Some new judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions

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R E S E A R C H

Open Access

Some new judgement theorems of Schur

geometric and Schur harmonic convexities

for a class of symmetric functions

Huan-Nan Shi

1

_{and Jing Zhang}

2*

*_{Correspondence:}

ldtzhangjing1@buu.edu.cn 2_{Basic Courses Department, Beijing}

Union University, Beijing, 100101, P.R. China

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

Abstract

The judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions are given. As their application, some analytic inequalities are established.

MSC: Primary 26D15; 05E05; 26B25

Keywords: Schur geometric convexity; Schur harmonic convexity; inequality; symmetric function

1 Introduction

Throughout this paper,Rdenotes the set of real numbers,x= (x,x, . . . ,xn) denotesn

-tuple (n-dimensional real vectors), the set of vectors can be written as

Rn₌_x_{= (}_x

, . . . ,xn) :xi∈R,i= , . . . ,n

,

Rn

+=

x= (x, . . . ,xn) :xi> ,i= , . . . ,n

.

In particular, the notationsRandR+denoteRandR+, respectively.

Letπ= (π(), . . . ,π(n)) be a permutation of (, . . . ,n), all permutations are totallyn!. The following conclusion is proved in [, pp.-].

Theorem A Let A⊂Rk_{be a symmetric convex set}_,_{and let}_ϕ_{be a Schur-convex function} deﬁned on A with the property that for each ﬁxed x, . . . ,xk,ϕ(z,x, . . . ,xk)is convex in z on {z: (z,x, . . . ,xk)∈A}.Then,for any n>k,

ψ(x, . . . ,xn) =

π

ϕ(xπ(), . . . ,xπ(k)) ()

is Schur-convex on

B=(x, . . . ,xn) : (xπ(), . . . ,xπ(k))∈A for all permutationsπ

.

Furthermore,the symmetric function

ψ(x) =

≤i<···<ik≤n

ϕ(xi, . . . ,xik) ()

is also Schur-convex on B.

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Theorem A is very eﬀective for judgement of the Schur-convexity of the symmetric func-tions of the form (), see the references [] and [].

The Schur geometrically convex functions were proposed by Zhang [] in . Further, the Schur harmonically convex functions were proposed by Chu and Lü [] in . The theory of majorization was enriched and expanded by using these concepts [–]. Re-garding Schur geometrically convex functions and Schur harmonically convex functions, the aim of this paper is to establish the following judgement theorems which are similar to Theorem A.

Theorem  Let A⊂Rk _{be a symmetric geometrically convex set}_, _{and let}_ϕ _{be a Schur} geometrically convex(concave)function deﬁned on A with the property that for each ﬁxed x, . . . ,xk,ϕ(z,x, . . . ,xk)is GA convex(concave)in z on{z: (z,x, . . . ,xk)∈A}.Then,for any n>k,

ψ(x, . . . ,xn) =

π

ϕ(xπ(), . . . ,xπ(k))

is Schur geometrically convex(concave)on

.

ψ(x) =

≤i<···<ik≤n

ϕ(xi, . . . ,xik)

is also Schur geometrically convex(concave)on B.

Theorem  Let A⊂Rk _{be a symmetric harmonically convex set}_,_{and let}_ϕ _{be a Schur} harmonically convex(concave)function deﬁned on A with the property that for each ﬁxed x, . . . ,xk,ϕ(z,x, . . . ,xk)is HA convex(concave)in z on{z: (z,x, . . . ,xk)∈A}.Then,for any n>k,

ψ(x, . . . ,xn) =

π

ϕ(xπ(), . . . ,xπ(k))

is Schur harmonically convex(concave)on

.

ψ(x) =

≤i<···<ik≤n

ϕ(xi, . . . ,xik)

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2 Deﬁnitions and lemmas

In order to prove some further results, in this section we recall useful deﬁnitions and lem-mas.

Deﬁnition [, ] Letx= (x, . . . ,xn) andy= (y, . . . ,yn)∈Rn.

(i) We sayymajorizesx(xis said to be majorized byy), denoted byx≺y, if

k i=x[i]≤

k

i=y[i]fork= , , . . . ,n– and

n i=xi=

n

i=yi, wherex[]≥ · · · ≥x[n]

andy[]≥ · · · ≥y[n]are rearrangements ofxandyin a descending order.

