Research ArticleComplex Convexity of Musielak-Orlicz Function SpacesEquipped with the 𝑝-Amemiya Norm

Lili Chen, Yunan Cui, and Yanfeng Zhao

Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China

Correspondence should be addressed to Lili Chen; cll2119@hotmail.com

Received 27 November 2013; Revised 5 April 2014; Accepted 13 April 2014; Published 8 May 2014

Academic Editor: Angelo Favini

Copyright © 2014 Lili Chen et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The complex convexity of Musielak-Orlicz function spaces equipped with the 𝑝-Amemiya norm is mainly discussed. It is obtainedthat, for any Musielak-Orlicz function space equipped with the 𝑝-Amemiya norm when 1 ≤ 𝑝 < ∞, complex strongly extremepoints of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in above spaces are given.Criteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced.

1. Introduction

Let (𝑋, ‖ ⋅ ‖) be a complex Banach space over the complexfield C, let 𝑖 be the complex number satisfying 𝑖2 = −1, andlet 𝐵(𝑋) and 𝑆(𝑋) be the closed unit ball and the unit sphereof 𝑋, respectively. In the sequel, N and R denote the set ofnatural numbers and the set of real numbers, respectively.

In the early 1980s, a huge number of papers in the area ofthe geometry of Banach spaces were directed to the complexgeometry of complex Banach spaces. It is well known that thecomplex geometric properties of complex Banach spaces haveapplications in various branches, among others in HarmonicAnalysis Theory, Operator Theory, Banach Algebras, 𝐶∗-Algebras, Differential EquationTheory, QuantumMechanicsTheory, and Hydrodynamics Theory. It is also known thatextreme points which are connected with strict convexity ofthe whole spaces are the most basic and important geometricpoints in geometric theory of Banach spaces (see [1–6]).

In [7], Thorp and Whitley first introduced the conceptsof complex extreme point and complex strict convexitywhen they studied the conditions under which the StrongMaximum Modulus Theorem for analytic functions alwaysholds in a complex Banach space.

A point 𝑥 ∈ 𝑆(𝑋) is said to be a complex extreme point of𝐵(𝑋) if for every nonzero𝑦 ∈ 𝑋 there holds sup

|𝜆|≤1‖𝑥+𝜆𝑦‖ >

1. A complex Banach space 𝑋 is said to be complex strictlyconvex if every element of 𝑆(𝑋) is a complex extreme pointof 𝐵(𝑋).

In [8], we further studied the notions of complex stronglyextreme point and complex midpoint locally uniform con-vexity in general complex spaces.

A point 𝑥 ∈ 𝑆(𝑋) is said to be a complex strongly extremepoint of 𝐵(𝑋) if for every 𝜀 > 0 we have Δ

𝑐(𝑥, 𝜀) > 0, where

Δ𝑐(𝑥, 𝜀) = inf {1 − |𝜆| : ∃𝑦 ∈ 𝑋s.t. 𝜆𝑥 ± 𝑦

≤ 1,

𝜆𝑥 ± 𝑖𝑦 ≤ 1,

𝑦 ≥ 𝜀} .

(1)

A complex Banach space 𝑋 is said to be complex midpointlocally uniformly convex if every element of 𝑆(𝑋) is a complexstrongly extreme point of 𝐵(𝑋).

Let (𝑇, Σ, 𝜇) be a nonatomic and complete measure spacewith 𝜇(𝑇) < ∞. By Φ we denote a Musielak-Orlicz function;that is, Φ : 𝑇 × [0, +∞) → [0, +∞] satisfies the following:

(1) for each 𝑢 ∈ [0,∞], Φ(𝑡, 𝑢) is a 𝜇-measurablefunction of 𝑡 on 𝑇;

(2) for 𝜇-a.e. 𝑡 ∈ 𝑇,Φ(𝑡, 0) = 0, lim𝑢→∞

Φ(𝑡, 𝑢) = ∞ andthere exists 𝑢 > 0 such thatΦ(𝑡, 𝑢) < ∞;

(3) for 𝜇-a.e. 𝑡 ∈ 𝑇, Φ(𝑡, 𝑢) is convex on the interval[0,∞) with respect to 𝑢.

