LiliChen,YunanCui,andYanfengZhao · Research Article Complex Convexity of Musielak-Orlicz Function...
Transcript of LiliChen,YunanCui,andYanfengZhao · Research Article Complex Convexity of Musielak-Orlicz Function...
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Research ArticleComplex Convexity of Musielak-Orlicz Function SpacesEquipped with the ð-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; [email protected]
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 ð-Amemiya norm is mainly discussed. It is obtainedthat, for any Musielak-Orlicz function space equipped with the ð-Amemiya norm when 1 †ð < â, 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 (ð, â â â) be a complex Banach space over the complexfield C, let ð be the complex number satisfying ð2 = â1, andlet ðµ(ð) and ð(ð) be the closed unit ball and the unit sphereof ð, 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, ð¶â-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 [1â6]).
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 ð¥ â ð(ð) is said to be a complex extreme point ofðµ(ð) if for every nonzeroðŠ â ð there holds sup
|ð|â€1âð¥+ððŠâ >
1. A complex Banach space ð is said to be complex strictlyconvex if every element of ð(ð) is a complex extreme pointof ðµ(ð).
In [8], we further studied the notions of complex stronglyextreme point and complex midpoint locally uniform con-vexity in general complex spaces.
A point ð¥ â ð(ð) is said to be a complex strongly extremepoint of ðµ(ð) if for every ð > 0 we have Î
ð(ð¥, ð) > 0, where
Îð(ð¥, ð) = inf {1 â |ð| : âðŠ â ðs.t. ð𥠱 ðŠ
†1,
ð𥠱 ððŠ †1,
ðŠ ⥠ð} .
(1)
A complex Banach space ð is said to be complex midpointlocally uniformly convex if every element of ð(ð) is a complexstrongly extreme point of ðµ(ð).
Let (ð, Σ, ð) be a nonatomic and complete measure spacewith ð(ð) < â. By Ί we denote a Musielak-Orlicz function;that is, Ί : ð à [0, +â) â [0, +â] satisfies the following:
(1) for each ð¢ â [0,â], Ί(ð¡, ð¢) is a ð-measurablefunction of ð¡ on ð;
(2) for ð-a.e. ð¡ â ð,Ί(ð¡, 0) = 0, limð¢ââ
Ί(ð¡, ð¢) = â andthere exists ð¢ > 0 such thatΊ(ð¡, ð¢) < â;
(3) for ð-a.e. ð¡ â ð, Ί(ð¡, ð¢) is convex on the interval[0,â) with respect to ð¢.
Let ð¿ð(ð) be the space of all ð-equivalence classes ofcomplex and Σ-measurable functions defined on ð. For eachð¥ â ð¿ð(ð), we define on ð¿ð(ð) the convex modular of ð¥ by
ðŒÎŠ(ð¥) = â«
ð
Ί (ð¡, |ð¥ (ð¡)|) ðð¡. (2)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 190203, 6 pageshttp://dx.doi.org/10.1155/2014/190203
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2 Abstract and Applied Analysis
We define supp ð¥ = {ð¡ â ð : |ð¥(ð¡)| Ìž= 0} and the Musielak-Orlicz space ð¿
Ίgenerated by the formula
ð¿ÎŠ= {ð¥ â ð¿
ð(ð) : ðŒ
Ί(ðð¥) < â for some ð > 0} . (3)
Set
ð (ð¡) = sup {ð¢ ⥠0 : Ί (ð¡, ð¢) = 0} ,
ðž (ð¡) = sup {ð¢ ⥠0 : Ί (ð¡, ð¢) < â} ;(4)
then ð(ð¡) and ðž(ð¡) are ð-measurable (see [9]).The notion of ð-Amemiya norm has been introduced in
[10]; for any 1 †ð †â and ð¢ > 0, define
ð ð(ð¢) = {
(1 + ð¢ð)1/ð
, for 1 †ð < â,max {1, ð¢} , for ð = â.