(ii) Let⊂Rn_{, a function}_ϕ_:_→_R_{is said to be a Schur-convex function on}_if

x≺yonimpliesϕ(x)≤ϕ(y). A functionϕis said to be a Schur-concave function onif and only if–ϕis Schur-convex function on.

Deﬁnition [, ] Letx= (x, . . . ,xn) andy= (y, . . . ,yn)∈Rn, ≤α≤. A set⊂Rnis

said to be a convex set ifx,y∈impliesαx+ ( –α)y= (αx+ ( –α)y, . . . ,αxn+ ( –α)yn)∈ .

Deﬁnition [, ]

(i) A set⊂Rnis called a symmetric set ifx∈impliesxP∈for everyn×n

permutation matrixP.

(ii) A functionϕ:→Ris called symmetric if for every permutation matrixP,

ϕ(xP) =ϕ(x)for allx∈.

Deﬁnition  Let⊂Rn

+,x= (x, . . . ,xn)∈andy= (y, . . . ,yn)∈.

(i) [, p.] A setis called a geometrically convex set if(xα

y

β

, . . . ,xαny

β

n)∈for all

x,y∈andα,β∈[, ]such thatα+β= .

(ii) [, p.] A functionϕ:→R+is said to be a Schur geometrically convex function

onif(logx, . . . ,logxn)≺(logy, . . . ,logyn)onimpliesϕ(x)≤ϕ(y). A function ϕis said to be a Schur geometrically concave function onif and only if–ϕis a Schur geometrically convex function.

Deﬁnition [] Let⊂Rn

+.

(i) A setis said to be a harmonically convex set if_λx_+(–xy_λ₎_y∈for everyx,y∈and

λ∈[, ], wherexy=n_i_=xiyiandx= (



x, . . . ,



xn).

(ii) A functionϕ:→R+is said to be a Schur harmonically convex function onif 

x≺



yimpliesϕ(x)≤ϕ(y). A functionϕis said to be a Schur harmonically concave

function onif and only if–ϕis a Schur harmonically convex function.

Deﬁnition [] LetI⊂R+,ϕ:I→R+be continuous.

(i) A functionϕis said to be a GA convex (concave) function onIif

ϕ(√xy)≤(≥)ϕ(x) +ϕ(y) 

for allx,y∈I.

(ii) A functionϕis said to be a HA convex (concave) function onIif

ϕ

xy x+y

≤(≥)ϕ(x) +ϕ(y) 

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Lemma [, p.] Let⊂Rn_{be a symmetric convex set with a nonempty interior}_.

ϕ:→Ris continuous onand diﬀerentiable on_._Then_ϕ_{is a Schur-convex}₍

Schur-concave)function if and only ifϕis symmetric onand

(x–x)

∂ϕ ∂x

– ∂ϕ

∂x

≥ (≤) ()

holds for anyx= (x, . . . ,xn)∈.

Lemma [, p.] Let⊂Rn₊be a symmetric geometrically convex set with a nonempty interior.Letϕ:→R+be continuous onand diﬀerentiable on.Thenϕis a Schur

geometrically convex(Schur geometrically concave)function if and only ifϕis symmetric

onand

(x–x)

x

∂ϕ ∂x

–x

∂ϕ ∂x

≥ (≤) ()

Lemma [, ] Let⊂Rn

+be a symmetric harmonically convex set with a nonempty

interior_._Let_ϕ_:_→_R

+be continuous onand diﬀerentiable on.Thenϕis a Schur

harmonically convex(Schur harmonically concave)function if and only ifϕis symmetric

onand

(x–x)

x_∂ϕ ∂x

–x_∂ϕ ∂x

≥ (≤) ()

Lemma [] Let I⊂R+be an open subinterval,and letϕ:I→R+be diﬀerentiable.

(i) ϕis GA-convex(concave)if and only ifxϕ(x)is increasing(decreasing). (ii) ϕis HA-convex(concave)if and only ifxϕ(x)is increasing(decreasing).