Let 𝐿𝑐(𝜇) be the space of all 𝜇-equivalence classes ofcomplex and Σ-measurable functions defined on 𝑇. For each𝑥 ∈ 𝐿𝑐(𝜇), we define on 𝐿𝑐(𝜇) the convex modular of 𝑥 by

𝐼Φ(𝑥) = ∫

𝑇

Φ (𝑡, |𝑥 (𝑡)|) 𝑑𝑡. (2)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 190203, 6 pageshttp://dx.doi.org/10.1155/2014/190203

2 Abstract and Applied Analysis

We define supp 𝑥 = {𝑡 ∈ 𝑇 : |𝑥(𝑡)| ̸= 0} and the Musielak-Orlicz space 𝐿

Φgenerated by the formula

𝐿Φ= {𝑥 ∈ 𝐿

𝑐(𝜇) : 𝐼

Φ(𝜆𝑥) < ∞ for some 𝜆 > 0} . (3)

Set

𝑒 (𝑡) = sup {𝑢 ≥ 0 : Φ (𝑡, 𝑢) = 0} ,

𝐸 (𝑡) = sup {𝑢 ≥ 0 : Φ (𝑡, 𝑢) < ∞} ;(4)

then 𝑒(𝑡) and 𝐸(𝑡) are 𝜇-measurable (see [9]).The notion of 𝑝-Amemiya norm has been introduced in

[10]; for any 1 ≤ 𝑝 ≤ ∞ and 𝑢 > 0, define

𝑠𝑝(𝑢) = {

(1 + 𝑢𝑝)1/𝑝

, for 1 ≤ 𝑝 < ∞,max {1, 𝑢} , for 𝑝 = ∞.

(5)

Furthermore, define 𝑠Φ,𝑝(𝑥) = 𝑠

𝑝∘𝐼Φ(𝑥) for all 1 ≤ 𝑝 ≤ ∞

and 𝑥 ∈ 𝐿𝑐(𝜇). Notice that the function 𝑠𝑝is increasing on 𝑅

+

for 1 ≤ 𝑝 < ∞; however, the function 𝑠∞is increasing on the

interval [1,∞) only.For 𝑥 ∈ 𝐿𝑐(𝜇), define the 𝑝-Amemiya norm by the

formula

‖𝑥‖Φ,𝑝 = inf𝑘>0

1

𝑘𝑠Φ,𝑝(𝑘𝑥) , 1 ≤ 𝑝 ≤ ∞. (6)

For convenience, from now on, we write 𝐿Φ,𝑝

=

(𝐿Φ, ‖ ⋅ ‖Φ,𝑝); it is easy to see that 𝐿

Φ,𝑝is a Banach space. For

1 ≤ 𝑝 < ∞, 𝑥 ∈ 𝐿Φ,𝑝

, set

𝐾𝑝(𝑥) = {𝑘 > 0 : ‖𝑥‖Φ,𝑝 =

1

𝑘{1 + 𝐼

𝑝

Φ(𝑘𝑥)}1/𝑝

} . (7)

2. Main Results

We begin this section from the following useful lemmas.

Lemma 1 (see [9]). For any 𝜀 > 0, there exists 𝛿 ∈ (0, 1/2)such that if 𝑢, V ∈ C and

|V| ≥𝜀

8max𝑒|𝑢 + 𝑒V| , (8)

then

|𝑢| ≤1 − 2𝛿

4Σ𝑒 |𝑢 + 𝑒V| , (9)

where

max𝑒|𝑢 + 𝑒V| = max {|𝑢 + V| , |𝑢 − V| , |𝑢 + 𝑖V| , |𝑢 − 𝑖V|} ,

Σ𝑒 |𝑢 + 𝑒V| = |𝑢 + V| + |𝑢 − V| + |𝑢 + 𝑖V| + |𝑢 − 𝑖V| .