(5)
Furthermore, define ð Ί,ð(ð¥) = ð
ðâðŒÎŠ(ð¥) for all 1 †ð †â
and ð¥ â ð¿ð(ð). Notice that the function ð ðis increasing on ð
+
for 1 †ð < â; however, the function ð âis increasing on the
interval [1,â) only.For ð¥ â ð¿ð(ð), define the ð-Amemiya norm by the
formula
âð¥âΊ,ð = infð>0
1
ðð Ί,ð(ðð¥) , 1 †ð †â. (6)
For convenience, from now on, we write ð¿ÎŠ,ð
=
(ð¿ÎŠ, â â âΊ,ð); it is easy to see that ð¿
Ί,ðis a Banach space. For
1 †ð < â, ð¥ â ð¿ÎŠ,ð
, set
ðŸð(ð¥) = {ð > 0 : âð¥âΊ,ð =
1
ð{1 + ðŒ
ð
Ί(ðð¥)}1/ð
} . (7)
2. Main Results
We begin this section from the following useful lemmas.
Lemma 1 (see [9]). For any ð > 0, there exists ð¿ â (0, 1/2)such that if ð¢, V â C and
|V| â¥ð
8maxð|ð¢ + ðV| , (8)
then
|ð¢| â€1 â 2ð¿
4Σð |ð¢ + ðV| , (9)
where
maxð|ð¢ + ðV| = max {|ð¢ + V| , |ð¢ â V| , |ð¢ + ðV| , |ð¢ â ðV|} ,
Σð |ð¢ + ðV| = |ð¢ + V| + |ð¢ â V| + |ð¢ + ðV| + |ð¢ â ðV| .
(10)
Lemma 2. If limð¢ââ
(Ί(ð¡, ð¢)/ð¢) = â for ð-a.e. ð¡ â ð, thenðŸð(ð¥) Ìž= 0 for any ð¥ â ð¿
Ί,ð\ {0}, where 1 †ð < â.
Proof. For any ð¥ â ð¿ÎŠ,ð
\ {0}, there exists ðŒ > 0 such thatð{ð¡ â ð : |ð¥(ð¡)| ⥠ðŒ} > 0. Let ð
1= {ð¡ â ð : |ð¥(ð¡)| ⥠ðŒ} and
define the subsets
ð¹ð= {ð¡ â ð
1:Ί (ð¡, ð¢)
ð¢â¥4âð¥âΊ,ð
ðŒ â ðð1
, ð¢ ⥠ð} (11)
for each ð â N. It is easy to see that ð¹1â F2â â â â â ð¹
ðâ â â â
and limðââ
ðð¹ð= ðð1. Then there exists ð
0â N such that
ðð¹ð0> (1/2)ðð
1.
It follows from the definition of ð-Amemiya norm thatthere is a sequence {ð
ð} satisfying
âð¥âΊ,ð = limðââ
1
ðð
{1 + ðŒð
Ί(ððð¥)}1/ð
. (12)
For any ð > 0, we have
1
ð{1 + ðŒ
ð
Ί(ðð¥)}1/ð
=1
ð{1 + [â«
ð
Ί (ð¡, ð |ð¥ (ð¡)|) ðð¡]
ð
}
1/ð
â¥
â«ð¹ð0
Ί (ð¡, ð |ð¥ (ð¡)|) ðð¡
ð
⥠â«ð¹ð0
Ί (ð¡, ððŒ)
ððð¡
= ðŒâ«ð¹ð0
Ί (ð¡, ððŒ)
ððŒðð¡.
(13)
If ð > ð0/ðŒ, notice that
1
ð{1 + ðŒ
ð
Ί(ðð¥)}1/ð
⥠ðŒ â 4âð¥âΊ,ð
ðŒ â ðð1
â ðð¹ð0> 2âð¥âΊ,ð, (14)
which means the sequence {ðð} is bounded. Hence, without
loss of generality, we assume that ððâ ð0as ð â â.
