa b s t r a c tIn this paper, some properties of log-convex function are researched, and integral inequal-

ities of log-convex functions are proved. As an application, an estimation formula of re- mainder terms in Taylor series expansion is given. Crown Copyright©2011 Published by Elsevier Ltd. All rights reserved.1. Introduction

Throughout the paper we assume thatR,R++andNrespectively stands for real number set, positive real number set

and natural number set. Recall that the definition of a log-convex function. Definition 1.Letf: [a,b] ⊆R→R++. Thenfis called a log-convex(concave) function, if f(αx+(1 holds for anyx,y∈ [a,b],α∈ [0,1].

The authors of [

1 ] proved the following results on the log-convex functions. Theorem 1.Let f: [a,b] ⊆R→R++be log-convex (concave), denote

M=

(b-a)(f(b)-f(a))lnf(b)-lnf(a),if f(a)̸=f(b); (b-a)f(a),if f(a)=f(b).

Then

b a

Corresponding author. Tel.: +86 13957359895.

E-mail addresses:zjzxm79@126.com(X. Zhang), jackjwd@163.com (W. Jiang).

0898-1221/$ - see front matter Crown Copyright©2011 Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.camwa.2011.12.019brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by Elsevier - Publisher Connector

1112X. Zhang, W. Jiang / Computers and Mathematics with Applications 63 (2012) 1111-1116

The aim of this paper is to show some results on the log-convex functions. In Section 2 , we give some integral properties

of the log-convex function, including a lower bound of its integral inequality. As an application, in Section

3 , an estimation

formula of remainder terms in Taylor series expansion ofex(x>0)is given. This result is better than the result in [2,3]

or [ 4

2. Some properties of the log-convex function

Theorem 2.(i)Let f: [a,b] → [0,+∞)be a strictly decreasing and differentiable function, f(x) >0for x∈(a,b]. Define

F(x)=x

af(t)dt with x∈(a,b]. Then F is a log-concave function.

(ii)Let f: [a,b] → [0,+∞)be a twice differentiable log-concave function, f(x) >0and f′(x) >0for x∈(a,b]. Define

F(x)=x

af(t)dt with x∈(a,b]. Then F is a log-concave function.

(iii)Let f: [a,b] → [0,+∞)be a twice differentiable log-convex function, f(x) >0and f′(x) >0for x∈(a,b],

lim x→a+f2(x)/f′(x)=0. Define F(x)=x af(t)dt with x∈(a,b]. Then F is log-convex function. Proof.We only prove (ii), the other proof is similar. Let

G(x):=F′′(x)F(x)-(F′(x))2f

′(x),x∈(a,b]. Then

G(x)=

x af(t)dt′′·x af(t)dt-f2(x)f ′(x) x a f(t)dt-f2(x)f ′(x). and G ′(x)=f(x)-2f(x)(f′(x))2-f2(x)f′′(x)(f′(x))2

ThenGis decreasing. We have

2(x)f and F

The proof of (ii) is completed.

Theorem 3.Let f: [a,b] ⊆R→R++be log-convex (concave), c∈(a,b), f′-(c)̸=0and f′+(c)̸=0. Then

b a ′-(c)