On Generalized Moduli of Quasi-Banach Space

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


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, 1−‖2 −1 (+)‖ > 0 whenever ‖‖ = ‖‖ = 1.
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.

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 B, the modulus of convexity is a function  B : (0, 2] → [0, 1] defined as A characteristic or related coefficient of this modulus is Definition 2. Let  ∈ (0, 1) and  ∈ [0, 2].For a quasi-Banach space B, the generalized modulus of convexity is a function  () B : (0, 2] → [0, 1] defined as A characteristic or related coefficient of this modulus is Definition 3.For a quasi-Banach space B, the modulus of smoothness is a function A characteristic or related coefficient of this modulus is Definition 4. Let  ∈ (0, 1) and  ∈ [0, 2].For a quasi-Banach space B, the generalized modulus of smoothness is a function where A characteristic or related coefficient of this modulus is B : (0, ∞) → (0, ∞) defined as A characteristic or related coefficient of this modulus is

Relations Concerning Generalized Modulus of Convexity
Lemma 6 (see [9]).Every convex function  with convex domain in R is continuous.
Corollary 10.Let   be the characteristic of generalized modulus of convexity  () B () of a quasi-Banach space B. Then Then Now by using Proposition 9. Hence we get

Relations Concerning Generalized Modulus of Smoothness
Proof.Throughout the proof of the first part, we take Hence, we get This completes the first part of the proof.
(3) This part is an immediate consequence of Theorem 14.
Theorem 16.For a nontrivial quasi-Banach space B with  ∈ [0, ∞) and  ∈ (0, 1), one has Proof.Let ,  ∈  B and  ∈ (0, 1).Then, we have ) This proves the first inequality; now to prove the second inequality, we proceed as which implies that