Journal of Function Spaces
Volume 2018 (2018), Article ID 7324783, 10 pages
https://doi.org/10.1155/2018/7324783

Research Article

On Generalized Moduli of Quasi-Banach Space

Young Chel Kwun,¹ Hussain Minhaj Uddin Ahmad Qadri,² Waqas Nazeer,³ Absar Ul Haq,⁴ and Shin Min Kang5,6

¹Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea
²Aitchison College, Lahore 54000, Pakistan
³Division of Science and Technology, University of Education, Lahore 54000, Pakistan
⁴Department of Mathematics, University of Management and Technology, Sialkot Campus, Sialkot 51410, Pakistan
⁵Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea
⁶Center for General Education, China Medical University, Taichung 40402, Taiwan

Correspondence should be addressed to Waqas Nazeer; nazeer.waqas@ue.edu.pk and Shin Min Kang; smkang@gnu.ac.kr

Received 23 November 2017; Accepted 7 March 2018; Published 3 May 2018

Academic Editor: P. Veeramani

Copyright © 2018 Young Chel Kwun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We shall discuss three generalized moduli such as generalized modulus of convexity, modulus of smoothness, and modulus of Zou-Cui of quasi-Banach spaces and give some important properties of these moduli. Furthermore, we establish relationships of these generalized moduli with each other.

1. Introduction

The study on Banach space geometry provides many fundamental notions and interesting aspects and sometimes has surprising results. The basic geometric properties such as convexity, smoothness, and nonsquareness have made great contributions to various fields of Banach space theory. Strict convexity of Banach spaces was first introduced in 1936 by Clarkson [1] (and independently by Akhiezer and Krein) as the property that the unit sphere contains no nontrivial line segments; that is, whenever . Clarkson [1] made use of these values to define the “uniform” version of convexity to look at how “convex” the unit ball is in a space. And the modulus of convexity provides a quantification of the geometric structure of the space from the viewpoint of convexity. A situation similar to this also occurs in smoothness and other properties. A Banach space is said to be smooth if each unit vector has a unique norm one support functional. In fact, this is equivalent to the statement that the norm is Gateaux differentiable. This allows us to quantify the geometric structure of the space from the viewpoint of smoothness, namely, the modulus of smoothness of a Banach space . An advantage of these quantifications is that the complete duality between uniform convexity and uniform smoothness can be easily deduced by the well-known Lindenstrauss formulas; that is, a Banach space is uniformly convex if and only if its dual space is uniformly smooth. The same statement still holds if is replaced with Thus quantifying geometric structures might lead to better results. Note that the same duality does not hold between strict convexity and smoothness in general, though one of those two properties of implies the other of . There are some other ideas to quantify geometric structures of Banach spaces.

In [2], the authors claim that modulus of convexity and generalized convexity mold have dual relationship, and generalized convexity mold has many excellent properties.

In [3], the authors study a generalized modulus of convexity where certain related geometrical properties of this modulus are analyzed in Banach spaces.

In [4, 5], the modulus of Yang-Wang was introduced in Banach spaces.

In [6], the modulus of Zuo-Cui was introduced in Banach spaces. The author proved many results with this special type of modulus.

The most recent research work at this topic can be consulted from [7, 8].

2. Preliminaries

There are lots of quantitative descriptions of geometrical properties of quasi-Banach spaces. The most common way for creating these descriptions is to define a real function (a modulus) and a suitable coefficient or constant closely related to this function, depending on the space structure under consideration. Some of the moduli and their related coefficients (or characteristics) for quasi-Banach spaces have also been investigated so far. These moduli are the attempts in order to get a better understanding of the two facts about the space:

(i)The shape of the unit ball of the concerned space.(ii)The conditions and relations for convergence of sequences.

The most recent research work with these moduli is investigated by [7, 8].

Definition 1. For a quasi-Banach space , the modulus of convexity is a function defined as

A characteristic or related coefficient of this modulus is

Definition 2. Let and . For a quasi-Banach space , the generalized modulus of convexity is a function defined as

A characteristic or related coefficient of this modulus is

Definition 3. For a quasi-Banach space , the modulus of smoothness is a function defined as

A characteristic or related coefficient of this modulus is

Definition 4. Let and . For a quasi-Banach space , the generalized modulus of smoothness is a function defined as

where , , , and
A characteristic or related coefficient of this modulus is

Definition 5. Let and . For a quasi-Banach space , the modulus of Zuo-Cui is a function defined as

A characteristic or related coefficient of this modulus is

3. Relations Concerning Generalized Modulus of Convexity

Lemma 6 (see [9]). Every convex function with convex domain in is continuous.

Proposition 7 (see [10]). Let be a uniformly convex space. Then for every , , and for arbitrary vectors, with , , and , where , there exists such that

Proposition 8 (see [10]). A nontrivial quasi-Banach space is uniformly nonsquare if and only if .

