The monotonicity and convexity of a function involving psi function with applications

R E S E A R C H

Open Access

The monotonicity and convexity of a

function involving psi function with

applications

Bang-Cheng Sun

_{, Zhi-Ming Liu}

_{, Qiang Li}

_{and Shen-Zhou Zheng}

*_{Correspondence: [email protected];}

[email protected]

1_{School of Mechanical, Electronic}

and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China

3_{Department of Mathematics,}

Beijing Jiaotong University, Beijing, 100044, China

Full list of author information is available at the end of the article

Abstract

In this paper, we prove that the function

x→exp

ψ

x+1

2

– 1 24

1 x2_{+ 7/40}

–x

is decreasing from (–1/2,∞) onto (0, 1/2) and convex on (–1/2,∞). As a consequence of the main theorem, various type of bounds for the psi function are presented, which essentially generalize or improve some known results.

MSC: Primary 33B15; 26A48; secondary 11B83; 26D15

Keywords: psi function; harmonic number; inequality

1 Introduction

Forx> , the classical Euler gamma functionand psi (digamma) functionψare defined by

(x) =

∞



tx–e–tdt, ψ(x) =

_(x)

(x), (.) respectively. Furthermore, the derivativesψ,ψ,ψ, . . . are known as polygamma func-tions.

As an important role played in many branches, such as mathematical physics, proba-bility, statistics, and engineering, the gamma and polygamma functions have attracted the attention of many scholars. Recently, many authors showed numerous interesting inequal-ities for the digamma (psi) functionψand the Euler constant defined by

γ = lim

n→∞

_n

k=



k–lnn

= .· · ·,

wheren_k=

k:=Hnis called the harmonic number. In particular, there has many

approx-imation formulas for psi function and harmonic number, which can be found in [–], and closely related references therein.

We would like to mention DeTemple and Wang’s paper [] for half-integer approxima-tion formulas

γ +ln n+ 

+ 

(n+ )<Hn<γ+ln n+

 

+ 

n (.)

and

γ +ln n+ 

+  n +

 



(n+ ) <Hn<γ +ln n+

 

+  n +

 



n. (.)

It was also proved in [, ], [], Lemma ., that

ln x+ 

<ψ(x+ )≤lnx+e–γ _(.)

forx> , where_ ande–γ _{are the best possible constants, andγ = .· · · is the}

Euler-Mascheroni constant. Thanks to formula (.) and the relationHn=γ +ψ(n+ ),

we have

γ +ln n+ 

<Hn≤γ +ln

n+e–γ–  (.)

for anyn∈N. In , Batir [] further proved that 

ln

x+x+e–γ≤ψ(x+ ) < ln x

_+x+



(.)

for allx> , wheree–γ _{and / are the best possible. As a direct consequence, he showed}

that, forn∈N,

γ + ln

n+n+e–γ – ≤Hn<γ +

 ln n

_+n+



. (.)

Batir [, ] provided another interesting bound for the psi function:

a–lne/x– <ψ(x) <b–lne/x– , (.) wherex> ,a≤ln, andb≥. Consequently, the double inequality

lnπ



 –ln

e/(n+)– <Hn<γ –ln

e/(n+)–  (.) for alln∈Nwas attained in Corollary . in []. Later, this inequality was sharpened to

 +ln(√e– ) –lne/(n+)– <Hn<γ–ln

e/(n+)–  (.) for alln∈Nby Alzer [].

Also, inequalities (.) are equivalent to

a≤eHn+_–eHn_{<b (.)}

with the best constantsa=e(√e– )≈. andb=eγ _{≈. (see also []). For}

a more general result, see [], Theorem .. Similarly, in the context, we call inequalities (.) theAlzer-type ones.

Batir [] further proved some new Batir-type inequalities for the psi functions and harmonic number, in particular,

 ln

x+b

e/(x+)_{– }≤ψ(x+ )≤

 ln

x+b

e/(x+)_{– } forx≥

with the best constantsa=  andb=e–γ_(e_{– ). This implies that, forn∈N, we have}

 ln

n+b

e/(n+)_{– }≤Hn≤

 ln

n+b e/(n+)_{– },

wherea=  andb=e–γ_{(e– ) – ≈. are the best possible. Obviously, it is}

equiv-alent to the double inequalities

eγ(n+ ) <eHn+_–eHn≤_eγ

(n+ .· · ·).

Clearly, it is an Alzer-type inequality.

