On the James type constant of \(l {p} l {1}\)

R E S E A R C H

Open Access

On the James type constant of

l

_p

–

l

_

Changsen Yang

_{and Haiying Li}

*_{Correspondence:}

[email protected] College of Mathematics and Information Science, 46 East of Construction Road, Xinxiang, Henan 453007, P.R. China

Abstract

For any

τ

τ

thelp–l1space is investigated. As an application, the exact value of the von

Neuman-Jordan type constant of thelp–l1space can also be obtained.

MSC: Primary 46B20; secondary 47H10

Keywords: James type constant;lp–l1space; von Neuman-Jordan type constant

1 Introduction and preliminaries

Throughout this paper, we shall assume thatXstands for a nontrivial Banach space,i.e.,

dimX≥. We will useSXandBXto denote the unit sphere and unit ball ofX, respectively.

A Banach spaceXis called uniformly non-square in the sense of James if there exists a positive numberδ<  such thatx+y_≤δorx–y_≤δ, wheneverx,y∈SX. The non-square

or James constant is defined by

J(X) =supminx+y,x–y,x,y∈SX

.

Obviously,Xis uniformly non-square in the sense of James if and only ifJ(X) <  (see []). The von Neumann-Jordan constant, introduced by Clarkson in [], is defined as follows:

CNJ(X) =sup

x+y+x–y

(x_+y₎ :x∈SX,y∈BX

.

It is well known that the von Neumann-Jordan constant is not larger than the James constant. This resultCNJ(X)≤J(X) was obtained by Takahashi-Kato in [], Wang in []

and Yang-Li in [] almost at the same time.

Recently, as a generalization of the James constant and the von Neumann-Jordan con-stant, Takahashi in [] introduced the James type constantJX,t(τ) and the von

Neumann-Jordan type constantCt(X), respectively, as follows: JX,t(τ) =sup

μt

x+τy,x–τy:x,y∈SX

,

whereτ ≥, –∞ ≤t< +∞. Here, we denoteμt(a,b) = (a

t_+bt

 ) 

t (t= ) and_μ(a,b) =

limt→μt(a,b) = √

abfor two positive numbersaandb. It is well known thatμt(a,b) is

nondecreasing andμ–∞(a,b) =limt→–∞μt(a,b) =min(a,b). Therefore,J(X) =JX,–∞(),

Ct(X) =sup

JX,t(τ)

 +τ : ≤τ≤

.

It is obvious thatC(X) =CNJ(X) and the James type constants include some known

constants such as Alonso-Llorens-Fuster’s constantT(X) in [], Baronti-Casini-Papini’s constantA(X) in [], Gao’s constantE(X) in [] and Yang-Wang’s modulusγX(t) in [].

These constants are defined byT(X) =JX,(),A(X) =JX,(),E(X) = JX, () andγX(t) = J

X,(t).

Now let us list some known results of the constantJX,t(τ); for more details, see [, –

].

() If–∞ ≤t≤t<∞, thenJX,t(τ)≤JX,t(τ)for anyτ≥.

() Lett≥,τ≥andX=l–l, then

JX,t(τ) =

( +τ)t + ( +_τ)t





t

. (.)

() LetXbe anl∞–lspace. If≤τ≤, then

JX,t(τ) =

(+(+_τ)t)t_{, t}≥_,

 +τ

, t≤.

() Let≤t≤p≤ ∞,≤pand≤τ≤. Then

JX,t(τ) =  + – 

p_τ,

whereXis anl∞–lpspace.

() Lett≥t≥and≤τ≤. Then, for any Banach spaceX,

Jt X,t(τ)≤J

t X,t(τ)≤

( +τ)t₊{_Jt

X,t(τ) – ( +τ) t}

t

t_

 . (.)

() JX,t(τ) =  +τ if and only ifJX,t(τ) =  +τ.

Forp≥, thelp–lspace is defined byX= Rwith the norm

x=(x,x)=

xp, xx≥, x, xx≤.

For anyτ≥ andp≥, we have calculated the exact value of the James type constant

Jlp–l,t(τ) fort≥. As an application, we also give the exact value of the von

Neumann-Jordan type constantCt(lp–l) for ≤t≤. In [], for  <p≤, it is known thatCNJ(lp– l) =  + 



p–_{was given. In this paper, forp}≥_{, (p– )}p–≤_{ andp> , (p– )}p–≥_,

the exact value of the von Neumann-Jordan constantCNJ(lp–l) is obtained.

2 Main results and their proofs

To give the value ofJX,t(τ) forX=lp–l, we need the following lemmas.