Proposition 9. Let be the modulus of convexity of a quasi-Banach space ; then

Proof. Let , , and . Then we have

and we have obtained the first inequality; now to prove the second one, for any , there exist with , , and such that ; then we have

This completes the proof.

Corollary 10. Let be the characteristic of generalized modulus of convexity of a quasi-Banach space . Then

Theorem 11. A quasi-Banach space is uniformly nonsquare if and only if

Proof. By Proposition 8, is uniformly nonsquare if and only if , where , and by Proposition 9, if and only if .
Combining both of these results we get the proof.

Lemma 12. Let be a quasi-Banach space and . Then the following statements hold:

(1) is convex and continuous function.(2) is a nondecreasing function.(3) is a nondecreasing function.

Proof. Let

Then

Now

by using Proposition 9. Hence we get

Since is convex, so it is continuous by Lemma 6.
Let and with . Let us consider

Then with ; we have

thus

which implies that

Let ; we have

Since , so we have

This completes the proof.

Theorem 13. Let be a uniformly convex space and . Then for every , and for arbitrary vectors, with , , and , there exists such that

Proof. Without loss of generality, assume that so

Now by Proposition 7, there exist such that, for ,

Therefore, we get

This completes the proof.

4. Relations Concerning Generalized Modulus of Smoothness

Theorem 14. Let be a quasi-Banach space. Then for every , , and ,

Proof. Throughout the proof of the first part, we take

Take

Hence, we get

This completes the first part of the proof.
To prove the second part, let us start as

Lemma 15. Let be a quasi-Banach space and . Then the following statements hold:

(1) is convex and continuous function.(2) is a nondecreasing function.(3) is a nondecreasing function.

Proof. Since is convex, so it is continuous by Lemma 6.
Let , , and
Now let

Now

From (38) and (40), we get

Similarly,

Therefore, we have

which implies that

thus we have

This shows that is a nondecreasing function.
This part is an immediate consequence of Theorem 14.

Theorem 16. For a nontrivial quasi-Banach space with and , one has

Proof. Let and Then, we have

This proves the first inequality; now to prove the second inequality, we proceed as

Also

Therefore,

This completes the second inequality. Now, combining both of the above inequalities and taking the max of the latter, we get

This completes the proof.

Theorem 17. A quasi-Banach space is uniformly smooth if and only if

Proof. One has

Hence we get

Dividing by “” and taking supremum on both sides, we get

thus

which implies that

Now

This can be expressed as

Dividing by “” and taking supremum on both sides

thus

Therefore, from (57) and (61) we get

This completes the proof.

4.1. Relations Concerning Modulus of Zuo-Cui

Theorem 18. Let be a nontrivial quasi-Banach space and . Then

Proof. By using convexity of the function on , one can easily obtain

so,

which implies that

For , we have .
For , we have , where

Proposition 19. For a Banach space ,

Proof. Consider a convex function defined by

Let and . Then

This shows that . Therefore, we have

Since the opposite inequality holds obviously, to get the first inequality let

Since is a convex and an even function, therefore . Now for we have

which obtains the second inequality. This completes the proof.

Lemma 20. For any quasi-Banach space and any , then the following statements hold:

(1) is a nondecreasing function.(2) is convex and continuous function.(3) is a nondecreasing function.

Proof. Let be a convex and even function. Let and . Then we have

which implies that

Hence . Since is convex, so it is continuous by Lemma 6.
Let , , and and . Then we have

This shows that is convex.
Let and with . Then

This shows that is nondecreasing. This completes the proof.

Proposition 21. Let be a quasi-Banach space, , and . Then the following conditions are equivalent:

(1)(2).

Proof. Suppose on the contrary that ; it is enough to take . Since

by using the definition of , for any , there exist such that

Applying convexity of the function , we get

therefore from (79)

Since is any arbitrary so

which leads a contradiction.
Suppose on the contrary that ; it is enough to take . Again using the definition of , for all , there exist such that

also using

thus

since is arbitrary, so

which implies that

So using the definition of we get , which leads to a contradiction.

Corollary 22. Let be a quasi-Banach space, , and . Then the following conditions are equivalent:

(1) is uniformly nonsquare.(2)(3)

Theorem 23. A quasi-Banach apace is uniformly smooth if

Proof. Suppose that

From Theorem 18, we know that

which implies that

dividing both sides by and applying the

So by definition is uniformly smooth.

Theorem 24. Let be a quasi-Banach space. Then for

Proof. If , then holds true.
Let and . Then we have

and we have

Therefore, we have

Theorem 25. Let be a quasi-Banach space. Then

Proof. Since

therefore, we have

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

This work was supported by the Dong-A University research fund.

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