(10)

Lemma 2. If lim𝑢→∞

(Φ(𝑡, 𝑢)/𝑢) = ∞ for 𝜇-a.e. 𝑡 ∈ 𝑇, then𝐾𝑝(𝑥) ̸= 0 for any 𝑥 ∈ 𝐿

Φ,𝑝\ {0}, where 1 ≤ 𝑝 < ∞.

Proof. For any 𝑥 ∈ 𝐿Φ,𝑝

\ {0}, there exists 𝛼 > 0 such that𝜇{𝑡 ∈ 𝑇 : |𝑥(𝑡)| ≥ 𝛼} > 0. Let 𝑇

1= {𝑡 ∈ 𝑇 : |𝑥(𝑡)| ≥ 𝛼} and

define the subsets

𝐹𝑛= {𝑡 ∈ 𝑇

1:Φ (𝑡, 𝑢)

𝑢≥4‖𝑥‖Φ,𝑝

𝛼 ⋅ 𝜇𝑇1

, 𝑢 ≥ 𝑛} (11)

for each 𝑛 ∈ N. It is easy to see that 𝐹1⊆ F2⊆ ⋅ ⋅ ⋅ ⊆ 𝐹

𝑛⊆ ⋅ ⋅ ⋅

and lim𝑛→∞

𝜇𝐹𝑛= 𝜇𝑇1. Then there exists 𝑛

0∈ N such that

𝜇𝐹𝑛0> (1/2)𝜇𝑇

1.

It follows from the definition of 𝑝-Amemiya norm thatthere is a sequence {𝑘

𝑛} satisfying

‖𝑥‖Φ,𝑝 = lim𝑛→∞

1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛𝑥)}1/𝑝

. (12)

For any 𝑘 > 0, we have

1

𝑘{1 + 𝐼

𝑝

Φ(𝑘𝑥)}1/𝑝

=1

𝑘{1 + [∫

𝑇

Φ (𝑡, 𝑘 |𝑥 (𝑡)|) 𝑑𝑡]

𝑝

}

1/𝑝

≥

∫𝐹𝑛0

Φ (𝑡, 𝑘 |𝑥 (𝑡)|) 𝑑𝑡

𝑘

≥ ∫𝐹𝑛0

Φ (𝑡, 𝑘𝛼)

𝑘𝑑𝑡

= 𝛼∫𝐹𝑛0

Φ (𝑡, 𝑘𝛼)

𝑘𝛼𝑑𝑡.

(13)

If 𝑘 > 𝑛0/𝛼, notice that

1

𝑘{1 + 𝐼

𝑝

Φ(𝑘𝑥)}1/𝑝

≥ 𝛼 ⋅4‖𝑥‖Φ,𝑝

𝛼 ⋅ 𝜇𝑇1

⋅ 𝜇𝐹𝑛0> 2‖𝑥‖Φ,𝑝, (14)

which means the sequence {𝑘𝑛} is bounded. Hence, without

loss of generality, we assume that 𝑘𝑛→ 𝑘0as 𝑛 → ∞.

We can also choose the monotonic increasing or decreasingsubsequence of {𝑘

𝑛} that converges to the number 𝑘

0. Apply-

ing Levi Theorem and Lebesgue Dominated ConvergenceTheorem, we obtain

‖𝑥‖Φ,𝑝 = lim𝑛→∞

1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛𝑥)}1/𝑝

=1

𝑘0

{1 + 𝐼𝑝

Φ(𝑘0𝑥)}1/𝑝

(15)

which implies 𝑘0∈ 𝐾𝑝(𝑥).

In order to exclude the case when such sets 𝐾𝑝(𝑥) are

empty for 𝑥 ∈ 𝐿Φ,𝑝\{0}, in the sequel we will assume, without

any special mention, that lim𝑢→∞

(Φ(𝑡, 𝑢)/𝑢) = ∞ for 𝜇-a.e.𝑡 ∈ 𝑇.

