Complex Convexity of Orlicz Modular Sequence Spaces

Abstract

The concepts of complex extreme points, complex strongly extreme points, complex strict convexity, and complex midpoint locally uniform convexity in general modular spaces are introduced. Then we prove that, for any Orlicz modular sequence space , is complex midpoint locally uniformly convex. As a corollary, is also complex strictly convex.

1. Introduction

In 1967, the notions of complex extreme points and complex strict convexity have been introduced by Thorp and Whitley [1]. They proved that the strong maximum modulus theorem for analytic functions with values in a complex Banach space holds true whenever each point of the unit sphere of is a complex extreme point. In 1975, Globevnik further introduced the notions of complex strict and uniform convexity of complex normed spaces and proved that the complex space is complex uniformly convex (see [2]). Davis et al. in [3] investigated the complex convexity of quasi-normed linear spaces. Dowling et al. in [4] studied the complex convexity of Lebesgue-Bochner function spaces. Blasco and Pavlović in [5] obtained sufficient and necessary conditions for a complex Banach space which is -uniformly PL-convex. Choi et al. in [6] obtained criteria for complex extreme points, complex rotundity, and complex uniform convexity in Orlicz-Lorentz spaces. Hudzik and Narloch in [7] considered relationships between monotonicity and complex rotundity; for instance, a point of the complexification of a real Köthe space is a complex extreme point if and only if is a point of upper monotonicity in . Lee in [8, 9] continued to study relationships between monotonicity and complex convexity in Banach lattices and Quasi-Banach lattices, respectively. Czerwińska and Kamińska in [10] discussed the complex rotundity and midpoint local uniform convexity in symmetric spaces of measurable operators. Recently, Czerwińska and Parrish in [11] characterized complex extreme points in Marcinkiewicz spaces.

In this paper, we introduce the concepts of complex extreme points, complex strongly extreme points, complex strict convexity, and complex midpoint locally uniform convexity in general modular spaces. Then we prove that, for any Orlicz modular sequence space , is complex midpoint locally uniformly convex. As a corollary, is also complex strictly convex.

Before starting with our results, we need to recall some basic concepts and facts of the theory of modular spaces and Orlicz spaces.

Let be a vector space over the complex field . A functional is called a modular provided that, for any ,(a) if and only if ;(b) for any with ;(c) for any with ; we can replace (c) by the following;(c′) for any with ; in this case the modular is said to be a convex modular. A modular space is defined by

Let be a modular space; then denote the closed unit ball and the unit sphere of , respectively. In the sequel , , and denote the set of natural numbers, the set of real numbers, and the set of complex numbers, respectively. Let be the complex number satisfying .

A map is said to be an Orlicz function if is vanishing at zero, even, convex, and continuous and satisfies and . For every Orlicz function , its complementary function is defined by the formula and the complementary function is also an Orlicz function.

DefineFor any , let denote the support set of . For a given -function , we define on a convex modular by which is said to be an Orlicz modular. The modular space induced by an Orlicz modular is said to be the Orlicz sequence space . Indeed, Orlicz sequence spaces and their kinds of generalizations belong to modular spaces. For the sake of simplicity, we denote and .

For more details on Orlicz spaces we refer to [12–24], and for more details on modular spaces one can consult [25–33].

2. Main Results

In this section, we always assume that is a convex modular. It is known that extreme points which are connected with strict convexity of the whole spaces are the most basic and important geometric points in geometric theory of spaces. In [1], Thorp and Whitley first introduced the concepts of complex extreme points and complex strict convexity when they studied the conditions under which the strong maximum modulus theorem for analytic functions always holds in a complex Banach space.

Definition 1 (see [1]). Let be a Banach space. A point is said to be a complex extreme point of if for every nonzero there holds . A Banach space is said to be complex strictly convex if every element of is a complex extreme point of .

In [12], we further studied the notions of complex strongly extreme points and complex midpoint locally uniform convexity in general complex spaces. For more details on notions of complex extreme points and complex strongly extreme points we refer to [12, 13].

