Relations between generalized von Neumann-Jordan and James constants for quasi-Banach spaces

Abstract

Let $\mathcal{C}_{NJ} ( \mathcal{B} )$ and $J ( \mathcal{B} )$ be the generalized von Neumann-Jordan and James constants of a quasi-Banach space $\mathcal{B}$, respectively. In this paper we shall show the relation between $\mathcal {C}_{NJ} ( \mathcal{B} )$, $J ( \mathcal{B} )$, and the modulus of convexity. Also, we show that if $\mathcal{B}$ is not uniform non-square then $J ( \mathcal{B} )=\mathcal{C}_{NJ} ( \mathcal{B} )=2$. Moreover, we give an equivalent formula for the generalized von Neumann-Jordan constant.

1 Introduction

Among various geometric constants of a Banach space $\mathcal{B}$, the von Neumann-Jordan constant $C_{NJ} (\mathcal{B})$, and the James constant $J(\mathcal{B})$ have been treated most widely. In connection with the famous work [1] (see also [2]) of Jordan and von Neumann concerning inner products, the von Neumann-Jordan constant $C_{NJ} (\mathcal{B})$ for a Banach space $\mathcal{B}$ was introduced by Clarkson [3] as the smallest constant C for which the estimates

$$\frac{1}{C}\leq\frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2({\Vert x_{1}\Vert }^{2}+{\Vert x_{2}\Vert }^{2})}\leq C $$

hold for any $x_{1},x_{2}\in\mathcal{B}$ with $(x_{1},x_{2})\neq(0,0)$. Equivalently

$$C_{NJ}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2({\Vert x_{1}\Vert }^{2}+{\Vert x_{2}\Vert }^{2})}: x_{1},x_{2}\in\mathcal{B} \text{ with } (x_{1},x_{2})\neq(0,0) \biggr\} . $$

The classical von Neumann-Jordan constant $C_{NJ} (\mathcal{B})$ was investigated in [4–8].

A Banach space $\mathcal{B}$ is said to be uniformly non-square in the sense of James if there exists a positive number $\delta<2$ such that for any $x_{1},x_{2}\in S_{\mathcal {B}}=\{x\in\mathcal{B}:\Vert x\Vert =1\}$ we have

$$\min\bigl(\Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2}\Vert \bigr)\leq\delta. $$

The James constant $J(\mathcal{B})$ of a Banach space $\mathcal{B}$ is defined by

$$J(\mathcal{B})=\sup\bigl\{ \min\bigl(\Vert x_{1}+x_{2} \Vert ,\Vert x_{1}-x_{2} \Vert \bigr): x_{1},x_{2}\in S_{\mathcal{B}}\bigr\} . $$

It is obvious that $\mathcal{B}$ is uniformly non-square if and only if $J(\mathcal{B})<2$.

In [9], the authors introduced the generalized von Neumann-Jordan constant $C_{NJ}^{(p)}(\mathcal{B})$, which is defined as

$$C_{NJ}^{p}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+x_{2} \Vert }^{p}+{\Vert x_{1}-x_{2}\Vert }^{p}}{2^{p-1}({\Vert x_{1}\Vert }^{p}+{\Vert x_{2}\Vert }^{p})}: x_{1},x_{2}\in\mathcal{B} \text{ with } (x_{1},x_{2})\neq(0,0) \biggr\} , $$

and obtained the relationship between $C_{NJ}^{(p)}(\mathcal{B})$ and $J(\mathcal{B})$. Furthermore, they used the constant $C_{NJ}^{(p)}(\mathcal{B})$ to establish some new equivalent conditions for the uniform non-squareness of a Banach space $\mathcal{B}$. Both the von Neumann-Jordan $\mathcal{C}_{NJ} ( \mathcal{B} )$ and the James constants $J ( \mathcal{B} )$ 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 $\mathcal{C}_{NJ} ( \mathcal{B} )$ and the James constant $J ( \mathcal{B} )$ and we also show that if $\mathcal {B}$ is not uniform non-square then $J ( \mathcal{B} )=\mathcal{C}_{NJ} ( \mathcal{B} )=2$. 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 2.1

[10]

A quasi-norm on $\Vert \cdot \Vert$ on vector space $\mathcal{B}$ over a field K ($\mathbb{R}$ or $\mathbb{C}$) is a map $\mathcal {B}\longrightarrow[0,\infty)$ with the properties:

• $\Vert x\Vert =0 \Longleftrightarrow x=0$.