3 Proofs of main results

Proof of Theorem To verify condition () of Lemma , denote by_π₍_i_,_j₎the summation over all permutationsπsuch thatπ(i) = ,π(j) = . Becauseϕis symmetric,

ψ(x, . . . ,xn)

=

i,j≤k i=j

π(i,j)

ϕ(x,x,xπ(), . . . ,xπ(i–),xπ(i+), . . . ,xπ(j–),xπ(j+), . . . ,xπ(k))

+

i≤k<j

π(i,j)

ϕ(x,xπ(), . . . ,xπ(i–),xπ(i+), . . . ,xπ(k))

+

j≤k<i

π(i,j)

ϕ(x,xπ(), . . . ,xπ(j–),xπ(j+), . . . ,xπ(k))

+

k<i,j i=j

π(i,j)

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Then

:=

x

∂ψ ∂x

–x

∂ψ ∂x

(x–x)

=

i,j≤k i=j

π(i,j)

xϕ()(x,x,xπ(), . . . ,xπ(i–),xπ(i+), . . . ,xπ(j–),xπ(j+), . . . ,xπ(k))

–xϕ()(x,x,xπ(), . . . ,xπ(i–),xπ(i+), . . . ,xπ(j–),xπ(j+), . . . ,xπ(k)) (x–x)

+

i≤k<j

π(i,j)

xϕ()(x,xπ(), . . . ,xπ(i–),xπ(i+), . . . ,xπ(k))

–xϕ()(x,xπ(), . . . ,xπ(i–),xπ(i+), . . . ,xπ(k)) (x–x).

Here,

(xϕ()–xϕ())(x–x)≥ (≤)

becauseϕis Schur geometrically convex (concave), and

xϕ()(x,z) –xϕ()(x,z)(x–x)≥ (≤)

becauseϕ(z,x, . . . ,xk) is GA convex (concave) in its ﬁrst argument on{z: (z,x, . . . ,xk)∈ A}. Accordingly,≥ (≤). This shows thatψis Schur geometrically convex (concave)

on

.

Notice that

ψ(x) =ψ(x)/k!(n–k)!.

Of course,ψ is Schur geometrically convex (concave) wheneverψis Schur geometrically convex (concave).

The proof of Theorem  is completed.

Proof of Theorem We only need to verify condition () of Lemma , the proof is similar to that of Theorem  and is omitted.

Remark  In most applications,Ahas the formIkfor some intervalI⊂Rand in this case

B=In. Notice that the convexity ofϕin its ﬁrst argument also implies thatϕis convex in each argument, the other arguments being ﬁxed, becauseϕis symmetric.

4 Applications Let

Ek

x  –x

=

≤i<···<ik≤n k

j=

xij

 –xij

. ()

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Theorem  The symmetric function Ek(_–xx),k= , . . . ,n,is Schur geometrically convex on (, )n_.

Now, we give a new proof of Theorem  by using Theorem . Furthermore, we prove the following theorem through Theorem .

Theorem  The symmetric function Ek(_–xx),k= , . . . ,n,is Schur harmonically convex on (, )n.

Proof of Theorem Letϕ(z) =k_i_=[zi/( –zi)]. Then

logϕ(z) =

k

i=

logzi–log( –zi)

and

∂ϕ(z)

∂z

=ϕ(z)



z

+   –z

, ∂ϕ(z)

∂z

=ϕ(z)



z

+   –z

, ()

:= (z–z)

z

∂ϕ(z)

∂z

–z

∂ϕ(z)

∂z

= (z–z)ϕ(z)

z

 –z

– z  –z

= (z–z)ϕ(z)

 ( –z)( –z)

.

This shows that≥ when  <zi< ,i= , . . . ,k. According to Lemma ,ϕis Schur

ge-ometrically convex onA={z:z∈(, )k_}_{. Let}_g₍_t_{) =} t

–t, thenh(t) :=tg(t) = t

(–t). From t∈(, ), it follows thath(t) = +t

(–t) ≥. According to Lemma (i),ϕis GA convex in its

single variable on (, ). SoEk(_–xx) is Schur geometrically convex on (, )

n_from

Theo-rem . The proof of TheoTheo-rem  is completed.

Proof of Theorem Letϕ(z) =k_i_=(zi/ –zi), then

logϕ(z) =

k

i=

logzi–log( –zi) .

From (), we get

:= (z–z)

z_∂ϕ(z) ∂z

–z_∂ϕ(z) ∂z

= (z–z)ϕ(z)

z–z+

z 

 –z

– z

 

 –z

= (z–z)ϕ(z)

 + z+z–zz ( –z)( –z)

.

This shows that ≥ when  <zi< ,i= , . . . ,k. According to Lemma ,ϕ is Schur

harmonically convex onA={z:z∈(, )k_}_{. Let}_g₍_t_{) =} t

–t, thenp(t) :=tg(t) = t

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t∈(, ), it follows thatp(t) = _(–t_t₎ ≥. According to Lemma (ii),ϕ is HA convex in

its single variable on (, ). SoEk(_–xx) is Schur harmonically convex on (, )nfrom Theo-rem . The proof of TheoTheo-rem  is completed.