On the other hand, Villarino [], Theorem ., proved that the sequence

dn=



ψ(n+ ) –ln(n+ /)–   n+

 



is increasing forn∈N, and meanwhile DeTemple and Wang [] by an approximation argument for the harmonic number showed, forn∈N, the following inequality:



(n+ /)_{+ /}<Hn–ln n+

 

–γ < 

(n+ /)_{+ /( –ln +ln –γ) – }

with the best constants _ and /( –ln +ln +ψ()) – ≈.. Yanget al.[], Theorem , further showed that the function

x→Fa(x) = 

x+aψ(x+ /) –lnx– 

is strictly completely monotonic on (,∞) if and only ifa≥/.

Motivated by all these recent papers, the aim of this paper is to investigate the mono-tonicity and convexity of the function related to the psi function and present some new general, Batir-type, and Alzer-type inequalities for the psi function and harmonic number.

2 Preliminaries

Lemma ([], pp.-) Let x> and n∈N.Then

ψ(n)(x+ ) –ψ(n)(x) =(–)

n_n!

xn+ . (.)

Lemma ([], pp.-) As x→ ∞,we have

ψ(x)∼lnx–  x–

 x +

 x–



x, (.) ψ(x)∼ 

x+

 x +

 x –

 x+



x, (.) ψ(x)∼–

x –



x –

 x+

 x–



x. (.)

Lemma ([]) Let f be a function on an interval I such thatlimx→∞f(x) = .If f(x+ ) –

f(x) > for all x∈(a,∞),then f(x) < .Conversely,if f(x+ ) –f(x) < for all x∈(a,∞),

then f(x) > .

Lemma ([], Lemma ) For n∈Nand m∈N∪{}with n>m,let Pn(t)be a polynomial with n degrees defined by

Pn(t) = n

i=m+

aiti– m

i=

aiti, (.)

where an,am> and ai≥for≤i≤n– with i=m.Then there exists an unique number tm+∈(,∞)satisfying Pn(t) = such that Pn(t) < for t∈(,tm+)and Pn(t) > for t∈

(tm+,∞).

Lemma  Let u be the function on(–∞,∞)defined by

u(x) = 

x_{+ }. (.)

Then,for x= –/,we have

p(x) = –  (x+ /) –u

_{(x+ ) +u(x) < . (.)}

Proof Differentiation leads to

u(x) = – 

x

(x_{+ )}. (.)

Factoring gives

p(x) = –  (x+ /) +

 

x+  ((x+ )_{+ )} –

 

x

(x_{+ )}

= –  (x+ /) –

 

,x_{+ ,x}_{+ ,x}_{+ ,x– }

Replacingxby (t– /) and factoring, we get

p(x) = –,,t

_{– ,t}_{+ ,,t}_{– ,t}_{+ ,}

t_{((t– /)}_{+ )}_{((t+ /)}_{+ )} ,

wheret=x+ /.

Note that the numerator of this fraction can be written as

p(t) = ,,t– ,t+ ,,t– ,t+ ,

= ,tt– +,,  t

₊



t– > , (.)

and the desired result easily follows.

Lemma  Let u and p be defined by(.)and(.).Suppose that q and r are defined on

(–/,∞)by

q(x) =  (x+ /) –u

_{(x+ ) +u(x),}

r(x) = –  (x+ /) –u

_{(x+ ) –u(x).}

Then,for x> –/,we have

S(x) := – 

(x+ /) +r(x+ ) +

q(x+ )

p(x+ )–r(x) –

q(x)

p(x)< . (.)

Proof An immediate computation yields

u(x) = 

x– 

(x_{+ )}. (.)

Then we get

q(x) =  (x+ /) –

 

(x+ )_{– }

((x+ )_{+ )} +

 

x_{– }

(x_{+ )}, (.)

r(x) = –  (x+ /) +

 

x+  ((x+ )_{+ )} +

 

x

(x_{+ )}. (.)

Substitutingp(x),q(x),r(x) intoS(x) and factoring it give

S(x) = –×





S(x)

S(x)

,

where

S(x) = (x+ )(x+ )

x+ x+ x+ 

×,,x+ ,,x+ ,,x+ ,,,x

+ ,,,x+ ,,

×,,x+ ,,x+ ,,x+ ,,x

+ ,,x+ ,,x+ ,,x

+ ,,x+ ,, (.)

