Relations between generalized von Neumann-Jordan ...

Kwun et al. Journal of Inequalities and Applications (2016) 2016:171 DOI 10.1186/s13660-016-1115-z

RESEARCH

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Relations between generalized von Neumann-Jordan and James constants for quasi-Banach spaces Young Chel Kwun1 , Qaisar Mehmood2 , Waqas Nazeer3 , Absar Ul Haq4 and Shin Min Kang5* *

Correspondence: [email protected] 5 Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract Let CNJ (B) and J(B) be the generalized von Neumann-Jordan and James constants of a quasi-Banach space B, respectively. In this paper we shall show the relation between CNJ (B), J(B), and the modulus of convexity. Also, we show that if B is not uniform non-square then J(B) = CNJ (B) = 2. Moreover, we give an equivalent formula for the generalized von Neumann-Jordan constant. MSC: Primary 46B20; secondary 46E30 Keywords: the generalized von Neumann-Jordan constant; James constant; uniform non-squareness; quasi-normed spaces

1 Introduction Among various geometric constants of a Banach space B , the von Neumann-Jordan constant CNJ (B ), and the James constant J(B ) have been treated most widely. In connection with the famous work [] (see also []) of Jordan and von Neumann concerning inner products, the von Neumann-Jordan constant CNJ (B ) for a Banach space B was introduced by Clarkson [] as the smallest constant C for which the estimates  x + x  + x – x  ≤C ≤ C (x  + x  ) hold for any x , x ∈ B with (x , x ) = (, ). Equivalently 

 x + x  + x – x  : x , x ∈ B with (x , x ) = (, ) . CNJ (B ) = sup (x  + x  ) The classical von Neumann-Jordan constant CNJ (B ) was investigated in [–]. A Banach space B is said to be uniformly non-square in the sense of James if there exists a positive number δ <  such that for any x , x ∈ SB = {x ∈ B : x = } we have   min x + x , x – x  ≤ δ. © 2016 Kwun et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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The James constant J(B ) of a Banach space B is defined by     J(B ) = sup min x + x , x – x  : x , x ∈ SB . It is obvious that B is uniformly non-square if and only if J(B ) < . (p) In [], the authors introduced the generalized von Neumann-Jordan constant CNJ (B ), which is defined as  p CNJ (B ) = sup

 x + x p + x – x p : x , x ∈ B with (x , x ) = (, ) , p– (x p + x p ) (p)

and obtained the relationship between CNJ (B ) and J(B ). Furthermore, they used the con(p) stant CNJ (B ) to establish some new equivalent conditions for the uniform non-squareness of a Banach space B . Both the von Neumann-Jordan CNJ (B ) and the James constants J(B ) play an important role in the description of various geometric structures. It is therefore worthwhile to clarify the relation between them. In this paper, we shall show the relation between the generalized von Neumann-Jordan constant CNJ (B ) and the James constant J(B ) and we also show that if B is not uniform non-square then J(B ) = CNJ (B ) = . In the second section, we present basic definitions and define the modulus of convexity of a quasi-Banach space. In the third section, we establish a relationship between the generalized von Neumann-Jordan constant and the modulus of convexity, the James constant and the modulus of convexity, the generalized von Neumann-Jordan constant and the James constant, and we give the equivalent formula of the generalized von Neumann-Jordan constant.

2 Preliminaries Definition . [] A quasi-norm on  ·  on vector space B over a field K (R or C) is a map B −→ [, ∞) with the properties: • x =  ⇐⇒ x = . • αx = |α|x if ∀α ∈ K , ∀x ∈ B . • There is a constant C ≥  such that if ∀x , x ∈ B we have   x + x  ≤ C x  + x  . (p)

Definition . The generalized von Neumann-Jordan constant CNJ (B ) for quasi-Banach spaces is defined by  (p)

CNJ (B ) = sup

 x + x p + x – x p : x , x ∈ B with (x , x ) =  (, ) ,     p– C p (x p + x p )

where  ≤ p < ∞. (p)

We will also use the following parametrized formula for the constant CNJ (B ):  (p)

CNJ (B ) = sup

 x + tx p + x – tx p : x , x ∈ S ,  ≤ t ≤  ,   B C p p– ( + t P )

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where SB is the unit sphere. By taking t =  and x = x , we obtain the estimate (p)

CNJ (B ) ≥

p  x p ≥ = . p p– C ( + ) C ( + ) C

Definition . In a quasi-Banach space B the James constant is defined as 

    min x + x , x – x  : x , x ∈ SB . J(B ) = sup C

(.)