Theorem 3. Assume 1 ≤ 𝑝 < ∞, 𝑥 ∈ 𝑆(𝐿Φ,𝑝). Then the

following assertions are equivalent:

(1) 𝑥 is a complex strongly extreme point of the unit ball𝐵(𝐿Φ,𝑝);

(2) 𝑥 is a complex extreme point of the unit ball 𝐵(𝐿Φ,𝑝);

(3) for any 𝑘 ∈ 𝐾𝑝(𝑥), 𝜇{𝑡 ∈ 𝑇 : 𝑘|𝑥(𝑡)| < 𝑒(𝑡)} = 0.

Proof. The implication (1) ⇒ (2) is trivial. Let 𝑥 be a complexextreme point of the unit ball 𝐵(𝐿

Φ,𝑝) and there exists 𝑘

0∈

𝐾𝑝(𝑥) such that 𝜇{𝑡 ∈ 𝑇 : 𝑘

0|𝑥(𝑡)| < 𝑒(𝑡)} > 0. Then we can

Abstract and Applied Analysis 3

find 𝑑 > 0 such that 𝜇{𝑡 ∈ 𝑇 : 𝑘0|𝑥(𝑡)| + 𝑑 < 𝑒(𝑡)} > 0.

Indeed, if 𝜇{𝑡 ∈ 𝑇 : 𝑘0|𝑥(𝑡)| + 𝑑 < 𝑒(𝑡)} = 0 for any 𝑑 > 0,

then we set 𝑑𝑛= 1/𝑛 and define the subsets

𝑇𝑛= {𝑡 ∈ 𝑇 : 𝑘

0 |𝑥 (𝑡)| + 𝑑𝑛 < 𝑒 (𝑡)} (16)

for each 𝑛 ∈ N. Notice that 𝑇1⊆ 𝑇2⊆ ⋅ ⋅ ⋅ ⊆ 𝑇

𝑛⊆ ⋅ ⋅ ⋅ and

{𝑡 ∈ 𝑇 : 𝑘0|𝑥(𝑡)| ≤ 𝑒(𝑡)} = ⋃

∞

𝑛=1𝑇𝑛; we can find that

𝜇 {𝑡 ∈ 𝑇 : 𝑘0 |𝑥 (𝑡)| ≤ 𝑒 (𝑡)} = 0 (17)

which is a contradiction.Let 𝑇0= {𝑡 ∈ 𝑇 : 𝑘

0|𝑥(𝑡)| + 𝑑 < 𝑒(𝑡)} and define 𝑦 =

(𝑑/𝑘0)𝜒𝑇0; we have 𝑦 ̸= 0 and for any 𝜆 ∈ C with |𝜆| ≤ 1,

𝑥 + 𝜆𝑦Φ,𝑝 ≤

1

𝑘0

{1 + 𝐼𝑝

Φ(𝑘0(𝑥 + 𝜆𝑦))}

1/𝑝

=1

𝑘0

{1 + [𝐼Φ(𝑘0𝑥𝜒𝑇\𝑇0

)

+𝐼Φ(𝑘0𝑥𝜒𝑇0+ 𝜆𝑑𝜒

𝑇0)]𝑝

}1/𝑝

≤1

𝑘0

{1 + [𝐼Φ(𝑘0𝑥𝜒𝑇\𝑇0

)

+𝐼Φ((𝑘0 |𝑥| + 𝑑) 𝜒𝑇0

)]𝑝

}1/𝑝

=1

𝑘0

{1 + [𝐼Φ(𝑘0𝑥𝜒𝑇\𝑇0

)]𝑝

}1/𝑝

=1

𝑘0

(1 + 𝐼𝑝

Φ(𝑘0𝑥))1/𝑝

= 1,

(18)

which shows that 𝑥 is not a complex extreme point of the unitball 𝐵(𝐿

Φ,𝑝).