• $\Vert \alpha x\Vert =\vert \alpha \vert \Vert x\Vert$ if $\forall\alpha\in K$, $\forall x\in\mathcal{B}$.

• There is a constant $C\geq1$ such that if $\forall x_{1},x_{2}\in\mathcal{B}$ we have

  $$\Vert x_{1}+x_{2}\Vert \leq C\bigl(\Vert x_{1}\Vert +\Vert x_{2}\Vert \bigr). $$

Definition 2.2

The generalized von Neumann-Jordan constant $\mathcal {C}_{NJ}^{(p)}(\mathcal{B})$ for quasi-Banach spaces is defined by

$$\mathcal{C}_{NJ}^{(p)}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+x_{2} \Vert }^{p}+{\Vert x_{1}-x_{2}\Vert }^{p}}{2^{p-1}C^{p}({\Vert x_{1}\Vert }^{p}+{\Vert x_{2}\Vert }^{p})}: x_{1},x_{2}\in\mathcal{B} \text{ with } (x_{1},x_{2})\neq (0,0) \biggr\} , $$

where $1\leq p< \infty$.

We will also use the following parametrized formula for the constant $\mathcal{C}_{NJ}^{(p)}(\mathcal{B})$:

$$\mathcal{C}_{NJ}^{(p)}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+tx_{2} \Vert }^{p}+{\Vert x_{1}-tx_{2}\Vert }^{p}}{C^{p}2^{p-1}(1+t^{P})}: x_{1},x_{2}\in S_{\mathcal{B}}, 0\leq t \leq1 \biggr\} , $$

where $S_{\mathcal{B}}$ is the unit sphere. By taking $t = 1$ and $x_{1} = x_{2}$, we obtain the estimate

$$\mathcal{C}_{NJ}^{(p)}(\mathcal{B})\geq\frac{{\Vert 2x_{1}\Vert }^{p}}{C2^{p}(1+1)}\geq \frac{2^{p}}{C2^{p-1}(1+1)}=\frac{1}{C}. $$

Definition 2.3

In a quasi-Banach space $\mathcal{B}$ the James constant is defined as

$$ J(\mathcal{B})=\sup \biggl\{ \frac{1}{C}\min\bigl(\Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr): x_{1},x_{2}\in S_{\mathcal{B}} \biggr\} . $$

(2.1)

Definition 2.4

A quasi-Banach space $\mathcal{B}$ is said to be uniformly non-square if there exists a positive number $\delta<2$ such that for any $x_{1},x_{2}\in S_{\mathcal{B}}$, we have

$$\min \biggl(\biggl\Vert \frac{x_{1}+x_{2}}{C}\biggr\Vert ,\biggl\Vert \frac {x_{1}+x_{2}}{C}\biggr\Vert \biggr)\leq\delta. $$

Remark 2.5

It is obvious that $\mathcal{B}$ is uniformly non-square if and only if $J(\mathcal{B})<2$.

Definition 2.6

The modulus of uniform smoothness of the quasi-Banach space $\mathcal {B}$ is defined as

$$\rho_{\mathcal{B}}(t)=\sup \biggl\{ \frac{{\Vert x_{1}+tx_{2} \Vert }+{\Vert x_{1}-tx_{2}\Vert }}{2C}-\frac{1}{C}: x_{1},x_{2}\in S_{\mathcal{B}}, t\geq0 \biggr\} . $$

It is clear that $\rho_{\mathcal{B}}(t)$ is a convex function on the interval $[0, \infty)$ satisfying $\rho_{\mathcal{B}}(0)=0$, whence it follows that $\rho_{\mathcal{B}}(t)$ is nondecreasing on $[0,\infty )$.