S(x) = ,,,x+ ,,,x+

,,,,  x



+,,,,  x

₊,,,,,

 x



+,,,,,  x

₊,,,,,

 x



+,,,,,  x

₊,,,,,,

, x



+,,,,,, , x

₊,,,,,,

, x



+,,,,,, , x

₊,,,,,,

, x



+,,,,,, , x

₊,,,,,,

, x



+,,,,,, , x

₊,,,,,,

,, x



+,,,,,, ,, x

₊,,,,,,

,, x



+,,,,,, ,, x+

,,,,,, ,,, .

We further prove thatS(x),S(x) >  forx> –/. In fact, replacingxby (t– /) in (.)

and arranging, we obtain

S(x) = t(t+ )

t– t+ t+ t+ 

×,,t+ ,,t+ ,,t+ ,,t

+ ,,t+ ,,t+ ,,t+ ,,t

+ ,,×p(t),

wherep(t) is defined by (.), andt=x+ / > . Clearly,S(x) >  forx> –/.

Similarly, we have

S(x) =S(t) +tS(t),

where

S(t) = ,,,t+ ,,,t+

,,,,  t



+,,,,  t

₊,,,,

 t

₊,,,,

 t

+,,,,,  t

₊,,,,

 t



+,,,,, , t

₊,,,,,

, t



+,,,,, , t

₊,,,,

, t



+,,,,, , t

₊,,,,

, t



+,,,,, , t

₊,,,,,

,, t +,,,,,,

,,, ,

S(t) =

,,,,, , t

_–,,,,,

,, t



–,,,, ,, t+

,,,,, ,, .

It is clear thatS(t) >  fort=x+ / > . To prove thatS(x) >  forx> –/, it suffices to

prove thatS(t) >  fort> . In fact, it is easy to check that

S_(t) =,,,,, , t

_–,,,,,

,, t –,,,,,

,,

has a positive zero pointt∈(/, /) such thatS(t) <  fort∈(,t) andS(t) <  for

t∈(t,∞). SinceS(),S(∞) >  and

S(t) =

,,,,, , t

 +

,,,,, ,, – ,,,,,

,, t

 +

,,,,, ,, t

>,,,,, ,

 



+,,,,, ,, – ,,,,,

,,

 



+,,,,, ,,

 

=,,,,,, ,,, > ,

so we getS(t) >  fort> , which proves thatS(x) >  forx> –/ and completes the

proof.

3 Monotonicity and convexity

Theorem  The function

x→f(x) =exp ψ x+

 

–  



x_{+ /}

–x

is decreasing from(–/,∞)onto(, /)and convex on(–/,∞).

Proof With (.) in hand,f(x) can be written as

f(x) =eψ(x+/)–u(x)–x.

Differentiation of this formula yields

f_(x) =ψ(x+ /) –u(x)eψ(x+/)–u(x)– , (.)

f_(x) =ψ

_{(x+ /) –u(x) + (ψ(x+ /) –u(x))}

e–ψ(x+/)+u(x)

:= g(x)

e–ψ(x+/)+u(x). (.)

By (.) this yields

g(x+ ) –g(x)

=ψ(x+ /, ) –u(x+ )+ψ(x+ /, ) –u(x+ ) –ψ(x+ /, ) –u(x) +ψ(x+ /, ) –u(x) =p(x)

ψ(x+ /, ) +r(x) +q(x)

p(x)

:=p(x)h(x), (.)

wherep(x),q(x),r(x) are defined by (.), (.), (.), respectively. Similarly, we get

h(x+ ) –h(x) = – 

(x+ /)+r(x+ ) +

q(x+ )

p(x+ )–r(x) –

q(x)

p(x)=S(x).

By Lemma  we haveh(x+ ) –h(x) < . It follows fromlim_x→∞h(x) =  and Lemma  that

h(x) >lim_x→∞h(x) = . Thanks to inequality (.),p(x) < , and Lemma , it follows that

g(x+ ) –g(x) =p(x)h(x) < ,

which implies by Lemma  thatg(x) >lim_x→∞g(x) = . Thus, in combination with (.), this leads tof_(x) > , that is,fis convex on (–/,∞), andfis increasing on (–/,∞).

Utilizing the asymptotic formulas (.)-(.), this yields

lim

x→∞f

Therefore, we get thatf_(x) <lim_x→∞f_(x) = , which implies thatf(x) is decreasing on

(–/,∞). Moreover, we conclude that

 = lim

x→∞f(x) <f(x) <x→lim–/+f(x) =

 ,

which completes the proof.