Lemma . Let x,x,y,y≥and p≥such that

If≤τ≤, ≤τy≤xand≤x≤τy,then

(x+τy)p+ (x+τy)p



p_+x

–τy+τy–x≤ +τ+

 +τp 

p_.

Proof It is readily seen that ≤x–τy+τy–x≤ +τ. Let us now consider two possible

cases.

Case . ≤x–τy+τy–x≤( +τp)/p. Hence

(x+τy)p+ (x+τy)p



p_+x

–τy+τy–x ≤xp_+xp_/p+τpyp_+τpyp_/p+ +τp



p

=  +τ+ +τp 

p_.

Case . ( +τp)/p_≤x

–τy+τy–x≤ +τ. By Minkowski’s inequality,

(x+τy)p+ (x+τy)p

/p

+x–τy+τy–x ≤xp_+τpy_p/p+τpyp_+xp_/p+x–τy+τy–x ≤xp_+τpyp_/p+τy+x+x–τy+τy–x ≤( +τ) + +τp/p,

where the second inequality follows from the fact · p≤ · . Consequently, the proof

is complete.

Lemma . Letτ∈(, ),t∈[, ]and p≥.Then

(a) τp+p–  –pτ≥;

(b)  –τp–– (p– )(τp––τp)≥; (c) the function

f(τ) = τ–τ

p–

( –τ)( +τ)t–

 +τp

t p–

is nondecreasing;moreover,≤f(τ)≤(p– )pt–t_.

Proof (a) Lettingh(τ) = τp+ (p– ) –pτ, we haveh(τ) = p(τp––τ)≤, andh(τ)≥

h() = .

(b) Lettingg(τ) =  –τp–– (p– )(τp––τp), we have

g(τ) = –(p– )τp–τp+p–  –pτ. Hence,g(τ)≤ by (a) andg(τ)≥g() = .

(c) By a basic calculation, then by use of (b), we have

f(τ) =  [( –τ)( +τ)t–_]

( –τ)( +τ)t– – (p– )τp– +τp

t p–

+τ–τp–(t–p)τp– +τp

–τ–τp– +τp

t

p–_{–( +τ)}t–_{+ ( –τ)(t– )( +τ)}t–

=( +τ

p₎pt–_{( +τ)}t–

[( –τ)( +τ)t–_]

( +τ) +τp – (p– )τp––τ+ (p– )τp–

+τ–τp–+ ( –τ)τ–τp–(t–p)( +τ)τp–– +τp(t– )

=( +τ

p₎pt–_{( +τ)}t–

[( –τ)( +τ)t–_]

 +τ –τp–– (p– )τp– –τ

+ ( –t)( –τ)τ–τp– –τp–≥.

Now fromlimτ→–f(τ) = (p– )

t

p–t_{, we have }≤_f(τ)≤_{(p– )}pt–t_.

Theorem . Let t≥,p≥,τ≥and X=lp–lspace.Then

JX,t(τ) =

( +τp)pt _{+ ( +τ)}t





t

. (.)

Proof AsJX,t(τ) =τJX,t(τ) is valid for anyτ > , we only consider the case ≤τ≤. We claim that the following inequality is valid for anyx,y∈Slp–l:

x+τy+x–τy ≤ +τp 

p_{+  +τ. (.)}

In fact, by the convexity of norm, we only need to show that this inequality is valid for any x,y∈ext(Slp–l), whereext(Slp–l) denotes the set of extreme points ofSlp–l. From

ext(Slp–l) ={(x,x) :x p +x

p

= ,xx≥}, we may assume thatx= (a,b),y= (c,d), where a,b,c,d≥ withap+bp=cp+dp= .

(I) If (a–cτ)(b–dτ)≥,

x+τy+x–τy=x+τyp+x–τyp ≤ +τ+|a–cτ|p+|b–dτ|p



p

≤ +τ+maxap+bpp_,(cτ)p_{+ (dτ)}pp

≤ +τ ≤ +τp



p_{+  +τ.}

(II) If (a–cτ)(b–dτ)≤.

We may assume thata–cτ>  andb–dτ≤. Then, by use of Lemma ., we also have

x+τy+x–τy=x+τyp+x–τy≤

 +τp 

p_{+  +τ.}

Thus (.) is valid.

Now, by takingx= (, ) andy= (, ), we have Jlp–l,(τ) = ( +τ

p₎p_{+  +τ. Therefore}

by (.) we have

J_X,tt (τ)≤( +τ)

t_{+ [J}

X,(τ) – ( +τ)]t

 =

( +τ)t+ ( +τp)pt

On the other hand, by takingx= (, ),y= (, ), we have

x+τy= +τp 

p_{, x–τy=  +τ,}

so

J_X,tt (τ)≥( +τ)

t_{+ ( +τ}p₎tp

 .