We can also choose the monotonic increasing or decreasingsubsequence of {ð
ð} that converges to the number ð
0. Apply-
ing Levi Theorem and Lebesgue Dominated ConvergenceTheorem, we obtain
âð¥âΊ,ð = limðââ
1
ðð
{1 + ðŒð
Ί(ððð¥)}1/ð
=1
ð0
{1 + ðŒð
Ί(ð0ð¥)}1/ð
(15)
which implies ð0â ðŸð(ð¥).
In order to exclude the case when such sets ðŸð(ð¥) are
empty for ð¥ â ð¿ÎŠ,ð\{0}, in the sequel we will assume, without
any special mention, that limð¢ââ
(Ί(ð¡, ð¢)/ð¢) = â for ð-a.e.ð¡ â ð.
Theorem 3. Assume 1 †ð < â, ð¥ â ð(ð¿ÎŠ,ð). Then the
following assertions are equivalent:
(1) ð¥ is a complex strongly extreme point of the unit ballðµ(ð¿ÎŠ,ð);
(2) ð¥ is a complex extreme point of the unit ball ðµ(ð¿ÎŠ,ð);
(3) for any ð â ðŸð(ð¥), ð{ð¡ â ð : ð|ð¥(ð¡)| < ð(ð¡)} = 0.
Proof. The implication (1) â (2) is trivial. Let ð¥ be a complexextreme point of the unit ball ðµ(ð¿
Ί,ð) and there exists ð
0â
ðŸð(ð¥) such that ð{ð¡ â ð : ð
0|ð¥(ð¡)| < ð(ð¡)} > 0. Then we can
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Abstract and Applied Analysis 3
find ð > 0 such that ð{ð¡ â ð : ð0|ð¥(ð¡)| + ð < ð(ð¡)} > 0.
Indeed, if ð{ð¡ â ð : ð0|ð¥(ð¡)| + ð < ð(ð¡)} = 0 for any ð > 0,
then we set ðð= 1/ð and define the subsets
ðð= {ð¡ â ð : ð
0 |ð¥ (ð¡)| + ðð < ð (ð¡)} (16)
for each ð â N. Notice that ð1â ð2â â â â â ð
ðâ â â â and
{ð¡ â ð : ð0|ð¥(ð¡)| †ð(ð¡)} = â
â
ð=1ðð; we can find that
ð {ð¡ â ð : ð0 |ð¥ (ð¡)| †ð (ð¡)} = 0 (17)
which is a contradiction.Let ð0= {ð¡ â ð : ð
0|ð¥(ð¡)| + ð < ð(ð¡)} and define ðŠ =
(ð/ð0)ðð0; we have ðŠ Ìž= 0 and for any ð â C with |ð| †1,
ð¥ + ððŠÎŠ,ð â€
1
ð0
{1 + ðŒð
Ί(ð0(ð¥ + ððŠ))}
1/ð
=1
ð0
{1 + [ðŒÎŠ(ð0ð¥ðð\ð0
)
+ðŒÎŠ(ð0ð¥ðð0+ ððð
ð0)]ð
}1/ð
â€1
ð0
{1 + [ðŒÎŠ(ð0ð¥ðð\ð0
)
+ðŒÎŠ((ð0 |ð¥| + ð) ðð0
)]ð
}1/ð
=1
ð0
{1 + [ðŒÎŠ(ð0ð¥ðð\ð0
)]ð
}1/ð
=1
ð0
(1 + ðŒð
Ί(ð0ð¥))1/ð
= 1,
(18)
which shows that ð¥ is not a complex extreme point of the unitball ðµ(ð¿
Ί,ð).
(3) â (1). Suppose that ð¥ â ð(ð¿ÎŠ,ð) is not a complex
strongly extreme point of the unit ball ðµ(ð¿ÎŠ,ð),; then there
exists ð0> 0 such thatÎ
ð(ð¥, ð0) = 0.That is, there existð
ðâ C
with |ðð| â 1 andðŠ
ðâ ð¿ÎŠ,ð
satisfying âðŠðâΊ,ð
⥠ð0such that
ððð¥0 ± ðŠðΊ,ð †1,
ððð¥0 ± ððŠðΊ,ð †1, (19)
which givesð¥0±ðŠð
ðð
Ί,ð
â€1ðð
,
ð¥0± ððŠð
ðð
Ί,ð
â€1ðð
. (20)
Setting ð§ð= ðŠð/ðð, we haveð§ðΊ,ð â¥
ðŠðΊ,ð ⥠ð0,
ð¥0 ± ð§ðΊ,ð â€
1ðð
,
ð¥0 ± ðð§ðΊ,ð â€
1ðð
.
