Pub Date : 2004-11-15DOI: 10.1080/0278107031000151347
T. Cheng, Zong-Xuan Chen
In this article we first investigate the zero of the solution of linear differential equation , where are all transcendental entire functions and their orders of growth are less than n. We treat the case where ζ1/ζ2 is not real. It is shown that the exponent of convergence to the zero-sequence of any meromorphic solution of the above equation is infinite.
{"title":"A Result for the convergence to the zero-sequence of the solution of certain linear differential equation","authors":"T. Cheng, Zong-Xuan Chen","doi":"10.1080/0278107031000151347","DOIUrl":"https://doi.org/10.1080/0278107031000151347","url":null,"abstract":"In this article we first investigate the zero of the solution of linear differential equation , where are all transcendental entire functions and their orders of growth are less than n. We treat the case where ζ1/ζ2 is not real. It is shown that the exponent of convergence to the zero-sequence of any meromorphic solution of the above equation is infinite.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133684181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-10-10DOI: 10.1080/02781070412331298633
Jyunji Inoue †, T. Nakazi
Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0 < p < ∞ , Hp (W) denotes a weighted Hardy space on the unit circle. When W ≡ 1, H p(W) is the usual Hardy space Hp . We are interested in Hp ( W)+ the set of all nonnegative functions in Hp ( W). If p ≥ 1/2, Hp + consists of constant functions. However Hp ( W)+ contains a nonconstant nonnegative function for some weight W. In this paper, if p ≥ 1/2 we determine W and describe Hp ( W)+ when the linear span of Hp ( W)+ is of finite dimension. Moreover we show that the linear span of Hp (W)+ is of infinite dimension for arbitrary weight W when 0 < p < 1/2.
设W是一个非负的可和函数，它的对数对于单位圆上的勒贝格测度也是可和的。当0 < p <∞时，Hp (W)表示单位圆上的加权Hardy空间。当W≡1时，Hp (W)是通常的Hardy空间Hp。我们感兴趣的是Hp (W)+ Hp (W)中所有非负函数的集合。如果p≥1/2，则Hp +由常数函数组成。然而，Hp (W)+包含一个对某权W的非常非负函数，当p≥1/2时，我们确定了W，并在Hp (W)+的线性张成空间是有限维时描述了Hp (W)+。进一步证明了当0 < p < 1/2时，对于任意权值W, Hp (W)+的线性张成空间是无限维的。
{"title":"Nonnegative functions in weighted hardy spaces","authors":"Jyunji Inoue †, T. Nakazi","doi":"10.1080/02781070412331298633","DOIUrl":"https://doi.org/10.1080/02781070412331298633","url":null,"abstract":"Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0 < p < ∞ , Hp (W) denotes a weighted Hardy space on the unit circle. When W ≡ 1, H p(W) is the usual Hardy space Hp . We are interested in Hp ( W)+ the set of all nonnegative functions in Hp ( W). If p ≥ 1/2, Hp + consists of constant functions. However Hp ( W)+ contains a nonconstant nonnegative function for some weight W. In this paper, if p ≥ 1/2 we determine W and describe Hp ( W)+ when the linear span of Hp ( W)+ is of finite dimension. Moreover we show that the linear span of Hp (W)+ is of infinite dimension for arbitrary weight W when 0 < p < 1/2.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131562329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-10-10DOI: 10.1080/02781070412331298589
P. Domínguez, Núria Fagella †
We apply the Shishikura surgery construction to transcendental maps in order to obtain examples of meromorphic functions with Herman rings, in a variety of possible arrangements. We give a sharp bound on the maximum possible number of such rings that a meromorphic function may have, in terms of the number of poles. Finally we discuss the possibility of having “unbounded” Herman rings (i.e., with an essential singularity in the boundary), and give some examples of maps with this property.
{"title":"Existence of herman rings for meromorphic functions","authors":"P. Domínguez, Núria Fagella †","doi":"10.1080/02781070412331298589","DOIUrl":"https://doi.org/10.1080/02781070412331298589","url":null,"abstract":"We apply the Shishikura surgery construction to transcendental maps in order to obtain examples of meromorphic functions with Herman rings, in a variety of possible arrangements. We give a sharp bound on the maximum possible number of such rings that a meromorphic function may have, in terms of the number of poles. Finally we discuss the possibility of having “unbounded” Herman rings (i.e., with an essential singularity in the boundary), and give some examples of maps with this property.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132845434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-10-10DOI: 10.1080/02781070410001701092
I. Laine, Jarkko Rieppo †
This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.