Definition 2 (see [12]). Let be a Banach space. A point is said to be a complex strongly extreme point of if, for every , one has , whereA Banach space is said to be complex midpoint locally uniformly convex if every element of is a complex strongly extreme point of .

Now we generalize the notions of complex extreme points and complex strongly extreme points of Banach spaces to the modular spaces.

Definition 3. Let be a modular space. A point is said to be a complex extreme point of if, for any with , there holds is said to be complex strictly convex if every element of is a complex extreme point of .

Definition 4. Let be a modular space. A point is said to be a complex strongly extreme point of if for every , whereand with . is said to be complex midpoint locally uniformly convex if every element of is a complex strongly extreme point of .

Proposition 5. Let , , and ; then the following conditions are equivalent:(i) for all with .(ii) and .

Proof. (i)⇒(ii) is trivial. For any with , without loss of generality, there exist such thatIt follows thatConsequently,

Remark 6. By Proposition 5, it is easy to see that the inequality in Definition 3is equivalent to the inequality .

Next, we give a new modulus which is associated with complex strongly extreme points.

Definition 7. Let be a modular space and . For each , one defines a function as follows:Observe that for all .

Proposition 8. Let be a modular space. A point is a complex strongly extreme point of if and only if for all .

Proof.  
Necessity. Let be a complex strongly extreme point of . Assume that there exists such that . By Definition 7, we can find a sequence satisfying andLet and . By (14) and noticing that and as , we obtainwhich implies , a contradiction.
Sufficiency. Suppose that is not a complex strongly extreme point of and for all . By Definition 4, then there exists such that . Hence, there are with and satisfying , such thatfor each . Setting , we have In view of Proposition 5, we deduce thatIt follows thatwhich contradicts the fact that for all .

In Banach spaces, we have shown that if is a complex strongly extreme point of , then is a complex extreme point of (see [12]). We will prove that it is also true in modular spaces.

Theorem 9. Let be a modular space. If is a complex strongly extreme point of , then is a complex extreme point of .

Proof. Suppose that is not a complex extreme point of the closed unit ball . Then there exists such thatHence, we haveLetting , then we obtain which is a contradiction.

The following result is an immediate corollary of the previous theorem.

Corollary 10. Let be a modular space. If is complex midpoint locally uniformly convex, it is also complex strictly convex.

Next, we will study complex convexity of Orlicz modular sequence spaces. Before starting with our results, let us recall a useful lemma.

Lemma 11 (see [13]). For any , there exists such that if and then where

In [34], we have given criteria for complex strict convexity and complex midpoint locally uniform convexity of Orlicz sequence spaces equipped with the -Amemiya norm.

Theorem 12 (see [34]). Assume ; then the following are equivalent:(i) is complex midpoint locally uniformly convex.(ii) is complex strictly convex.(iii).

Theorem 13 (see [34]). If , then is complex strictly convex if and only if(i),(ii).

Theorem 14 (see [34]). If , then is complex midpoint locally uniformly convex if and only if(i),(ii),(iii)for any , there exists such that, for every , there holds , where , , , ,   for any , and

We will show that the Orlicz modular sequence space is complex midpoint locally uniformly convex without any condition.

Theorem 15. Let be an Orlicz modular sequence space. Then is complex midpoint locally uniformly convex.

Proof. Suppose that is not a complex strongly extreme point of the unit ball . By Definition 4, then there exists such that . In a similar way to Proposition 8, we can find a sequence satisfying and For the above , by Lemma 11, there exists , such that if and thenFor every , let It is easy to see that for every , and for large enough since as . Consequently, we obtainwhich shows that . Furthermore, we have Notice that Hence, In view of as , letting , then we get a contradictionwhich completes the proof.

Combining Corollary 10 and Theorem 15 we obtain immediately the following result.

Corollary 16. Let be an Orlicz modular sequence space. Then is complex strictly convex.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 11401141, Heilongjiang Provincial Natural Science Foundation for Youths under Grant QC2013C001, Heilongjiang Provincial Postdoctoral Science Foundation under Grant LBH-Z15098, and Natural Science Foundation of Heilongjiang Educational Committee under Grant 12531099.