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Research ArticleComplex Convexity of Musielak-Orlicz Function SpacesEquipped with the 119901-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 cll2119hotmailcom
Received 27 November 2013 Revised 5 April 2014 Accepted 13 April 2014 Published 8 May 2014
Academic Editor Angelo Favini
Copyright copy 2014 Lili Chen et alThis 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 119901-Amemiya norm is mainly discussed It is obtainedthat for any Musielak-Orlicz function space equipped with the 119901-Amemiya norm when 1 le 119901 lt infin complex strongly extremepoints of the unit ball coincide with complex extreme points of the unit ball Moreover criteria for them in above spaces are givenCriteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced
1 Introduction
Let (119883 sdot ) be a complex Banach space over the complexfield C let 119894 be the complex number satisfying 1198942 = minus1 andlet 119861(119883) and 119878(119883) be the closed unit ball and the unit sphereof 119883 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 119862lowast-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 [1ndash6])
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 119909 isin 119878(119883) is said to be a complex extreme point of119861(119883) if for every nonzero119910 isin 119883 there holds sup
|120582|le1119909+120582119910 gt
1 A complex Banach space 119883 is said to be complex strictlyconvex if every element of 119878(119883) is a complex extreme pointof 119861(119883)
In [8] we further studied the notions of complex stronglyextreme point and complex midpoint locally uniform con-vexity in general complex spaces
A point 119909 isin 119878(119883) is said to be a complex strongly extremepoint of 119861(119883) if for every 120576 gt 0 we have Δ
1003817100381710038171003817120582119909 plusmn 1198941199101003817100381710038171003817 le 1
10038171003817100381710038171199101003817100381710038171003817 ge 120576
(1)
A complex Banach space 119883 is said to be complex midpointlocally uniformly convex if every element of 119878(119883) is a complexstrongly extreme point of 119861(119883)
Let (119879 Σ 120583) be a nonatomic and complete measure spacewith 120583(119879) lt infin By Φ we denote a Musielak-Orlicz functionthat is Φ 119879 times [0 +infin) rarr [0 +infin] satisfies the following
(1) for each 119906 isin [0infin] Φ(119905 119906) is a 120583-measurablefunction of 119905 on 119879
(2) for 120583-ae 119905 isin 119879Φ(119905 0) = 0 lim119906rarrinfin
(3) for 120583-ae 119905 isin 119879 Φ(119905 119906) is convex on the interval[0infin) with respect to 119906
Let 119871119888(120583) be the space of all 120583-equivalence classes ofcomplex and Σ-measurable functions defined on 119879 For each119909 isin 119871119888(120583) we define on 119871119888(120583) the convex modular of 119909 by
119868Φ(119909) = int
119879
Φ (119905 |119909 (119905)|) 119889119905 (2)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 190203 6 pageshttpdxdoiorg1011552014190203
2 Abstract and Applied Analysis
We define supp 119909 = 119905 isin 119879 |119909(119905)| = 0 and the Musielak-Orlicz space 119871
Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent
(1) 119871Φ119901
is complex midpoint locally uniformly convex
(2) 119871Φ119901
is complex strictly convex
(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879
Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901
is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element
10038171003817100381710038171199101003817100381710038171003817Φ119901 le
1
1198961 + [int
119905isin1198790
Φ(119905119890 (119905)
2) 119889119905
+int1198791198790
Φ (119905 119896119909 (119905)) 119889119905]
119901
1119901
=1
119896(1 + 119868
119901
Φ(119896119909))1119901
= 1
(40)
Thus 119910Φ119901
= (1119896)(1 + 119868119901
Φ(119896119910))1119901
= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871
Φ119901)
is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3
(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex
strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3
that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896
0isin 119870119901(119909)
consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction
Remark 5 If 119901 = infin then 119901-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 (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)
References
[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003
[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004
[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005
[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005
[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007
[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010
[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967
[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009
[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996
[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008
Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent
(1) 119871Φ119901
is complex midpoint locally uniformly convex
(2) 119871Φ119901
is complex strictly convex
(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879
Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901
is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element
10038171003817100381710038171199101003817100381710038171003817Φ119901 le
1
1198961 + [int
119905isin1198790
Φ(119905119890 (119905)
2) 119889119905
+int1198791198790
Φ (119905 119896119909 (119905)) 119889119905]
119901
1119901
=1
119896(1 + 119868
119901
Φ(119896119909))1119901
= 1
(40)
Thus 119910Φ119901
= (1119896)(1 + 119868119901
Φ(119896119910))1119901
= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871
Φ119901)
is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3
(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex
strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3
that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896
0isin 119870119901(119909)
consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction
Remark 5 If 119901 = infin then 119901-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 (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)
References
[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003
[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004
[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005
[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005
[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007
[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010
[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967
[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009
[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996
[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008
Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent
(1) 119871Φ119901
is complex midpoint locally uniformly convex
(2) 119871Φ119901
is complex strictly convex
(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879
Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901
is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element
10038171003817100381710038171199101003817100381710038171003817Φ119901 le
1
1198961 + [int
119905isin1198790
Φ(119905119890 (119905)
2) 119889119905
+int1198791198790
Φ (119905 119896119909 (119905)) 119889119905]
119901
1119901
=1
119896(1 + 119868
119901
Φ(119896119909))1119901
= 1
(40)
Thus 119910Φ119901
= (1119896)(1 + 119868119901
Φ(119896119910))1119901
= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871
Φ119901)
is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3
(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex
strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3
that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896
0isin 119870119901(119909)
consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction
Remark 5 If 119901 = infin then 119901-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 (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)
References
[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003
[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004
[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005
[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005
[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007
[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010
[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967
[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009
[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996
[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008
Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent
(1) 119871Φ119901
is complex midpoint locally uniformly convex
(2) 119871Φ119901
is complex strictly convex
(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879
Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901
is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element
10038171003817100381710038171199101003817100381710038171003817Φ119901 le
1
1198961 + [int
119905isin1198790
Φ(119905119890 (119905)
2) 119889119905
+int1198791198790
Φ (119905 119896119909 (119905)) 119889119905]
119901
1119901
=1
119896(1 + 119868
119901
Φ(119896119909))1119901
= 1
(40)
Thus 119910Φ119901
= (1119896)(1 + 119868119901
Φ(119896119910))1119901
= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871
Φ119901)
is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3
(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex
strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3
that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896
0isin 119870119901(119909)
consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction
Remark 5 If 119901 = infin then 119901-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 (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)
References
[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003
[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004
[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005
[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005
[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007
[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010
[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967
[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009
[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996
[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008
Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent
(1) 119871Φ119901
is complex midpoint locally uniformly convex
(2) 119871Φ119901
is complex strictly convex
(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879
Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901
is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element
10038171003817100381710038171199101003817100381710038171003817Φ119901 le
1
1198961 + [int
119905isin1198790
Φ(119905119890 (119905)
2) 119889119905
+int1198791198790
Φ (119905 119896119909 (119905)) 119889119905]
119901
1119901
=1
119896(1 + 119868
119901
Φ(119896119909))1119901
= 1
(40)
Thus 119910Φ119901
= (1119896)(1 + 119868119901
Φ(119896119910))1119901
= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871
Φ119901)
is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3
(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex
strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3
that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896
0isin 119870119901(119909)
consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction
Remark 5 If 119901 = infin then 119901-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 (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)
References
[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003
[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004
[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005
[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005
[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007
[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010
[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967
[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009
[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996
[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008
This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)
References
[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003
[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004
[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005
[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005
[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007
[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010
[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967
[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009
[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996
[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008