Complex Convexity of Orlicz Modular Sequence Spaces

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.


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  1 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  Φ, ,  Φ, is complex midpoint locally uniformly convex.As a corollary,  Φ, 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 C. A functional  :  → [0, ∞] is called a modular provided that, for any ,  ∈ , (c  ) ( + ) ≤ () + () for any ,  ≥ 0 with  +  = 1; in this case the modular  is said to be a convex modular.A modular space   is defined by 2

Journal of Function Spaces
Let   be a modular space; then denote the closed unit ball and the unit sphere of   , respectively.In the sequel N, R, and C denote the set of natural numbers, the set of real numbers, and the set of complex numbers, respectively.Let  be the complex number satisfying  2 = −1.
A map Φ :  → [0, ∞] is said to be an Orlicz function if Φ is vanishing at zero, even, convex, and continuous and satisfies lim →0 (Φ()/) = 0 and lim →∞ (Φ()/) = ∞.For every Orlicz function Φ, its complementary function Ψ :  → [0, ∞] is defined by the formula and the complementary function Ψ is also an Orlicz function.Define For any  ∈   , let supp() = { ∈ N : () ̸ = 0} denote the support set of .For a given -function Φ, 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  Φ .Indeed, Orlicz sequence spaces and their kinds of generalizations belong to modular spaces.For the sake of simplicity, we denote  Φ, = ( Φ ,  Φ ) and  Φ = ( Φ , ‖⋅‖).

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 sup ||≤1 ‖ + ‖ > 1.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  > 0, one has Δ  (, ) > 0, where A 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  ̸ = 0, there holds sup 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 Δ , (, ) > 0 for every  > 0, where and  ∈ C with 0 < || ≤ 1.   is said to be complex midpoint locally uniformly convex if every element of (  ) is a complex strongly extreme point of (  ).
Next, we give a new modulus which is associated with complex strongly extreme points.Definition 7. Let   be a modular space and  ∈ (  ).For each  > 0, one defines a function as follows: Observe that  , (, ) ≥ 0 for all  > 0.
Proposition 8. Let   be a modular space.A point  ∈ (  ) is a complex strongly extreme point of (  ) if and only if  , (, ) > 0 for all  > 0. Proof.
In view of Proposition 5, we deduce that sup It follows that which contradicts the fact that  , (, ) > 0 for all  > 0.
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 (  ).
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  > 0, there exists  ∈ (0, 1/2) such that if , V ∈ C 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 15.Let  Φ, be an Orlicz modular sequence space.Then  Φ, is complex midpoint locally uniformly convex.
Combining Corollary 10 and Theorem 15 we obtain immediately the following result.
Corollary 16.Let  Φ, be an Orlicz modular sequence space.Then  Φ, is complex strictly convex.