(21)
For the above ð0> 0, by Lemma 1, there exists ð¿
0â
(0, 1/2) such that if ð¢, V â C and
|V| â¥ð0
8maxð|ð¢ + ðV| , (22)
then
|ð¢| â€1 â 2ð¿
0
4Σð |ð¢ + ðV| . (23)
For each ð â N, let
ðŽð= {ð¡ â ð :
ð§ð (ð¡) â¥
ð0
8maxð
ð¥0 (ð¡) + ðð§ð (ð¡)} ,
ð§(1)
ð: ð§(1)
ð(ð¡) = ð§
ð(ð¡) (ð¡ â ðŽ
ð) , ð§
(1)
ð(ð¡) = 0 (ð¡ â ðŽ
ð) ,
ð§(2)
ð: ð§(2)
ð(ð¡) = 0 (ð¡ â ðŽ
ð) , ð§
(2)
ð(ð¡) = ð§
ð(ð¡) (ð¡ â ðŽ
ð) .
(24)
It is easy to see that ð§ð= ð§(1)
ð+ð§(2)
ð,âð â N. Since |ð
ð| â 1
when ð â â, the following inequalities
ð§(1)
ð
Ί,ð
= infð>0
1
ð{1 + [â«
ð¡âðŽð
Ί(ð¡, ðð§ð (ð¡)
) ðð¡]
ð
}
1/ð
†infð>0
1
ð{1+
[â«ð¡âðŽð
Ί(ð¡, ðð0
8maxð
ð¥0 (ð¡) + ðð§ð (ð¡)) ðð¡]
ð
}
1/ð
â€ð0
8
maxð
ð¥0 + ðð§ð
Ί,ð
â€ð0
8
Σðð¥0 + ðð§ð
Ί,ð
â€ð0
2ðð
<3ð0
4
(25)
hold for ð large enough.Therefore, âð§(2)
ðâΊ,ð
> ð0/4 which shows that ð(ðŽ
ð) > 0
for ð large enough. Furthermore, for any ð¡ â ðŽð, we have
ð¥0 (ð¡) â€
1 â 2ð¿0
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡) .
(26)
To complete the proof, we consider the following twocases.
(I) One has ððâ â(ð â â), where ð
ðâ ðŸð((1/4)
Σð|ð¥0+ ðð§ð|).
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4 Abstract and Applied Analysis
For each ð â N, we get
1 =ð¥0
Ί,ð
â€1
ðð
{1 + ðŒð
Ί(ððð¥0)}1/ð
=1
ðð
{1 + [â«ð¡âðŽð
Ί(ð¡, ðð
ð¥0 (ð¡)) ðð¡
+â«ð¡âðŽð
Ί(ð¡, ðð
ð¥0 (ð¡)) ðð¡]
ð
}
1/ð
â€1
ðð
{1 + [â«ð¡âðŽð
Ί(ð¡, ðð(1 â 2ð¿
0
4
ÃΣð
ð¥0 (ð¡) + ðð§ð (ð¡) )) ðð¡
+ â«ð¡âðŽð
Ί(ð¡, ðð
à (1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡]
ð
}
1/ð
â€1
ðð
{1
+ [ (1 â 2ð¿0)
à â«ð¡âðŽð
Ί(ð¡, ðð(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡
+ â«ð¡âðŽð
Ί(ð¡, ðð
Ã(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡]
ð
}
1/ð
=1
ðð
{1
+ [ðŒÎŠ(ðð(1
4Σð
ð¥0 + ðð§ð)) â 2ð¿0
à â«ð¡âðŽð
Ί(ð¡, ðð
Ã(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡]
ð
}
1/ð
.