Therefore, (.) is valid fort≥.

Theorem . Let p= ,t≥or p> ,t∈[, ],and X be an lp–lspace. For p and t such that(p– )pt–t≤_,then

Ct(X) =

pt–t _{+ }t





t

. (.)

For p and t such that(p– )pt–t_{> ,then}

Ct(X) =

  +τ_

( +τ)t+ ( +τp)

t p





t

,

whereτis the unique solution of the equation

(τ–τp–_{)( +τ}p₎pt–

( –τ)( +τ)t– = . (.) Proof By (.), we have

Ct(X) =

suph(τ) : ≤τ ≤



t_{, whereh(τ) =}( +τ)

t_{+ ( +τ}p₎pt

( +τ₎t

.

A simple computation yields

h(τ) = t( –τ)( +τ)

t–

( +τ₎t+

 –(τ–τ

p–_{)( +τ}p₎pt–

( –τ)( +τ)t–

.

Ifp= ,t≥ orp> ,t∈[, ] such that (p– )pt–t≤_{, Lemma . impliesh(τ)}≥_{, so}

thathis nondecreasing. Hence

Ct(X) =h() 

t ₌

pt–t _{+ }t





t

.

Otherwise, letτ∈(, ) be the unique solution to equation (.). It then follows from

Lemma . that h(τ)≥ forτ ∈[,τ] andh(τ)≤ forτ ∈[τ, ]. In other words,h

attains its maximum atτ. Hence

Ct(X) =

  +τ_

( +τ)t+ ( +τp)

t p





t

For  <p≤,CNJ(lp–l) =  +  

p– _{(see []). Now, by takingt=  in Theorem .,}

as a generalization, we can obtain the following corollary on the von Neumann-Jordan constant oflp–lspace.

Corollary . Let X be the lp–lspace.

(a) Ifp≥and(p– )p–≤_,thenC

NJ(X) =  +  

p–_.

(b) Ifp> and(p– )p–≥_,then

CNJ(X) =

 +

 –τ_p

(τ–τp–)

,

whereτ∈(, )is the unique solution to the equation

(τ–τp–)( +τp)p–

 –τ = .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper, and read and approved the final manuscript.

Acknowledgements

The research was supported by the National Natural Science Foundation of China (Nos. 11271112; 11201127) and IRTSTHN (14IRTSTHN023).

Received: 8 October 2014 Accepted: 13 February 2015 References

1. Gao, J, Lau, KS: On two classes of Banach spaces with uniform normal structure. Stud. Math.99(1), 41-56 (1991) 2. Clarkson, JA: The von Neumann-Jordan constant for the Lebesgue space. Ann. Math.38, 114-115 (1937) 3. Takahashi, Y, Kato, M: A simple inequality for the von Neumann-Jordan and James constants of a Banach space.

J. Math. Anal. Appl.359(2), 602-609 (2009)

4. Wang, F: On the James and von Neumann-Jordan constants in Banach spaces. Proc. Am. Math. Soc.138(2), 695-701 (2010)

5. Yang, C, Li, H: An inequality between Jordan-von Neumann constant and James constant. Appl. Math. Lett.23(3), 277-281 (2010)

6. Takahashi, Y: Some geometric constants of Banach spaces - a unified approach. In: Banach and Function Spaces II, pp. 191-220. Yokohama Publishers, Yokohama (2008)

7. Alonso, J, Llorens-Fuster, E: Geometric mean and triangles inscribed in a semicircle in Banach spaces. J. Math. Anal. Appl.340(2), 1271-1283 (2008)

8. Baronti, M, Casini, E, Papini, PL: Triangles inscribed in semicircle, in Minkowski planes, and in normed spaces. J. Math. Anal. Appl.252(1), 124-146 (2000)

9. Gao, J: A Pythagorean approach in Banach spaces. J. Inequal. Appl.2006, Article ID 94982 (2006)

10. Yang, C, Wang, F: On a new geometric constant related to the von Neumann-Jordan constant. J. Math. Anal. Appl. 324(1), 555-565 (2006)

11. Dhompongsa, S, Piraisangjun, P, Saejung, S: Generalised Jordan-von Neumann constants and uniform normal structure. Bull. Aust. Math. Soc.67(2), 225-240 (2003)

12. Yang, C: An inequality between the James type constant and the modulus of smoothness. J. Math. Anal. Appl.398(2), 622-629 (2013)