{"title":"Differential polynomials generated by linear differential equations","authors":"I. Laine, Jarkko Rieppo †","doi":"10.1080/02781070410001701092","DOIUrl":"https://doi.org/10.1080/02781070410001701092","url":null,"abstract":"This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114603732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-09-22DOI: 10.1080/02781070500139781
N. Nikolov, W. Zwonek
In this article, we study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.
本文研究了多极Lempert函数的积性质问题，并主要考虑了该函数在平面域上的一些性质。
{"title":"On the product property for the Lempert function","authors":"N. Nikolov, W. Zwonek","doi":"10.1080/02781070500139781","DOIUrl":"https://doi.org/10.1080/02781070500139781","url":null,"abstract":"In this article, we study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132532217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-09-15DOI: 10.1080/02781070412331298570
L. Bragg
An averaging operator over the roots of unity is defined on a class of analytic functions and its algebraic and analytic properties are investigated. A Cauchy like integral formula for this is obtained. This operator and its properties are then employed to solve higher order Cauchy problems, to derive addition formulas for hypergeometric functions and to obtain integral representations for special classes of hypergeometric functions.
{"title":"Functions averages over the roots of unity, cauchy problems and addition formulas","authors":"L. Bragg","doi":"10.1080/02781070412331298570","DOIUrl":"https://doi.org/10.1080/02781070412331298570","url":null,"abstract":"An averaging operator over the roots of unity is defined on a class of analytic functions and its algebraic and analytic properties are investigated. A Cauchy like integral formula for this is obtained. This operator and its properties are then employed to solve higher order Cauchy problems, to derive addition formulas for hypergeometric functions and to obtain integral representations for special classes of hypergeometric functions.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116707395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-09-15DOI: 10.1080/02781070412331298552
Wenjie Wang, Ping Li
We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.
{"title":"Unicity of entire functions and their linear differential polynomials","authors":"Wenjie Wang, Ping Li","doi":"10.1080/02781070412331298552","DOIUrl":"https://doi.org/10.1080/02781070412331298552","url":null,"abstract":"We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126210768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-09-15DOI: 10.1080/02781070412331298624
Wei-Chuan Lin, H. Yi
In this article, we deal with the uniqueness problems on meromorphic functions concerning differential polynomials that share fixed-points. Moreover, we greatly improve a former result.
本文讨论了具有不动点的微分多项式的亚纯函数的唯一性问题。而且，我们大大改进了以前的结果。
{"title":"Uniqueness theorems for meromorphic functions concerning fixed-points","authors":"Wei-Chuan Lin, H. Yi","doi":"10.1080/02781070412331298624","DOIUrl":"https://doi.org/10.1080/02781070412331298624","url":null,"abstract":"In this article, we deal with the uniqueness problems on meromorphic functions concerning differential polynomials that share fixed-points. Moreover, we greatly improve a former result.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128362661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-08-15DOI: 10.1080/02781070410001731684
Ivan H. Feschiev, S. Gocheva-Ilieva
In this paper there is proved a generalization of a theorem of Stein and Weiss concerning metric properties of the conjugate characteristic functions of given sets on the interval [0, 2π]. As an application for the Hilbert transform of a bounded function, the optimal correlation where the Favard's constant : is also established.
{"title":"On the extension of a theorem of Stein and Weiss and its application","authors":"Ivan H. Feschiev, S. Gocheva-Ilieva","doi":"10.1080/02781070410001731684","DOIUrl":"https://doi.org/10.1080/02781070410001731684","url":null,"abstract":"In this paper there is proved a generalization of a theorem of Stein and Weiss concerning metric properties of the conjugate characteristic functions of given sets on the interval [0, 2π]. As an application for the Hilbert transform of a bounded function, the optimal correlation where the Favard's constant : is also established.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121310811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-06-10DOI: 10.1080/02781070410001731738
Björn Bennewitz, John L. Lewis †
In this note we point out that theorems of David–Jerison and Semmes for harmonic measure on the boundary of an NTA Ahlfors regular type domain can be extended to more general domains.
{"title":"On weak reverse hölder inequalities for nondoubling harmonic measures","authors":"Björn Bennewitz, John L. Lewis †","doi":"10.1080/02781070410001731738","DOIUrl":"https://doi.org/10.1080/02781070410001731738","url":null,"abstract":"In this note we point out that theorems of David–Jerison and Semmes for harmonic measure on the boundary of an NTA Ahlfors regular type domain can be extended to more general domains.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122134278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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