Definition 2.7

A quasi-Banach space $\mathcal{B}$ is said to be uniformly smooth if $(\rho_{\mathcal{B}})'_{+}(0)=\lim_{t\rightarrow0^{+}}\frac{\rho _{\mathcal{B}}(t)}{t}=0$.

Definition 2.8

Given any quasi-Banach space $\mathcal{B}$ and a number $p\in[0,\infty )$, another function $J_{\mathcal{B},p}(t)$ is defined by

$$J_{\mathcal{B},p}(t)=\sup \biggl\{ \biggl(\frac{{\Vert x_{1}+tx_{2} \Vert }^{p}+{\Vert x_{1}-tx_{2}\Vert }^{p}}{2C^{p}} \biggr)^{\frac {1}{p}}: x_{1},x_{2}\in S_{\mathcal{B}} \biggr\} . $$

Definition 2.9

The modulus of convexity of a quasi-Banach space $\mathcal{B}$ is defined as

$$\delta(\epsilon)=\inf \biggl\{ 1-\biggl\Vert \frac{x_{1}+x_{2}}{2C}\biggr\Vert : \biggl\Vert \frac{x_{1}-x_{2}}{C}\biggr\Vert \geq\epsilon ; \forall x_{1},x_{2}\in S_{\mathcal{B}} \biggr\} ,\quad 0\leq\epsilon \leq2. $$

3 Main results

Lemma 3.1

For any number $0\leq a\leq2$, $0\leq b\leq2$ we have

$$ \biggl(\frac{a+b}{2C}-\frac{1}{C} \biggr)^{2}+ \frac{1}{C^{2}}\geq\frac{a^{2}+b^{2}}{4C^{2}} . $$

(3.1)

Proof

$$\begin{aligned} \biggl(\frac{a+b}{2C}-\frac{1}{C} \biggr)^{2}+ \frac{1}{C^{2}} =& \frac {a^{2}+b^{2}+2ab}{4C^{2}}+\frac{1}{C^{2}}-\frac{(a+b)}{C^{2}}+ \frac{1}{C^{2}} \\ =& \frac{a^{2}+b^{2}}{4C^{2}}+\frac{ab}{2C^{2}}+\frac{1}{C^{2}}-\frac {(a+b)}{C^{2}}+ \frac{1}{C^{2}} \\ =& \frac{a^{2}+b^{2}}{4C^{2}}+\frac{1}{C^{2}} \biggl[\frac{ab}{2}+2-(a+b) \biggr] \\ \geq& \frac{a^{2}+b^{2}}{4C^{2}}. \end{aligned}$$

 □

Lemma 3.2

Let a be a real number and let $b>0$, then

$$\frac{at^{2}+bt}{1+t^{2}}\leq\frac{a+\sqrt{a^{2}+b^{2}}}{2}, \quad \forall t\geq0. $$

The first theorem is a relation between $\mathcal{C}_{NJ}(\mathcal{B})$ and the modulus of uniform smoothness.

Theorem 3.3

Let $\mathcal{B}$ be a quasi-Banach space, then

$$ \mathcal{C}_{NJ}(\mathcal{B})\leq1+C\rho_{\mathcal{B}}(1) \bigl[ \sqrt{\bigl\{ 1-C\rho_{\mathcal{B}}(1)\bigr\} ^{2}+1}-\bigl\{ 1-C \rho_{\mathcal{B}}(1)\bigr\} \bigr] . $$

(3.2)

Proof

We know that

$$ \mathcal{C}_{NJ}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})}: \forall x_{1},x_{2}\in S_{\mathcal{B}} \text{ with } (x_{1},x_{2})\neq (0,0), 0\leq t\leq1 \biggr\} . $$