Theorem  The function

x→f(x) =expψ x+

 

–xexp 

 

x_{+ /}

is decreasing from(,∞)onto(,e–γ_{/)and convex on(/,∞).}

Proof We have

f(x) =exp

 



x_{+ /}

exp

ψ x+ 

–  



x_{+ /}

–x

=exp 

 

x_{+ /}

f(x) :=f(x)f(x).

Noting that

f_(x) = –  

x

(x_{+ /)}exp

 



x_{+ /}

≤ forx≥,

f_(x) =  ,

,x_{+ ,x}_{– }

(x_{+ /)} exp

 



x_{+ /}

>  forx≥ 

, we have

f_(x) =f_(x)f(x) +f(x)f(x) <  forx≥,

f_(x) =f_(x)f(x) + f(x)f(x) +f(x)f(x) >  forx≥

 . A simple calculation leads to

lim

x→∞f(x) =x→∞limexp

 



x_{+ /}

lim

x→∞f(x) =  and x→lim+f(x) =

 e

–γ

,

which completes the proof.

Theorem  Let a≥.Then the function

x→f(x) =ψ x+

 

–lnx–  



x_+a

Proof The necessity is obvious; it follows from the inequalitylim_x→∞x_f

(x)≤. Indeed,

using the asymptotic formulas (.), we have

lim

x→∞x

_f (x) =

 –

 a≤, which yieldsa≥/.

We now are a position to prove the sufficiency. By differentiation we have

f_(x) =ψ x+ 

–

x+

 

x

(x_+a),

f_(x) =ψ x+ 

+ 

x +

 

 (x_+a) –

 

x

(x_+a).

Using (.), we have

f_(x+ ) –f_(x) =ψ x+ 

+  (x+ )+

 



((x+ )_+a) –

 

x

((x+ )_+a)

–ψ x+ 

– 

x–

 

 (x_+a)+

 

x

(x_+a)

=  (x+ /) +

 (x+ ) +

 

 ((x+ )_+a)

– 

x

((x+ )_+a)–



x–

 

 (x_+a) +

 

x

(x_+a)

= – P(x)

x_{(x+ )}_(x_+a)_{(x+ )}_(x_{+ x+a+ )},

where

P(x) = (a– )x+ (a– )x+ (,a+ ,a– ,)x

+ (,a+ ,a– ,)x+ (,a+ ,a+ ,a– ,)x

+ (,a+ ,a+ ,a– )x

+ (,a+ ,a+ ,a+ ,a– )x

+ (,a+ ,a+ ,a+ a– )x

+ (a+ ,a+ ,a+ ,a+ a– )x

+ (,a+ ,a+ ,a+ a+ a)x

+ (a+ a+ ,a+ a+ a+a)x

+ (a+ a+ a+ a)x+ (a+ a+ a+ a).

It is easy to check that the coefficients of the polynomial P(x) are nonnegative for

a≥/. Then we havef_(x+ ) –f_(x) <  forx> . By Lemma  we get thatf_(x) >

lim_x→∞f_(x) = , which implies that f_(x) is increasing on (,∞). Therefore, f_(x) <

4 Inequalities for the psi function and harmonic number

DenoteFi(x) =fi(x+ /) (i= , , ) in Theorems  and . Then, sinceFis decreasing,

F() =e–γ–/, andlimx→∞F(x) = /, we get the following conclusion.

Corollary 

(i) Forx> –/,we have

ψ(x+ ) >ln x+ 

+  



x_{+x+ /}. (ii) Forx≥,we have

ln x+ 

+  



x_{+x+ /}<ψ(x+ ) <ln(x+α) +

 



x_{+x+ /}, (.)

where the constants/andα=e–γ–/≈.are the best possible.

Remark  Comparing (.) with (.), we find that, forx> ,

ln x+ 

<ln x+ 

+  



x_{+x+ /}<ψ(x+ )

<ln(x+α) +

 



x_{+x+ /}<ln

x+e–γ.

In fact, it suffices to show that the last inequality is valid forx> . Let

D(x) =ln(x+α) +

 



x_{+x+ /}<ln

x+e–γ.