(3) ⇒ (1). Suppose that 𝑥 ∈ 𝑆(𝐿Φ,𝑝) is not a complex

strongly extreme point of the unit ball 𝐵(𝐿Φ,𝑝),; then there

exists 𝜀0> 0 such thatΔ

𝑐(𝑥, 𝜀0) = 0.That is, there exist𝜆

𝑛∈ C

with |𝜆𝑛| → 1 and𝑦

𝑛∈ 𝐿Φ,𝑝

satisfying ‖𝑦𝑛‖Φ,𝑝

≥ 𝜀0such that

𝜆𝑛𝑥0 ± 𝑦𝑛Φ,𝑝 ≤ 1,

𝜆𝑛𝑥0 ± 𝑖𝑦𝑛Φ,𝑝 ≤ 1, (19)

which gives𝑥0±𝑦𝑛

𝜆𝑛

Φ,𝑝

≤1𝜆𝑛

,

𝑥0± 𝑖𝑦𝑛

𝜆𝑛

Φ,𝑝

≤1𝜆𝑛

. (20)

Setting 𝑧𝑛= 𝑦𝑛/𝜆𝑛, we have𝑧𝑛Φ,𝑝 ≥

𝑦𝑛Φ,𝑝 ≥ 𝜀0,

𝑥0 ± 𝑧𝑛Φ,𝑝 ≤

1𝜆𝑛

,

𝑥0 ± 𝑖𝑧𝑛Φ,𝑝 ≤

1𝜆𝑛

.

(21)

For the above 𝜀0> 0, by Lemma 1, there exists 𝛿

0∈

(0, 1/2) such that if 𝑢, V ∈ C and

|V| ≥𝜀0

8max𝑒|𝑢 + 𝑒V| , (22)

then

|𝑢| ≤1 − 2𝛿

0

4Σ𝑒 |𝑢 + 𝑒V| . (23)

For each 𝑛 ∈ N, let

𝐴𝑛= {𝑡 ∈ 𝑇 :

𝑧𝑛 (𝑡) ≥

𝜀0

8max𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡)} ,

𝑧(1)

𝑛: 𝑧(1)

𝑛(𝑡) = 𝑧

𝑛(𝑡) (𝑡 ∉ 𝐴

𝑛) , 𝑧

(1)

𝑛(𝑡) = 0 (𝑡 ∈ 𝐴

𝑛) ,

𝑧(2)

𝑛: 𝑧(2)

𝑛(𝑡) = 0 (𝑡 ∉ 𝐴

𝑛) , 𝑧

(2)

𝑛(𝑡) = 𝑧

𝑛(𝑡) (𝑡 ∈ 𝐴

𝑛) .

(24)

It is easy to see that 𝑧𝑛= 𝑧(1)

𝑛+𝑧(2)

𝑛,∀𝑛 ∈ N. Since |𝜆

𝑛| → 1

when 𝑛 → ∞, the following inequalities

𝑧(1)

𝑛

Φ,𝑝

= inf𝑘>0

1

𝑘{1 + [∫

𝑡∉𝐴𝑛

Φ(𝑡, 𝑘𝑧𝑛 (𝑡)

) 𝑑𝑡]

𝑝

}

1/𝑝

≤ inf𝑘>0

1

𝑘{1+

[∫𝑡∉𝐴𝑛

Φ(𝑡, 𝑘𝜀0

8max𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡)) 𝑑𝑡]

𝑝

}

1/𝑝

≤𝜀0

8

max𝑒

𝑥0 + 𝑒𝑧𝑛

Φ,𝑝

≤𝜀0

8

Σ𝑒𝑥0 + 𝑒𝑧𝑛

Φ,𝑝

≤𝜀0

2𝜆𝑛

<3𝜀0

4

(25)

hold for 𝑛 large enough.Therefore, ‖𝑧(2)

𝑛‖Φ,𝑝

> 𝜀0/4 which shows that 𝜇(𝐴

𝑛) > 0

for 𝑛 large enough. Furthermore, for any 𝑡 ∈ 𝐴𝑛, we have

𝑥0 (𝑡) ≤

1 − 2𝛿0

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡) .

(26)

To complete the proof, we consider the following twocases.

(I) One has 𝑘𝑛→ ∞(𝑛 → ∞), where 𝑘

𝑛∈ 𝐾𝑝((1/4)

Σ𝑒|𝑥0+ 𝑒𝑧𝑛|).

4 Abstract and Applied Analysis

For each 𝑛 ∈ N, we get

1 =𝑥0

Φ,𝑝

≤1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛𝑥0)}1/𝑝

=1

𝑘𝑛

{1 + [∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛

𝑥0 (𝑡)) 𝑑𝑡

+∫𝑡∉𝐴𝑛

Φ(𝑡, 𝑘𝑛

𝑥0 (𝑡)) 𝑑𝑡]

𝑝

}

1/𝑝

≤1

𝑘𝑛

{1 + [∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛(1 − 2𝛿

0

4

×Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡) )) 𝑑𝑡

+ ∫𝑡∉𝐴𝑛

Φ(𝑡, 𝑘𝑛

× (1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡]

𝑝

}

1/𝑝

≤1

𝑘𝑛

{1

+ [ (1 − 2𝛿0)

× ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡

+ ∫𝑡∉𝐴𝑛

Φ(𝑡, 𝑘𝑛

×(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡]

𝑝

}

1/𝑝

=1

𝑘𝑛

{1

+ [𝐼Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛)) − 2𝛿0

× ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛

×(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡]

𝑝

}

1/𝑝

.

(27)

Furthermore, we notice that

∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡

≥ ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛

𝑧𝑛 (𝑡)) 𝑑𝑡

≥ [𝑘𝑝

𝑛

𝑧(2)

𝑛

𝑝

Φ,𝑝− 1]1/𝑝

≥ [(𝜀0

4𝑘𝑛)

𝑝

− 1]

1/𝑝

.

(28)

Since 𝑘𝑛→ ∞when 𝑛 → ∞, we can see that the inequality

((𝜀0/4)𝑘𝑛)𝑝− 1 > 0 holds for 𝑛 large enough. Moreover, we

find that

𝐼Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛))

≥ 𝐼Φ(𝑘𝑛𝑥0)

≥ (𝑘𝑝

𝑛− 1)1/𝑝

≥ 2𝛿0((𝜀0

4𝑘𝑛)

𝑝

− 1)

1/𝑝

.

(29)

Then we deduce

1 =𝑥0

Φ,𝑝

≤1

𝑘𝑛

{1 + [𝐼Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛)) − 2𝛿0

× ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛

×(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡]

𝑝

}

1/𝑝

≤1

𝑘𝑛

{1 + [𝐼Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛))

−2𝛿0((𝜀0

4𝑘𝑛)

𝑝

− 1)

1/𝑝

]

𝑝

}

1/𝑝

<1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛))}

1/𝑝

= 1,

(30)

a contradiction.(II) One has 𝑘

𝑛→ 𝑘

0(𝑛 → ∞), where 𝑘

𝑛∈

𝐾𝑝((1/4)Σ

𝑒|𝑥0+ 𝑒𝑧𝑛|), 𝑘0∈ R.

Then for each 𝑛 ∈ N,

1

𝜆𝑛

≥

1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛

Φ,𝑝

=1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛)}

1/𝑝

≥1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛𝑥0)}1/𝑝

≥𝑥0

Φ,𝑝 = 1.

(31)

Let 𝑛 → ∞, we deduce that 1 = (1/𝑘0){1 + 𝐼

𝑝

Φ(𝑘0𝑥0)}1/𝑝

=

‖𝑥0‖Φ,𝑝

.

Abstract and Applied Analysis 5

From (I), we obtain that

1 =‖ 𝑥0‖Φ,𝑝

≤1

𝑘𝑛

{1+ [𝐼Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛)) − 2𝛿0

× ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛

× (1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡]

𝑝

}

1/𝑝

.