By using Lemma 3.1

$$ \begin{aligned} &\frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{4C^{2}}\leq \biggl\{ \biggl(\frac{{\Vert x_{1}+x_{2} \Vert }+{\Vert x_{1}-x_{2}\Vert }}{2C}-\frac{1}{C} \biggr)^{2}+\frac {1}{C^{2}} \biggr\} , \\ &{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}\leq 4C^{2} \biggl\{ \rho_{\mathcal{B}}^{2}(1)+\frac{1}{C^{2}} \biggr\} . \end{aligned} $$

(3.3)

Also we have

$$ {\Vert x_{1}+x_{2} \Vert }+{\Vert x_{1}-x_{2}\Vert }\leq2C \biggl\{ \rho_{\mathcal{B}}(1)+ \frac{1}{C} \biggr\} . $$

(3.4)

Since

$$\begin{aligned}& {\Vert x_{1}+tx_{2} \Vert }={\bigl\Vert t(x_{1}+x_{2})+(1-t)x_{1} \bigr\Vert }\leq C \bigl(t{\Vert x_{1}+x_{2} \Vert }+(1-t) \bigr), \\& {\Vert x_{1}-tx_{2} \Vert }={\bigl\Vert t(x_{1}-x_{2})+(1-t)x_{1} \bigr\Vert }\leq C \bigl(t{\Vert x_{1}-x_{2} \Vert }+(1-t) \bigr), \end{aligned}$$

we have

$$\begin{aligned}& {\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2} \leq C^{2} \bigl[ \bigl(t{\Vert x_{1}+x_{2} \Vert }+(1-t) \bigr)^{2}+ \bigl(t{\Vert x_{1}-x_{2} \Vert }+(1-t) \bigr)^{2} \bigr] \\& \hphantom{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}} = C^{2}\bigl[ \bigl({\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2} \Vert }^{2} \bigr)t^{2} \\& \hphantom{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}=} {}+ \bigl({\Vert x_{1}+x_{2} \Vert }+{\Vert x_{1}-x_{2} \Vert } \bigr)2t(1-t)+2(1-t)^{2} \bigr] \\& \hphantom{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}} \leq C^{2}\biggl[ 4C^{2} \biggl\{ \rho_{\mathcal{B}}^{2}(1)+ \frac{1}{C^{2}} \biggr\} t^{2} \\& \hphantom{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}=} {}+ 2\biggl(2C\biggl(\frac{1}{C}+\rho_{\mathcal{B}}(1)\biggr)t(1-t)+2(1-t)^{2} \biggr)\biggr] \\& \hphantom{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}= C^{2}\bigl[4Ct^{2}\rho_{\mathcal{B}}(1) \bigl(C \rho_{\mathcal{B}}(1)-1\bigr)+4Ct\rho _{\mathcal{B}}(1)+\bigl(1+t^{2} \bigr)\bigr], \\& \frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})} \leq \frac{2Ct^{2}\rho_{\mathcal{B}}(1)(C\rho _{\mathcal{B}}(1)-1)+2Ct\rho_{\mathcal{B}}(1)+2(1+t^{2})}{1+t^{2}} \\& \hphantom{\frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})}} \leq C\rho_{\mathcal{B}}(1)\bigl\{ C\rho_{\mathcal{B}}(1)-1\bigr\} \\& \hphantom{\frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})}=}{} + \sqrt {C^{2}\rho^{2}_{\mathcal{B}}(1)\bigl\{ C \rho_{\mathcal{B}}(1)-1\bigr\} ^{2}+C^{2} \rho^{2}_{\mathcal{B}}(1)}+1 \\& \hphantom{\frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})}=} (\text{by using Lemma~3.2}) \\& \hphantom{\frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})}} = 1+C\rho_{\mathcal{B}}(1)\sqrt{\bigl\{ C\rho_{\mathcal{B}}(1)-1\bigr\} ^{2}+1}-\bigl\{ 1-C\rho_{\mathcal{B}}(1)\bigr\} . \end{aligned}$$

 □

The next result is the relation between the James constant and the modulus of convexity.