By differentiation we have

D_(x) = 

x+α

–  

x+  (x_{+x+ /)} –



x+β

= D(x)

(x+α)(x+β)(x+ x+ )

,

where

D(x) = ,(e–γ–α)x+ (e–γ – α– )x– (α– e–γ+ )x

– (α– e–γ + αe–γ)x– (α– e–γ + αe–γ)

:=ax+ax+ax+ax+a.

It is easy to check thata,a>  anda,a,a< . Then by Lemma  we see that there

isx>  such thatD(x) <  forx∈(,x) andD(x) >  forx∈(x,∞). This indicates

thatDis decreasing on (,x) and increasing on (x,∞). Therefore, we conclude that, for

x> ,

D(x) <max

D(),D(∞)

Remark  Similarly, we get the following inequalities: 

ln

x+x+e–γ<ln x+ 

+  



x_{+x+ /}<ψ(x+ )

<  ln x

_+x+



<ln(x+α) +

 



x_{+x+ /}

forx> . A direct computation shows that

lim

x→∞

ψ(x+ ) –_ln(x_+x+ )

x– = –



, (.)

lim

x→∞

ψ(x+ ) –ln(x+_) –_ _x_{+x+/}

x– =

,

,, (.) which implies that the approximation formula of the psi function given in (.) is superior to (.).

SinceF() =e/–γ, by the relationψ(n+ ) =Hn–γ we deduce the following:

Corollary  For n∈N,we have

γ +  



n_{+n+ /}+ln(n+ /) <Hn<γ+

 



n_{+n+ /}+ln(n+α),

where/andα=e/–γ≈.are the best possible.

SinceFis decreasing on (–/,∞),limx→∞F(x) = , and

F() =e–γ–

 e

/₌

β, F() =e

–γ_–

e

/₌

β, we get the following:

Corollary  For x≥,we have

ln x+ 

+  



x_{+x+ /}<ψ(x+ ) <ln x+

 

+  



x_{+x+ /}

+ln

 + β

x+ exp –  



x_{+x+ /}

,

whereβ= e–γ–e/≈.is the best constant.

Corollary  For n∈N,we have

γ +ln n+ 

+  



n_{+n+ /}<Hn<γ +ln n+

 

+  



n_{+n+ /}

+ln

 + β

n+ exp –  



n_{+n+ /}

,

Fora= /, sinceFis decreasing on (–/,∞),limx→∞F(x) = , and

F() =ln –



–γ =δ, F() =   –ln



–γ =δ, we deduce the following:

Corollary  For x≥,we have

ln x+ 

+  



x_{+x+ /}+δ

∗

<ψ(x+ ) <ln x+

 

+  



x_{+x+ /}+δ,

whereδ∗_= andδ=ln – / –γ≈.are the best constants.

Corollary  For n∈N,we have

γ+ln n+ 

+  



n_{+n+ /}+δ

∗

<Hn<γ+ln n+

 

+  



n_{+n+ /}+δ,

whereδ∗_= andδ= / –ln(/) –γ≈.are the best constants.

From Theorem  we can obtain the following Batir-type inequalities for the psi function and harmonic number.

Corollary  For x≥,we have



(x_{+x+ /)}–ln

eQ(x)– +c<ψ(x+ ) <



(x_{+x+ /)}–ln

eQ(x)– +c∗_

with the best constants c=ln(e,/,– ) – / –γ ≈–.and c∗= ,where

Q(x) =,x

_{+ ,x}_{+ ,x}_{+ x+ ,}

(x+ )(x_{+ x+ )(x}_{+ x+ )} .

Proof LetG(x) =F(x+ ) –F(x) =f(x+ /) –f(x+ /). Sincefis convex on (–/,∞),

we have

G_(x) =f_(x+ /) –f_(x+ /) =f_(x+ / +θ) > ,

which means thatGis increasing on (,∞). Considering

G(x) =exp ψ(x+ ) –

 

 (x+ /)_{+ /}

–exp ψ(x+ ) –  

 (x+ /)_{+ /}

– 

=exp ψ(x+ ) –  

 (x+ /)_{+ /}

eQ(x)– – ,

G() =exp

  –γ

–e – –γ

–  and lim

we have

G() <exp ψ(x+ ) –

 

 (x+ /)_{+ /}

eQ(x)– –  < ,

which attains the desired inequality. The increasing property ofGand

G() =exp

, ,–γ

–exp 

 –γ

– 

yield a sharp bound of harmonic number.