(32)

Now we consider the following two subcases.(a) Consider ∫

𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛((1/4)Σ

𝑒|𝑥0(𝑡) + 𝑒𝑧

𝑛(𝑡)|))𝑑𝑡 > 0

for some 𝑛 ∈ N.Then we obtain

1 =‖ 𝑥0‖Φ,𝑝

≤1

𝑘𝑛

{1 + [𝐼Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛)) − 2𝛿0

× ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛

×(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡]

𝑝

}

1/𝑝

<1

𝑘𝑛

{1 + 𝐼𝑝

Φ(𝑘𝑛(1

4Σ𝑒

𝑥0 + 𝑒𝑧𝑛))}

1/𝑝

= 1,

(33)

a contradiction.(b) Consider ∫

𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛((1/4)Σ

𝑒|𝑥0(𝑡) + 𝑒𝑧

𝑛(𝑡)|))𝑑𝑡 = 0

for any 𝑛 ∈ N.For 𝑛 large enough, we observe that

𝑘𝑛>1 − 2𝛿

0

1 − 𝛿0

𝑘0. (34)

Hence, we get

0 = ∫𝑡∈𝐴𝑛

Φ(𝑡, 𝑘𝑛(1

4Σ𝑒

𝑥0 (𝑡) + 𝑒𝑧𝑛 (𝑡))) 𝑑𝑡

≥ ∫𝑡∈𝐴𝑛

Φ(𝑡,

𝑘𝑛

1 − 2𝛿0

𝑥0(𝑡)

) 𝑑𝑡

≥ ∫𝑡∈𝐴𝑛

Φ(𝑡,

𝑘0

1 − 𝛿0

𝑥0(𝑡)

) 𝑑𝑡 ≥ 0.

(35)

Thus, we see the equality∫𝑡∈𝐴𝑛

Φ(𝑡, |(𝑘0/(1−𝛿

0))𝑥0(𝑡)|) 𝑑𝑡 = 0

holds for 𝑛 large enough. It follows that

𝑘0

1 − 𝛿0

𝑥0(𝑡)

≤ 𝑒 (𝑡) , 𝜇-a.e. 𝑡 ∈ 𝐴

𝑛 (36)

since 𝜇(𝐴𝑛) > 0 for 𝑛 large enough.

On the other hand, by (3), we deduce that

𝑘0𝑥0 (𝑡) ≥ 𝑒 (𝑡) , 𝜇-a.e. 𝑡 ∈ 𝐴𝑛. (37)

Hence, we get a contradiction:

𝑘0

1 − 𝛿0

𝑥0(𝑡)

≥

𝑒 (𝑡)

1 − 𝛿0

> 𝑒 (𝑡) , 𝜇-a.e. 𝑡 ∈ 𝐴𝑛. (38)

Theorem 4. Assume 1 ≤ 𝑝 < ∞; then the following assertionsare equivalent:

(1) 𝐿Φ,𝑝

is complex midpoint locally uniformly convex;

(2) 𝐿Φ,𝑝

is complex strictly convex;

(3) 𝑒(𝑡) = 0 for 𝜇-a.e. 𝑡 ∈ 𝑇.

Proof. The implication (1) ⇒ (2) is trivial. Now assume that𝐿Φ,𝑝

is complex strictly convex. If 𝜇{𝑡 ∈ 𝑇 : 𝑒(𝑡) > 0} > 0, let𝑇0= {𝑡 ∈ 𝑇 : 𝑒(𝑡) > 0} and it is not difficult to find an element

𝑥 ∈ 𝑆(𝐿Φ,𝑝) satisfying supp 𝑥 = 𝑇 \ 𝑇

0. Take 𝑘 ∈ 𝐾

𝑝(𝑥), and

define

𝑦 (𝑡) ={

{

{

𝑒 (𝑡)

2𝑘for 𝑡 ∈ 𝑇

0,

𝑥 (𝑡) for 𝑡 ∈ 𝑇 \ 𝑇0.

