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On the integral means spectrum of univalent functions with quasconformal extensions 论具有准共形扩展的单值函数的积分手段谱
arXiv - MATH - Complex Variables Pub Date : 2024-07-27 DOI: arxiv-2407.19240
Jianjun Jin
{"title":"On the integral means spectrum of univalent functions with quasconformal extensions","authors":"Jianjun Jin","doi":"arxiv-2407.19240","DOIUrl":"https://doi.org/arxiv-2407.19240","url":null,"abstract":"In this note we show that the integral means spectrum of any univalent\u0000function admitting a quasiconformal extension to the extended complex plane is\u0000strictly less than the universal integral means spectrum. This gives an\u0000affirmative answer to a question raised in our recent paper.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866085","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}
引用次数: 0
A note on meromorphic functions on a compact Riemann surface having poles at a single point 关于紧凑黎曼曲面上有单点极点的微函数的说明
arXiv - MATH - Complex Variables Pub Date : 2024-07-25 DOI: arxiv-2407.18286
V V Hemasundar Gollakota
{"title":"A note on meromorphic functions on a compact Riemann surface having poles at a single point","authors":"V V Hemasundar Gollakota","doi":"arxiv-2407.18286","DOIUrl":"https://doi.org/arxiv-2407.18286","url":null,"abstract":"On a compact Riemann surface $X$ of genus $g$, one of the questions is the\u0000existence of meromorphic functions having poles at a point $P$ on $X$. One of\u0000the theorems is the Weierstrass gap theorem that determines a sequence of $g$\u0000numbers $1 < n_k < 2g$, $1 leq k leq g$ for which a meromorphic function with\u0000the order with $n_k$ fails to exist at $P$. In this note, we give proof of the\u0000Weierstrass gap theorem in cohomology terminology. We see that an interesting\u0000combinatorial problem may be formed as a byproduct from the statement of the\u0000Weierstrass gap theorem.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866086","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}
引用次数: 0
All Teichmuller spaces are not starlike 所有的 Teichmuller 空间都不是星形的
arXiv - MATH - Complex Variables Pub Date : 2024-07-25 DOI: arxiv-2407.18239
Samuel L. Krushkal
{"title":"All Teichmuller spaces are not starlike","authors":"Samuel L. Krushkal","doi":"arxiv-2407.18239","DOIUrl":"https://doi.org/arxiv-2407.18239","url":null,"abstract":"This paper is the final step in solving the problem of starlikeness of\u0000Teichmuller spaces in Bers' embedding. This step concerns the case of finite\u0000dimensional Teichmuller spaces ${mathbf T}(g, n)$ of positive dimension\u0000(corresponding to punctured Riemann surfaces of finite conformal type $(g, n)$\u0000with $2g - 2 + n > 0$).","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772585","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}
引用次数: 0
Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains 凸多边形域边界上的$L^q$规范中的图兰型振荡不等式
arXiv - MATH - Complex Variables Pub Date : 2024-07-25 DOI: arxiv-2407.18404
Polina Glazyrina, Szilárd Gy. Révész
{"title":"Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains","authors":"Polina Glazyrina, Szilárd Gy. Révész","doi":"arxiv-2407.18404","DOIUrl":"https://doi.org/arxiv-2407.18404","url":null,"abstract":"In 1939 P'al Tur'an and J'anos ErH{o}d initiated the study of lower\u0000estimations of maximum norms of derivatives of polynomials, in terms of the\u0000maximum norms of the polynomials themselves, on convex domains of the complex\u0000plane. As a matter of normalization they considered the family\u0000$mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given\u0000convex, compact subset $KSubset {mathbb C}$. While Tur'an obtained the first\u0000results for the interval $I:=[-1,1]$ and the disk $D:={ zin {mathbb C}~:~\u0000|z|le 1}$, ErH{o}d extended investigations to other compact convex domains,\u0000too. The order of the optimal constant was found to be $sqrt{n}$ for $I$ and\u0000$n$ for $D$. It took until 2006 to clarify that all compact convex\u0000emph{domains} (with nonempty interior), follow the pattern of the disk, and\u0000admit an order $n$ inequality. For $L^q(partial K)$ norms with any $1le q <infty$ we obtained order $n$\u0000results for various classes of domains. Further, in the generality of all\u0000convex, compact domains we could show a $c n/log n$ lower bound together with\u0000an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all\u0000compact convex domains admit an order $n$ Tur'an type inequality. Here we\u0000prove this for all emph{polygonal} convex domains and any $0< q <infty$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866087","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}
引用次数: 0
Parametric Symplectic Jet Interpolation 参数交映射流插值
arXiv - MATH - Complex Variables Pub Date : 2024-07-24 DOI: arxiv-2407.17581
Rafael B. Andrist, Gaofeng Huang, Frank Kutzschebauch, Josua Schott
{"title":"Parametric Symplectic Jet Interpolation","authors":"Rafael B. Andrist, Gaofeng Huang, Frank Kutzschebauch, Josua Schott","doi":"arxiv-2407.17581","DOIUrl":"https://doi.org/arxiv-2407.17581","url":null,"abstract":"We prove a parametric jet interpolation theorem for symplectic holomorphic\u0000automorphisms of $mathbb{C}^{2n}$ with parameters in a Stein space. Moreover,\u0000we provide an example of an unavoidable set for symplectic holomorphic maps.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772672","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}