(27)
Furthermore, we notice that
â«ð¡âðŽð
Ί(ð¡, ðð(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡
⥠â«ð¡âðŽð
Ί(ð¡, ðð
ð§ð (ð¡)) ðð¡
⥠[ðð
ð
ð§(2)
ð
ð
Ί,ðâ 1]1/ð
⥠[(ð0
4ðð)
ð
â 1]
1/ð
.
(28)
Since ððâ âwhen ð â â, we can see that the inequality
((ð0/4)ðð)ðâ 1 > 0 holds for ð large enough. Moreover, we
find that
ðŒÎŠ(ðð(1
4Σð
ð¥0 + ðð§ð))
⥠ðŒÎŠ(ððð¥0)
⥠(ðð
ðâ 1)1/ð
⥠2ð¿0((ð0
4ðð)
ð
â 1)
1/ð
.
(29)
Then we deduce
1 =ð¥0
Ί,ð
â€1
ðð
{1 + [ðŒÎŠ(ðð(1
4Σð
ð¥0 + ðð§ð)) â 2ð¿0
à â«ð¡âðŽð
Ί(ð¡, ðð
Ã(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡]
ð
}
1/ð
â€1
ðð
{1 + [ðŒÎŠ(ðð(1
4Σð
ð¥0 + ðð§ð))
â2ð¿0((ð0
4ðð)
ð
â 1)
1/ð
]
ð
}
1/ð
<1
ðð
{1 + ðŒð
Ί(ðð(1
4Σð
ð¥0 + ðð§ð))}
1/ð
= 1,
(30)
a contradiction.(II) One has ð
ðâ ð
0(ð â â), where ð
ðâ
ðŸð((1/4)Σ
ð|ð¥0+ ðð§ð|), ð0â R.
Then for each ð â N,
1
ðð
â¥
1
4Σð
ð¥0 + ðð§ð
Ί,ð
=1
ðð
{1 + ðŒð
Ί(ðð
4Σð
ð¥0 + ðð§ð)}
1/ð
â¥1
ðð
{1 + ðŒð
Ί(ððð¥0)}1/ð
â¥ð¥0
Ί,ð = 1.
(31)
Let ð â â, we deduce that 1 = (1/ð0){1 + ðŒ
ð
Ί(ð0ð¥0)}1/ð
=
âð¥0âΊ,ð
.
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Abstract and Applied Analysis 5
From (I), we obtain that
1 =â ð¥0âΊ,ð
â€1
ðð
{1+ [ðŒÎŠ(ðð(1
4Σð
ð¥0 + ðð§ð)) â 2ð¿0
à â«ð¡âðŽð
Ί(ð¡, ðð
à (1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡]
ð
}
1/ð
.
(32)
Now we consider the following two subcases.(a) Consider â«
ð¡âðŽð
Ί(ð¡, ðð((1/4)Σ
ð|ð¥0(ð¡) + ðð§
ð(ð¡)|))ðð¡ > 0
for some ð â N.Then we obtain
1 =â ð¥0âΊ,ð
â€1
ðð
{1 + [ðŒÎŠ(ðð(1
4Σð
ð¥0 + ðð§ð)) â 2ð¿0
à â«ð¡âðŽð
Ί(ð¡, ðð
Ã(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡]
ð
}
1/ð
<1
ðð
{1 + ðŒð
Ί(ðð(1
4Σð
ð¥0 + ðð§ð))}
1/ð
= 1,
(33)
a contradiction.(b) Consider â«
ð¡âðŽð
Ί(ð¡, ðð((1/4)Σ
ð|ð¥0(ð¡) + ðð§
ð(ð¡)|))ðð¡ = 0
for any ð â N.For ð large enough, we observe that
ðð>1 â 2ð¿
0
1 â ð¿0
ð0. (34)
Hence, we get
0 = â«ð¡âðŽð
Ί(ð¡, ðð(1
4Σð
ð¥0 (ð¡) + ðð§ð (ð¡))) ðð¡
⥠â«ð¡âðŽð
Ί(ð¡,
ðð
1 â 2ð¿0
ð¥0(ð¡)
) ðð¡
⥠â«ð¡âðŽð
Ί(ð¡,
ð0
1 â ð¿0
ð¥0(ð¡)
) ðð¡ ⥠0.