(39)

Obviously, ‖𝑦‖Φ,𝑝

≥ ‖𝑥‖Φ,𝑝

= 1. On the other hand,

𝑦Φ,𝑝 ≤

1

𝑘{1 + [∫

𝑡∈𝑇0

Φ(𝑡,𝑒 (𝑡)

2) 𝑑𝑡

+∫𝑇\𝑇0

Φ (𝑡, 𝑘𝑥 (𝑡)) 𝑑𝑡]

𝑝

}

1/𝑝

=1

𝑘(1 + 𝐼

𝑝

Φ(𝑘𝑥))1/𝑝

= 1.

(40)

Thus, ‖𝑦‖Φ,𝑝

= (1/𝑘)(1 + 𝐼𝑝

Φ(𝑘𝑦))1/𝑝

= 1. However, for 𝑡 ∈𝑇0, we find 𝑘|𝑦(𝑡)| = 𝑒(𝑡)/2 < 𝑒(𝑡), which implies 𝑦 ∈ 𝑆(𝐿

Φ,𝑝)

is not a complex extreme point of 𝐵(𝐿Φ,𝑝) fromTheorem 3.

(3) ⇒ (1). Suppose that 𝑥 ∈ 𝑆(𝐿Φ,𝑝) is not a complex

strongly extreme point of 𝐵(𝐿Φ,𝑝). It follows fromTheorem 3

that 𝜇{𝑡 ∈ 𝑇 : 𝑘0|𝑥(𝑡)| < 𝑒(𝑡)} > 0 for some 𝑘

0∈ 𝐾𝑝(𝑥),

consequently 𝜇{𝑡 ∈ 𝑇 : 𝑒(𝑡) > 0} > 0which is a contradiction.

Remark 5. If 𝑝 = ∞ then 𝑝-Amemiya norm equals Luxem-burg norm, the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8].

Conflict of Interests

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

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (no.QC2013C001), Natural Science Foundation of HeilongjiangEducational Committee (no. 12531099), Youth Science Fundof Harbin University of Science and Technology (no.2011YF002), and Tianyuan Funds of the National NaturalScience Foundation of China (no. 11226127).

References

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[2] C. Choi, A. Kamińska, and H. J. Lee, “Complex convexity ofOrlicz-Lorentz spaces and its applications,” Bulletin of the PolishAcademy of Sciences:Mathematics, vol. 52, no. 1, pp. 19–38, 2004.

[3] H. Hudzik and A. Narloch, “Relationships betweenmonotonic-ity and complex rotundity properties with some consequences,”Mathematica Scandinavica, vol. 96, no. 2, pp. 289–306, 2005.

[4] H. J. Lee, “Monotonicity and complex convexity in Banachlattices,” Journal of Mathematical Analysis and Applications, vol.307, no. 1, pp. 86–101, 2005.

[5] H. J. Lee, “Complex convexity and monotonicity in Quasi-Banach lattices,” Israel Journal of Mathematics, vol. 159, no. 1, pp.57–91, 2007.

[6] M. M. Czerwińska and A. Kamińska, “Complex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operators,” Studia Mathematica, vol. 201, no. 3, pp.253–285, 2010.

[7] E. Thorp and R. Whitley, “The strong maximum modulustheorem for analytic functions into a Banach space,” Proceedingsof the AmericanMathematical Society, vol. 18, pp. 640–646, 1967.

[8] L. Chen, Y. Cui, and H. Hudzik, “Criteria for complex stronglyextreme points of Musielak-Orlicz function spaces,” NonlinearAnalysis: Theory, Methods & Applications, vol. 70, no. 6, pp.2270–2276, 2009.

[9] S. Chen, Geometry of Orlicz Spaces, Dissertationes Mathemati-cae, 1996.

[10] Y. Cui, L. Duan, H. Hudzik, and M. Wisła, “Basic theory of p-Amemiya norm in Orlicz spaces (1 ≤ 𝑝 ≤ ∞): extreme pointsand rotundity in Orlicz spaces endowed with these norms,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no.5-6, pp. 1796–1816, 2008.

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