By using Theorem A, the following conclusion is proved in [, p.]. The symmetric function

ψ(x) =

≤i<···<ik≤n

xi+· · ·+xik xi· · ·xik

()

is Schur-convex onRn

+.

Now we use Theorem  and Theorem , respectively, to study Schur geometric convexity and Schur harmonic convexity ofψ(x).

Theorem  The symmetric functionψ(x)is Schur geometrically convex and Schur har-monically concave onRn

+.

Proof Letϕ(y) =k_i_=yi/

k

i=yi, thenlogϕ(y) =log(

k i=yi) –

k

i=logyi. Thus, ∂ϕ(y)

∂y

=ϕ(y)

 k

i=yi

– 

y

, ∂ϕ(y)

∂y

=ϕ(y)

 k

i=yi

– 

y

,

:= (y–y)

y

∂ϕ(y)

∂y

–y

∂ϕ(y)

∂y

= (y–y)ϕ(y)

y–y

k i=yi

=(y–y)



k i=yi

≥.

According to Lemma ,ϕ(y) is Schur geometrically convex onRk

+. Letg(z) =ϕ(z,x, . . . ,

xk) =z_bz+a=_b+_bza, wherea=

k i=xi,b=

k

i=xi, thenh(z) :=zg(z) = –_bza. Fromz∈R+, it

follows thath(z) =_bza ≥. According to Lemma (i),ϕis GA convex in its single variable

onR+. Soψ(x) is Schur geometrically convex onR+from Theorem .

It is easy to check that

:= (y–y)

y_∂ϕ(y) ∂y

–y_∂ϕ(y) ∂y

=(y–y)

₍_y +y–

k i=yi)

k i=yi

≤.

According to Lemma ,ϕ(y) is Schur harmonically concave onRk

+. Leth(z) :=zg(z) = –ab. h(z) =  whenz∈R+. According to Lemma (ii),ϕis HA concave in its single variable on

R+. Soψ(x) is Schur harmonically concave onRn+from Theorem .

Remark  Let

H=_nn

i=xi

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wherexi> ,i= , . . . ,n. Then

(logG, . . . ,logG)≺(logx, . . . ,logxn), ()



H, . . . ,



H

≺



x

, . . . , 

xn

. ()

From Theorem , it follows that

kCk n Hk–≥

≤i<···<ik≤n

xi+· · ·+xik xi· · ·xik

≥ kCkn

Gk–. ()

By using Theorem A, the following conclusion is proved in [, p.]. The symmetric function

ψ(x) =

≤i<···<ik≤n

xi· · ·xik xi+· · ·+xik

is Schur-concave onRn

+.

By applying Theorem , we further obtain the following result.

Theorem  The symmetric functionψ(x)is Schur harmonically convex onRn₊.

Proof Letλ(y) =k_i_=yi/

k

i=yi. According to the proof of Theorem ,ϕ(y) is Schur

har-monically concave onRk

+. Letλ(y) =ϕ(y). From the deﬁnition of Schur harmonically

con-vex, it follows thatλ(y) is Schur harmonically convex onRk

+. Letg(z) =λ(z,x, . . . ,xk) = _zbz₊_a,

where a=k_i_=xi, b=

k

i=xi. Thenh(z) :=zg(z) = z _ab

(z+a). With the fact thath(z) =

za_b

(z+a) ≥ forz∈R+, it follows thatϕis HA convex in its single variable onR+. So, from

Theorem ,ψ(x) is Schur harmonically convex onRn₊. Remark  From Theorem  and (), it follows that

≤i<···<ik≤n

xi· · ·xik xi+· · ·+xik

≥Hk–Cnk

k , ()

wherexi> ,i= , . . . ,n.

Remark  It needs further discussion thatψ(x) is Schur geometrically convex onRn

+.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors co-authored this paper together. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Electronic Information, Teacher’s College, Beijing Union University, Beijing, 100011, P.R. China.}2_Basic

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Acknowledgements

The work was supported by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407) and the National Natural Science Foundation of China (Grant No. 11101034).

Received: 22 July 2013 Accepted: 18 October 2013 Published:11 Nov 2013 References

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