Definition . A quasi-Banach space B is said to be uniformly non-square if there exists a positive number δ <  such that for any x , x ∈ SB , we have

 x + x x + x , min C C ≤ δ. Remark . It is obvious that B is uniformly non-square if and only if J(B ) < . Definition . The modulus of uniform smoothness of the quasi-Banach space B is defined as   x + tx  + x – tx   – : x  , x  ∈ SB , t ≥  . ρB (t) = sup C C It is clear that ρB (t) is a convex function on the interval [, ∞) satisfying ρB () = , whence it follows that ρB (t) is nondecreasing on [, ∞). Definition . A quasi-Banach space B is said to be uniformly smooth if (ρB ) + () = limt→+ ρBt(t) = . Definition . Given any quasi-Banach space B and a number p ∈ [, ∞), another function JB,p (t) is defined by

   x + tx p + x – tx p p : x  , x  ∈ SB . JB,p (t) = sup C p Definition . The modulus of convexity of a quasi-Banach space B is defined as   x + x x – x : ≥ ; ∀x , x ∈ SB , δ() = inf  – C C

 ≤  ≤ .

3 Main results Lemma . For any number  ≤ a ≤ ,  ≤ b ≤  we have 

Proof 

a+b  – C C

a+b  – C C

 +

 a  + b ≥ .  C C 

+

 a + b + ab  (a + b)  = + – +  C C  C C C



=

a  + b ab  (a + b)  + + – +     C C C C C

(.)

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 a  + b  ab +  – (a + b) = +  C  C  ≥

a  + b . C 



Lemma . Let a be a real number and let b > , then at  + bt a + ≤  + t

√

a  + b , 

∀t ≥ .

The first theorem is a relation between CNJ (B ) and the modulus of uniform smoothness. Theorem . Let B be a quasi-Banach space, then

CNJ (B ) ≤  + CρB ()

     – CρB () +  –  – CρB () .

(.)

Proof We know that   x + tx  + x – tx  CNJ (B ) = sup : ∀x , x ∈ SB with (x , x ) = (, ),  ≤ t ≤  . C  ( + t  ) By using Lemma . 

x + x  + x – x   – C C    x + x  + x – x  ≤ C  ρB () +  . C x + x  + x – x  ≤ C 

 +

  , C

(.)

Also we have    x + x  + x – x  ≤ C ρB () + . C Since   x + tx  = t(x + x ) + ( – t)x ≤ C tx + x  + ( – t) ,   x – tx  = t(x – x ) + ( – t)x ≤ C tx – x  + ( – t) , we have

     x + tx  + x – tx  ≤ C  tx + x  + ( – t) + tx – x  + ( – t)

  = C  x + x  + x – x  t     + x + x  + x – x  t( – t) + ( – t)         ≤ C C ρB () +  t C

 

 + ρB () t( – t) + ( – t) +  C C

    = C  Ct  ρB () CρB () –  + CtρB () +  + t  ,

(.)

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x + tx  + x – tx  Ct  ρB ()(CρB () – ) + CtρB () + ( + t  ) ≤ C  ( + t  )  + t   ≤ CρB () CρB () –     + C  ρB () CρB () –  + C  ρB () +  (by using Lemma .)     =  + CρB () CρB () –  +  –  – CρB () .



The next result is the relation between the James constant and the modulus of convexity. Theorem . Let B be a quasi-Banach space then   ε . J(B ) = sup ε : δ(ε) ≤  –  Proof Let   ε . α = sup ε : δ(ε) ≤  –  We shall show that J(B ) ≤ α. For this purpose, if α = , then there is nothing to prove. So, we may assume that α < . For any β > α, we have for any x , x ∈ SB and x C–x  ≥ β δ(β) >  –

β . 

From the definition of δ, we have –

x + x  β >– , C 

which implies that x + x  < β; C therefore 

x + x  x – x  min , ≤ β. C C As J(B ) ≤ β and since β was arbitrary we have J(B ) ≤ α. For the reverse we use the definition of δ, so ∀γ >  there exist x , x ∈ SB such that x – x  ≥ε C and –

x + x  ≤ δ(ε) + γ , C

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where we have ε = α – γ , so x + x  >  – δ(ε) – γ ; C therefore 

x + x  x – x  J(B ) ≥ min , C C     ≥ min   – δ(ε) – γ , ε ≥ min(ε – γ , ε) = ε – γ = α – γ , where γ was arbitrary, so we have J(B ) ≥ α.



Corollary . For any quasi-Banach space B , we have J(B ) ≥

√

.