引用次数: 0
Some variants of the generalized Borel Theorem and applications 广义伯勒定理的一些变体及其应用
arXiv - MATH - Complex Variables Pub Date : 2024-07-23 DOI: arxiv-2407.16163
Dinh Tuan Huynh
{"title":"Some variants of the generalized Borel Theorem and applications","authors":"Dinh Tuan Huynh","doi":"arxiv-2407.16163","DOIUrl":"https://doi.org/arxiv-2407.16163","url":null,"abstract":"In the first part of this paper, we establish some results around generalized\u0000Borel's Theorem. As an application, in the second part, we construct example of\u0000smooth surface of degree $dgeq 19$ in $mathbb{CP}^3$ whose complements is\u0000hyperbolically embedded in $mathbb{CP}^3$. This improves the previous\u0000construction of Shirosaki where the degree bound $d=31$ was gave. In the last\u0000part, for a Fermat-Waring type hypersurface $D$ in $mathbb{CP}^n$ defined by\u0000the homogeneous polynomial [ sum_{i=1}^m h_i^d, ] where $m,n,d$ are positive\u0000integers with $mgeq 3n-1$ and $dgeq m^2-m+1$, where $h_i$ are homogeneous\u0000generic linear forms on $mathbb{C}^{n+1}$, for a nonconstant holomorphic\u0000function $fcolonmathbb{C}rightarrowmathbb{CP}^n$ whose image is not\u0000contained in the support of $D$, we establish a Second Main Theorem type\u0000estimate: [ big(d-m(m-1)big),T_f(r)leq N_f^{[m-1]}(r,D)+S_f(r). ] This\u0000quantifies the hyperbolicity result due to Shiffman-Zaidenberg and Siu-Yeung.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772732","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}
引用次数: 0
Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning 双曲圆盘上概率分布的共形自然族与几何深度学习视角
arXiv - MATH - Complex Variables Pub Date : 2024-07-23 DOI: arxiv-2407.16733
Vladimir Jacimovic, Marijan Markovic
{"title":"Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning","authors":"Vladimir Jacimovic, Marijan Markovic","doi":"arxiv-2407.16733","DOIUrl":"https://doi.org/arxiv-2407.16733","url":null,"abstract":"We introduce the novel family of probability distributions on hyperbolic\u0000disc. The distinctive property of the proposed family is invariance under the\u0000actions of the group of disc-preserving conformal mappings. The\u0000group-invariance property renders it a convenient and tractable model for\u0000encoding uncertainties in hyperbolic data. Potential applications in Geometric\u0000Deep Learning and bioinformatics are numerous, some of them are briefly\u0000discussed. We also emphasize analogies with hyperbolic coherent states in\u0000quantum physics.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772602","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}
引用次数: 0
Kobayashi hyperbolicity in Riemannian manifolds 黎曼流形中的小林双曲性
arXiv - MATH - Complex Variables Pub Date : 2024-07-22 DOI: arxiv-2407.15976
Hervé Gaussier, Alexandre Sukhov
{"title":"Kobayashi hyperbolicity in Riemannian manifolds","authors":"Hervé Gaussier, Alexandre Sukhov","doi":"arxiv-2407.15976","DOIUrl":"https://doi.org/arxiv-2407.15976","url":null,"abstract":"We study the boundary behavior of the Kobayashi-Royden metric and the\u0000Kobayashi hyperbolicity of domains in Riemannian manifolds.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772648","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}
引用次数: 0
Unbounded operators and the uncertainty principle 无界算子和不确定性原理
arXiv - MATH - Complex Variables Pub Date : 2024-07-22 DOI: arxiv-2407.15803
Friedrich Haslinger
{"title":"Unbounded operators and the uncertainty principle","authors":"Friedrich Haslinger","doi":"arxiv-2407.15803","DOIUrl":"https://doi.org/arxiv-2407.15803","url":null,"abstract":"We study a variant of the uncertainty principle in terms of the annihilation\u0000and creation operator on generalized Segal Bargmann spaces, which are used for\u0000the FBI-Bargmann transform. In addition, we compute the Berezin transform of\u0000these operators and indicate how to use spaces of entire functions in one\u0000variable to study the SzegH{o} kernel for hypersurfaces in $mathbb C^2.$","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772593","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}
引用次数: 0
Geometric subfamily of functions convex in some direction and Blaschke products 沿某一方向凸的函数几何亚族和布拉什克积
arXiv - MATH - Complex Variables Pub Date : 2024-07-20 DOI: arxiv-2407.14922
Liulan Li, Saminthan Ponnusamy
{"title":"Geometric subfamily of functions convex in some direction and Blaschke products","authors":"Liulan Li, Saminthan Ponnusamy","doi":"arxiv-2407.14922","DOIUrl":"https://doi.org/arxiv-2407.14922","url":null,"abstract":"Consider the family of locally univalent analytic functions $h$ in the unit\u0000disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the\u0000condition $${real} left( frac{z h''(z)}{alpha h'(z)}right) <frac{1}{2}\u0000~mbox{ for $zin ID$,} $$ where $0<alphaleq1$. The aim of this article is\u0000to show that this family has several elegant properties such as involving\u0000Blaschke products, Schwarzian derivative and univalent harmonic mappings.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772587","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}
引用次数: 0
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