Theorem 3.4

Let $\mathcal{B}$ be a quasi-Banach space then

$$ J(\mathcal{B})=\sup \biggl\{ \varepsilon: \delta(\varepsilon)\leq1- \frac {\varepsilon}{2} \biggr\} . $$

Proof

Let

$$\alpha=\sup \biggl\{ \varepsilon: \delta(\varepsilon)\leq1-\frac {\varepsilon}{2} \biggr\} . $$

We shall show that $J(\mathcal{B})\leq\alpha$. For this purpose, if $\alpha=2$, then there is nothing to prove. So, we may assume that $\alpha< 2$. For any $\beta> \alpha$, we have for any $x_{1},x_{2}\in S_{\mathcal{B}}$ and $\frac{\Vert x_{1}-x_{2}\Vert }{C}\geq\beta$

$$\delta(\beta)>1-\frac{\beta}{2}. $$

From the definition of δ, we have

$$1- \frac{\Vert x_{1}+x_{2}\Vert }{2C}> 1-\frac{\beta}{2}, $$

which implies that

$$\frac{\Vert x_{1}+x_{2}\Vert }{C}< \beta; $$

therefore

$$\min \biggl( \frac{\Vert x_{1}+x_{2}\Vert }{C}, \frac{\Vert x_{1}-x_{2}\Vert }{C} \biggr)\leq\beta. $$

As $J(\mathcal{B})\leq\beta$ and since β was arbitrary we have $J(\mathcal{B})\leq\alpha$.

For the reverse we use the definition of δ, so $\forall \gamma>0$ there exist $x_{1},x_{2}\in S_{\mathcal{B}}$ such that

$$\frac{\Vert x_{1}-x_{2}\Vert }{C}\geq\varepsilon $$

and

$$1-\frac{\Vert x_{1}+x_{2}\Vert }{2C}\leq\delta(\varepsilon)+\gamma, $$

where we have $\varepsilon=\alpha-\gamma$, so

$$\frac{\Vert x_{1}+x_{2}\Vert }{C}> 2-2\delta(\varepsilon)-2\gamma; $$

therefore

$$\begin{aligned} J(\mathcal{B}) \geq& \min \biggl( \frac{\Vert x_{1}+x_{2}\Vert }{C}, \frac{\Vert x_{1}-x_{2}\Vert }{C} \biggr) \\ \geq&\min \bigl( 2\bigl(1-\delta(\varepsilon)-\gamma\bigr),\varepsilon \bigr) \\ \geq&\min(\varepsilon-2\gamma, \varepsilon) \\ =& \varepsilon-2\gamma \\ =&\alpha-3\gamma, \end{aligned}$$

where γ was arbitrary, so we have $J(\mathcal{B})\geq\alpha$. □

Corollary 3.5

For any quasi-Banach space $\mathcal{B}$, we have

$$J(\mathcal{B})\geq\sqrt{2}. $$

Corollary 3.6

Let $\mathcal{B}$ be any quasi-Banach space and $J(\mathcal{B})\leq2$, then

$$\delta\bigl(J(\mathcal{B})\bigr)=1-\frac{J(\mathcal{B})}{2}. $$

Now, we are going to give an equivalent formula of the generalized von Neumann-Jordan constant.

Theorem 3.7

We have

$$ \mathcal{C}_{NJ}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+x_{2} \Vert }^{p}+{\Vert x_{1}-x_{2}\Vert }^{p}}{2^{p-1}C^{p}({\Vert x_{1}\Vert }^{p}+{\Vert x_{2}\Vert }^{p})}: x_{1},x_{2}\in\mathcal{B} \textit{ with } \Vert x_{1}\Vert =1, \Vert x_{2}\Vert \leq1 \biggr\} , $$

(3.5)

where $1\leq p< \infty$.