Corollary . Let B be any quasi-Banach space and J(B ) ≤ , then   J(B ) δ J(B ) =  – .  Now, we are going to give an equivalent formula of the generalized von Neumann-Jordan constant. Theorem . We have   x + x p + x – x p : x CNJ (B ) = sup p– p , x ∈ B with x  = , x  ≤  ,      C (x p + x p ) where  ≤ p < ∞. Proof If  = x  ≥ x  p x x + x + x p = x p x  x  ,   p x p p x , – x – x  = x  x  x  p p  x x x p p p x , + x + x  + x – x  = x  + – x  x  x  x  p

p

x x x x x + x p + x – x p  x  + x   +  x  – x   = . p– C p (x p + x p ) p– C p ( + ( xx  )p )

(.)

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This shows that   x + x p + x – x p CNJ (B ) = sup p– p : x , x ∈ B with x  = , x  ≤  .  C (x p + x p )



The next theorems show the relation between the generalized von Neumann-Jordan and James constants. Theorem . For any quasi-Banach space B , we have J  (B )  J(B ) ≤ CNJ (B ) ≤ .  (J(B ) – ) + 

(.)

Proof For any x , x ∈ SB , we have 

   x + x  + x – x    min x + x , x – x  ≤   

 x + x  + x – x  ≤    x + x  + x – x  · C  x  + x     C (x  + x  )   ≤ C  CNJ (B ) x  + x  =

= C  CNJ (B ), 

  min x + x , x – x  ≤ C  CNJ (B ),

    min x + x , x – x ) ≤  CNJ (B ). C Hence

     min x + x , x – x  ≤  CNJ (B ). sup C Therefore   J (B ) ≤ CNJ (B ).  To prove the right hand side we use Theorem ., so we only take x  =  and x  ≤ . Case : If x  = t ≥ J(B ) – , then 



    C x + x  + x – x  ≤ C x  + x  + min x + x , x – x  C  

       min x + x , x – x  = C x  + x  + C

   x + x  + x – x  ≤ C  x  + x  + J  (B ) , 



x + x  + x – x  (x  + x ) + J  (B ) ≤ C  (x  + x  ) (x  + x  ) =

( + t) + J  (B ) . ( + t  )

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The function f (t) =

( + t) + J  (B ) ( + t  )

is increasing on (, μ) and decreasing on (μ, ) where μ=

–J  (B ) +



J  (B ) + 



.

Since J(B ) –  ≥ μ and J(B ) –  ≤ t, we have f (t) ≤ f (J(B ) – ) x + x  + x – x  J  (B ) ≤ ,    C (x  + x  )  + (J(B ) – ) taking the supremum, we get

CNJ (B ) ≤

J  (B ) .  + (J(B ) – )

(.)

Case : If x  = t ≤ J(B ) – , then   x + x  + x – x  ≤ C  x  + x  = C  ( + t) , x + x  + x – x  ( + t) ≤ . C  (x  + x  ) ( + t  ) Let g(t) =

( + t) , ( + t  )

since g is increasing on (, ]   g(t) ≤ g J(B ) –  , hence

CNJ (B ) ≤

J  (B ) .  + (J(B ) – )

(.) 

Corollary . If B is not uniformly non-square then J(B ) = CNJ (B ) = . Theorem . For any quasi-Banach space B , we have 

CJ(B ) CNJ (B ) ≤  + 

 .

(.)

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Proof Since CNJ (B ) = sup{CNJ (t, B ) : t ∈ [, ]} where   x + tx  + x – tx  CNJ (t, B ) = sup , x ∈ S : ∀x   B . C  ( + t  ) First we prove that

CNJ (t, B ) ≤  +

C  t  J  (B ) + Ct( – t)J(B ) . ( + t  )

(.)

For this purpose

    x + tx  + x – tx  ≤ C( + t) + min x + tx , x – tx  

≤ C  ( + t) + J  (t, B ) , x + tx  + x – tx  ( + t) + J  (t, B ) ≤ . C  ( + t  ) ( + t  ) Taking the supremum we get

CNJ (t, B ) ≤

( + t) + J  (t, B ) . ( + t  )

(.)

Also note that     min x + tx  + x – tx  ≤ C min tx + x  + ( – t), tx – x  + ( – t) , J(t, B ) ≤ tCJ(B ) + ( – t).

(.)

Using (.) and (.), we get

CNJ (t, B ) ≤

[CtJ(B ) + ( – t)] + ( + t) ( + t  )

= +

C  t  J  (B ) + t( – t)J(B ) . ( + t  )

(.) (.)

Taking the supremum over t, we get 

CNJ (B ) ≤  +

CJ(B ) 

 .



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 Department of Mathematics, Dong-A University, Busan, 604-714, Korea. 2 Department of Mathematics, Science College, Wahdat Road, Lahore, Pakistan. 3 Division of Science and Technology, University of Education, Lahore, Pakistan. 4 Abdus Salam School of Mathmatical Sciences, GC University, Lahore, 54000, Pakistan. 5 Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea.

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