(35)
Thus, we see the equalityâ«ð¡âðŽð
Ί(ð¡, |(ð0/(1âð¿
0))ð¥0(ð¡)|) ðð¡ = 0
holds for ð large enough. It follows that
ð0
1 â ð¿0
ð¥0(ð¡)
†ð (ð¡) , ð-a.e. ð¡ â ðŽ
ð (36)
since ð(ðŽð) > 0 for ð large enough.
On the other hand, by (3), we deduce that
ð0ð¥0 (ð¡) ⥠ð (ð¡) , ð-a.e. ð¡ â ðŽð. (37)
Hence, we get a contradiction:
ð0
1 â ð¿0
ð¥0(ð¡)
â¥
ð (ð¡)
1 â ð¿0
> ð (ð¡) , ð-a.e. ð¡ â ðŽð. (38)
Theorem 4. Assume 1 †ð < â; then the following assertionsare equivalent:
(1) ð¿ÎŠ,ð
is complex midpoint locally uniformly convex;
(2) ð¿ÎŠ,ð
is complex strictly convex;
(3) ð(ð¡) = 0 for ð-a.e. ð¡ â ð.
Proof. The implication (1) â (2) is trivial. Now assume thatð¿ÎŠ,ð
is complex strictly convex. If ð{ð¡ â ð : ð(ð¡) > 0} > 0, letð0= {ð¡ â ð : ð(ð¡) > 0} and it is not difficult to find an element
ð¥ â ð(ð¿ÎŠ,ð) satisfying supp ð¥ = ð \ ð
0. Take ð â ðŸ
ð(ð¥), and
define
ðŠ (ð¡) ={
{
{
ð (ð¡)
2ðfor ð¡ â ð
0,
ð¥ (ð¡) for ð¡ â ð \ ð0.
(39)
Obviously, âðŠâΊ,ð
⥠âð¥âΊ,ð
= 1. On the other hand,
ðŠÎŠ,ð â€
1
ð{1 + [â«
ð¡âð0
Ί(ð¡,ð (ð¡)
2) ðð¡
+â«ð\ð0
Ί (ð¡, ðð¥ (ð¡)) ðð¡]
ð
}
1/ð
=1
ð(1 + ðŒ
ð
Ί(ðð¥))1/ð
= 1.
(40)
Thus, âðŠâΊ,ð
= (1/ð)(1 + ðŒð
Ί(ððŠ))1/ð
= 1. However, for ð¡ âð0, we find ð|ðŠ(ð¡)| = ð(ð¡)/2 < ð(ð¡), which implies ðŠ â ð(ð¿
Ί,ð)
is not a complex extreme point of ðµ(ð¿ÎŠ,ð) fromTheorem 3.
(3) â (1). Suppose that ð¥ â ð(ð¿ÎŠ,ð) is not a complex
strongly extreme point of ðµ(ð¿ÎŠ,ð). It follows fromTheorem 3
that ð{ð¡ â ð : ð0|ð¥(ð¡)| < ð(ð¡)} > 0 for some ð
0â ðŸð(ð¥),
consequently ð{ð¡ â ð : ð(ð¡) > 0} > 0which is a contradiction.
Remark 5. If ð = â then ð-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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[7] E. Thorp and R. Whitley, âThe strong maximum modulustheorem for analytic functions into a Banach space,â Proceedingsof the AmericanMathematical Society, vol. 18, pp. 640â646, 1967.
[8] L. Chen, Y. Cui, and H. Hudzik, âCriteria for complex stronglyextreme points of Musielak-Orlicz function spaces,â NonlinearAnalysis: Theory, Methods & Applications, vol. 70, no. 6, pp.2270â2276, 2009.
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