Proof

If $0\neq \Vert x_{1}\Vert \geq \Vert x_{2}\Vert$

$$\begin{aligned}& {\Vert x_{1}+x_{2}\Vert }^{p}={\Vert x_{1}\Vert }^{p}{\biggl\Vert \frac{x_{1}}{\Vert x_{1}\Vert }+ \frac{x_{2}}{\Vert x_{1}\Vert }\biggr\Vert }^{p}, \\& {\Vert x_{1}-x_{2}\Vert }^{p}={\Vert x_{1}\Vert }^{p}{\biggl\Vert \frac{x_{1}}{\Vert x_{1}\Vert }- \frac{x_{2}}{\Vert x_{1}\Vert }\biggr\Vert }^{p}, \\& {\Vert x_{1}+x_{2}\Vert }^{p}+{\Vert x_{1}-x_{2}\Vert }^{p}={\Vert x_{1} \Vert }^{p} \biggl[{\biggl\Vert \frac{x_{1}}{\Vert x_{1}\Vert }+ \frac{x_{2}}{\Vert x_{1}\Vert }\biggr\Vert }^{p}+{\biggl\Vert \frac{x_{1}}{\Vert x_{1}\Vert }- \frac{x_{2}}{\Vert x_{1}\Vert }\biggr\Vert }^{p} \biggr], \\& \frac{{\Vert x_{1}+x_{2} \Vert }^{p}+{\Vert x_{1}-x_{2}\Vert }^{p}}{2^{p-1}C^{p}({\Vert x_{1}\Vert }^{p}+{\Vert x_{2}\Vert }^{p})}=\frac{{\Vert \frac{x_{1}}{\Vert x_{1}\Vert }+\frac{x_{2}}{\Vert x_{1}\Vert } \Vert }^{p}+{\Vert \frac{x_{1}}{\Vert x_{1}\Vert }-\frac{x_{2}}{\Vert x_{1}\Vert }\Vert }^{p}}{2^{p-1}C^{p}{ (1+ (\frac{x_{2}}{\Vert x_{1}\Vert } )^{p} )}}. \end{aligned}$$

This shows that

$$\mathcal{C}_{NJ}(\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+x_{2} \Vert }^{p}+{\Vert x_{1}-x_{2}\Vert }^{p}}{2^{p-1}C^{p}({\Vert x_{1}\Vert }^{p}+{\Vert x_{2}\Vert }^{p})}: x_{1},x_{2}\in\mathcal{B} \text{ with } \Vert x_{1}\Vert =1, \Vert x_{2}\Vert \leq1 \biggr\} . $$

 □

The next theorems show the relation between the generalized von Neumann-Jordan and James constants.

Theorem 3.8

For any quasi-Banach space $\mathcal{B}$, we have

$$ \frac{1}{2}J(\mathcal{B})^{2}\leq\mathcal{C}_{NJ}( \mathcal{B})\leq\frac {J^{2}(\mathcal{B})}{(J(\mathcal{B})-1)^{2}+1}. $$

(3.6)

Proof

For any $x_{1},x_{2}\in S_{\mathcal{B}}$, we have

$$\begin{aligned}& 2\bigl(\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \bigr)^{2} \leq 2 \biggl(\frac{{\Vert x_{1}+x_{2}\Vert }+{\Vert x_{1}-x_{2}\Vert }}{2} \biggr)^{2} \\& \hphantom{2\bigl(\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \bigr)^{2}} \leq 2 \biggl(\frac{{\Vert x_{1}+x_{2}\Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2} \biggr) \\& \hphantom{2\bigl(\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \bigr)^{2}}= \frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2C^{2}({\Vert x_{1}\Vert }^{2}+{\Vert x_{2}\Vert }^{2})}\cdot {2C^{2}\bigl({\Vert x_{1}\Vert }^{2}+{\Vert x_{2}\Vert }^{2}}\bigr) \\& \hphantom{2\bigl(\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \bigr)^{2}} \leq 2C^{2}\mathcal{C}_{NJ}(\mathcal{B}) \bigl({\Vert x_{1}\Vert }^{2}+{\Vert x_{2}\Vert }^{2}\bigr) \\& \hphantom{2\bigl(\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \bigr)^{2}} = 4C^{2}\mathcal{C}_{NJ}(\mathcal{B}), \\& \bigl(\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \bigr)^{2} \leq 2 C^{2}\mathcal{C}_{NJ}(\mathcal{B}), \\& \frac{1}{C}\min\bigl\{ \Vert x_{1}+x_{2}\Vert , \Vert x_{1}-x_{2} \Vert )\bigr\} \leq 2\sqrt{ \mathcal{C}_{NJ}(\mathcal{B})}. \end{aligned}$$