Corollary  For n∈N,we have

γ + 

(n_{+n+ /)}–ln

eQ(n)– +c

<Hn<γ +



(n_{+n+ /)}–ln

eQ(n)– +c∗_,

where c= / –γ +ln(e,/,– )≈–.and c∗= are the best

con-stants.

Remark  Note that sinceψ(n+ ) =Hn–γ,G(n) is written as

G(n) =exp Hn+–γ –

 

 (n+ /)_{+ /}

–exp Hn–γ–

 

 (n+ /)_{+ /}

– .

Then byG()≤G(n) <G(∞) we have the following Alzer-type inequalities:

.≈exp ,

,

–exp 



<eHn+–un+_–eHn–un_<eγ _≈

.,

where

un=

 



(n+ /)_{+ /}. (.)

Remark  Similarly, it is easy to check that

G(x) =F(x+ ) –F(x) =f(x+ /) –f(x+ /)

is increasing on (,∞). Then, from

–.≈  e

/_–

e

/_+e–γ

we derive other Alzer-type inequalities:

n+ 

un+– n+

 

un+d<eHn+–eHn< n+

 

un+– n+

 

un+d,

whered=  andd= e// – e// +e–γ(e/– )≈–. are the best

con-stants withunas in (.).

Using the increasing property of F_, and noting that F_(–+_{) = –, F}

() = (π/ +

/)e–γ–/_{– , andF}

(∞) = , we get

 π

₊



e–γ–/_{–  <ψ(x+ ) –u(x)e}ψ(x+)–u(x)_{–  < ,}

which implies the following:

Corollary 

(i) Forx> –,we have

ψ(x+ ) < –  

x+ /

(x_{+x+ /)} +exp –ψ(x+ ) +



(x_{+x+ /)}

.

(ii) Forx≥,we have the double inequalities

–  

x+ /

(x_{+x+ /)}+λexp –ψ(x+ ) +



(x_{+x+ /)}

<ψ(x+ ) < –  

x+ / (x_{+x+ /)}

+λexp –ψ(x+ ) +



(x_{+x+ /)}

,

whereλ= (π/ + /)e–γ–/≈.andλ= are the best constants.

Remark  Elezovicet al.[] proved the inequality

ψ(x) <e–ψ(x) (.) forx> . It has been improved by Batir [] as

x+a∗e–ψ(x)<ψ(x+ ) <x+b∗e–ψ(x)

forx>  with the best constantsa∗= / andb∗=π_e–γ_{/. The last corollary gives}

an-other improvement of (.).

From the proof of Theorem  we see thatg(x) >  forx> –/, which can be written as the following corollary.

Corollary  For x> ,we have

ψ(x) –u(x– /) +ψ(x) –u(x– /)> , (.)

Remark  Batir [] showed that, forx> ,

ψ(x)+ψ(x) > . (.) Therefore, inequality (.) can be written as

ψ(x) +ψ(x)>(x), where

(x) = – 

x–  (x_{– x+ )} × ψ(x) – ,x

_{– ,x}_{+ ,x}_{– ,x+ ,}

(x– )(x_{– x+ )}

.

Indeed, this result is optimal due to

ψ(x) – 

(x– /)_{– }

((x– /)_{+ )} + ψ

_{(x) +}



(x– /) ((x– /)_{+ )}



=ψ(x) +ψ(x)– 



(x_{– x+ )}(x),

where

(x) = –(x– )ψ(x) +,x

_{– ,x}_{+ ,x}_{– ,x+ ,}

(x_{– x+ )} .

A numeric computation shows that(x) >  for  <x< / andx> /, and so inequality (.) is better than (.).

From the inequalitiesf_(x) <  andf_(x) >  on (,∞) fora= /, which are given in the proof of Theorem , we have the following:

Corollary  For x> ,we have the following inequalities:

ψ x+ 

< 

,x_{+ ,x}_{+ }

x(x_{+ )} , ψ x+



> – 

,x+ ,x+ ,x+ ,

x_(x_{+ )} ,

ψ x+ 

< ,,x

_{+ ,x}_{+ ,x}_{+ ,x}_{+ ,}

x_(x_{+ )} .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China.}2_Tangshan

Railway Vehicle Co. LTD, Tangshan, 063035, China.3_{Department of Mathematics, Beijing Jiaotong University, Beijing,}

Acknowledgements

This paper was supported by the Natural Science Foundation of China under Grant 11371050. The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper.

Received: 28 February 2016 Accepted: 19 May 2016 References

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