Hence

$$\sup \biggl(\frac{1}{C}\min\bigl\{ \Vert x_{1}+x_{2} \Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \biggr)\leq2 \sqrt{\mathcal{C}_{NJ}(\mathcal{B})}. $$

Therefore

$$\frac{1}{2}J^{2}(\mathcal{B})\leq\mathcal{C}_{NJ}( \mathcal{B}). $$

To prove the right hand side we use Theorem 3.7, so we only take $\Vert x_{1}\Vert =1$ and $\Vert x_{2}\Vert \leq1$.

Case 1: If $\Vert x_{2}\Vert =t\geq J(\mathcal{B})-1$, then

$$\begin{aligned}& {\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2} \leq \bigl[C\bigl( \Vert x_{1}\Vert +\Vert x_{2}\Vert \bigr) \bigr]^{2}+ \biggl(\frac{C}{C}\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \biggr)^{2} \\& \hphantom{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2} }= C^{2} \biggl[\bigl(\Vert x_{1}\Vert +\Vert x_{2}\Vert \bigr)^{2}+ \biggl(\frac{1}{C}\min\bigl\{ \Vert x_{1}+x_{2}\Vert ,\Vert x_{1}-x_{2} \Vert \bigr\} \biggr)^{2} \biggr] \\& {\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2} \leq C^{2} \bigl[\bigl(\Vert x_{1}\Vert +\Vert x_{2}\Vert \bigr)^{2}+J^{2}(\mathcal{B}) \bigr], \\& \frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2C^{2}(\Vert x_{1}\Vert ^{2}+\Vert x_{2}\Vert ^{2})} \leq \frac{(\Vert x_{1}\Vert +\Vert x_{2}\Vert )^{2}+J^{2}(\mathcal{B})}{2(\Vert x_{1}\Vert ^{2}+\Vert x_{2}\Vert ^{2})} \\& \hphantom{\frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2C^{2}(\Vert x_{1}\Vert ^{2}+\Vert x_{2}\Vert ^{2})}} = \frac{(1+t)^{2}+J^{2}(\mathcal{B})}{2(1+t^{2})}. \end{aligned}$$

The function

$$f(t)=\frac{(1+t)^{2}+J^{2}(\mathcal{B})}{2(1+t^{2})} $$

is increasing on $(0,\mu)$ and decreasing on $(\mu,1)$ where

$$\mu=\frac{-J^{2}(\mathcal{B})+\sqrt{J^{4}(\mathcal{B})+4}}{2}. $$

Since $J(\mathcal{B})-1\geq\mu$ and $J(\mathcal{B})-1\leq t$, we have $f(t)\leq f(J(\mathcal{B})-1)$

$$\frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2C^{2}(\Vert x_{1}\Vert ^{2}+\Vert x_{2}\Vert ^{2})}\leq\frac{J^{2}(\mathcal {B})}{1+(J(\mathcal{B})-1)^{2}}, $$

taking the supremum, we get

$$ \mathcal{C}_{NJ}(\mathcal{B})\leq\frac{J^{2}(\mathcal{B})}{1+(J(\mathcal {B})-1)^{2}}. $$

(3.7)

Case 2: If $\Vert x_{2}\Vert =t\leq J(\mathcal{B})-1$, then

$$\begin{aligned}& {\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2} \leq 2C^{2} \bigl(\Vert x_{1}\Vert +\Vert x_{2}\Vert \bigr)^{2} \\& \hphantom{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}} = 2C^{2}(1+t)^{2}, \\& \frac{{\Vert x_{1}+x_{2} \Vert }^{2}+{\Vert x_{1}-x_{2}\Vert }^{2}}{2C^{2}(\Vert x_{1}\Vert ^{2}+\Vert x_{2}\Vert ^{2})} \leq \frac{(1+t)^{2}}{(1+t^{2})}. \end{aligned}$$

Let

$$g(t)=\frac{(1+t)^{2}}{(1+t^{2})}, $$

since g is increasing on $(0,1]$

$$g(t)\leq g\bigl(J(\mathcal{B})-1\bigr), $$

hence

$$ \mathcal{C}_{NJ}(\mathcal{B})\leq\frac{J^{2}(\mathcal{B})}{1+(J(\mathcal {B})-1)^{2}}. $$

(3.8)

 □

Corollary 3.9

If $\mathcal{B}$ is not uniformly non-square then

$$J(\mathcal{B})=\mathcal{C}_{NJ}(\mathcal{B})=2. $$

Theorem 3.10

For any quasi-Banach space $\mathcal{B}$, we have

$$ \mathcal{C}_{NJ}(\mathcal{B})\leq1+ \biggl(\frac{CJ(\mathcal {B})}{2} \biggr)^{2}. $$

(3.9)

Proof

Since $\mathcal{C}_{NJ}(\mathcal{B})=\sup\{\mathcal{C}_{NJ}(t,\mathcal {B}): t\in[0,1]\}$ where

$$\mathcal{C}_{NJ}(t,\mathcal{B})=\sup \biggl\{ \frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})}: \forall x_{1},x_{2}\in S_{\mathcal{B}} \biggr\} . $$

First we prove that

$$ \mathcal{C}_{NJ}(t,\mathcal{B})\leq1+\frac{C^{2}t^{2}J^{2}(\mathcal {B})+2Ct(1-t)J(\mathcal{B})}{2(1+t^{2})}. $$

(3.10)

For this purpose

$$\begin{aligned}& {\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2} \leq \bigl[C(1+t) \bigr]^{2}+ \bigl(\min\bigl\{ {\Vert x_{1}+tx_{2} \Vert },{\Vert x_{1}-tx_{2}\Vert }\bigr\} \bigr)^{2} \\& \hphantom{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}} \leq C^{2}\bigl[(1+t)^{2}+J^{2}(t,\mathcal{B}) \bigr], \\& \frac{{\Vert x_{1}+tx_{2} \Vert }^{2}+{\Vert x_{1}-tx_{2}\Vert }^{2}}{2C^{2}(1+t^{2})} \leq \frac{(1+t)^{2}+J^{2}(t,\mathcal{B})}{2(1+t^{2})}. \end{aligned}$$

Taking the supremum we get

$$ \mathcal{C}_{NJ}(t,\mathcal{B})\leq\frac{ (1+t)^{2}+J^{2}(t,\mathcal {B})}{2(1+t^{2})}. $$

(3.11)

Also note that

$$ \begin{aligned} &\min\bigl\{ {\Vert x_{1}+tx_{2} \Vert }+{\Vert x_{1}-tx_{2}\Vert }\bigr\} \leq C\min\bigl\{ t{\Vert x_{1}+x_{2} \Vert }+(1-t),t{\Vert x_{1}-x_{2} \Vert }+(1-t)\bigr\} , \\ & J(t,\mathcal{B})\leq tCJ(\mathcal{B})+(1-t). \end{aligned} $$

(3.12)

Using (3.11) and (3.12), we get

$$\begin{aligned} \mathcal{C}_{NJ}(t,\mathcal{B}) \leq& \frac{[CtJ(\mathcal {B})+(1-t)]^{2}+(1+t)^{2}}{2(1+t^{2})} \end{aligned}$$

(3.13)

$$\begin{aligned} =& 1+\frac{C^{2}t^{2}J^{2}(\mathcal{B})+2t(1-t)J(\mathcal{B})}{2(1+t^{2})}. \end{aligned}$$

(3.14)

Taking the supremum over t, we get

$$ \mathcal{C}_{NJ}(\mathcal{B})\leq1+ \biggl(\frac{CJ(\mathcal {B})}{2} \biggr)^{2}. $$

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