摘要：
It is known that the inverse of a quasiconformal homeomorphism of domains in R n is also quasiconformal. This paper focuses on the inverse problem for free quasiconformality in Banach spaces. We first show that the inverse of a fully semisolid homeomorphism is fully semisolid under an additional coarsely Lipschitz condition in the quasihyperbolic metric. This gives several partial answers to two open problems posed by V & auml;is & auml;l & auml;. Next, we prove that the inverse of a locally quasisymmetric homeomorphism is also locally quasisymmetric. As applications, we obtain new characterizations of freely quasiconformal mappings in Banach spaces, and study the relation between freely quasiconformal mappings and quasisymmetric mappings between uniform domains.
期刊：
Complex Variables and Elliptic Equations,2024年:1-15 ISSN：1747-6933
通讯作者：
Li, LL
作者机构：
[Li, Liulan; Li, LL] Hengyang Normal Univ, Coll Math & Stat, Hengyang, Hunan, Peoples R China.;[Qian, Tao] Macau Univ Sci & Technol, Macao Ctr Math Sci, Macau, Peoples R China.;[Wang, Shilin] Unitedhlth Grp, Med Informat Dept, Cypress, CA USA.
通讯机构：
[Li, LL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang, Hunan, Peoples R China.
摘要：
In this paper, we introduce a new member to the universal families, called universal quotient Blaschke product, which is a formal quotient of two formal infinite Blaschke products. A formal infinite Blaschke product is of the form B(z)=Pi(infinity)(k=1) z-z(k)/1-(z) over bar (k)z' where {zk}(k=1)(infinity) is a sequence of points in the unit disk but may not satisfy the Blaschke condition: Sigma(infinity)(k=1)(1 - vertical bar z(k vertical bar)) < infinity. A partial quotient of a universal quotient Blaschke product is the quotient of two finite Blaschke products. We show that the set of partial quotients of a universal quotient Blaschke product is dense in the set of continuous self-mappings on the unit circle in the complex plane. Meanwhile, subsequences of the partial quotients of a universal quotient Blaschke product can be used to approximate any holomorphic functions bounded by one on the unit disk. Moreover, we prove that the set of universal quotient Blaschke products is huge in the sense of Baire category.
作者机构：
佛山科学技术学院数学与大数据学院 佛山528000;衡阳师范学院 衡阳421001;[李希宁] 中山大学数学系 珠海510970;[Saminathan PONNUSAMY] Indian Institute of Technology Madras India 600036;[Saminathan PONNUSAMY] Department of Mathematics,Petrozavodsk State University,Lenina 33,185910 Petrozavodsk,Russia
期刊：
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY,2023年151(11):4781-4792 ISSN：0002-9939
通讯作者：
Zhou, QS
作者机构：
[He, Yuehui; Zhou, QS; Zhou, Qingshan] Foshan Univ, Sch Math & Big Data, Foshan 528000, Guangdong, Peoples R China.;[Li, Liulan] Hengyang Normal Univ, Coll Math & Stat, Hengyang, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.;[Ponnusamy, Saminathan] Lomonosov Moscow State Univ, Moscow Ctr Fundamental & Appl Math, Moscow, Russia.
通讯机构：
[Zhou, QS ] F;Foshan Univ, Sch Math & Big Data, Foshan 528000, Guangdong, Peoples R China.
关键词：
Relatively quasimobius maps;quasihyperbolic metric;distance ratio metric;uniform domains;natural domains;invariance
摘要：
<p>In this paper, we explore relatively quasimöbius invariance of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi">
<mml:semantics>
<mml:mi>φ<!-- φ --></mml:mi>
<mml:annotation encoding="application/x-tex">\varphi</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-uniform domains and natural domains. Firstly, we prove that the control function of relatively quasimöbius mappings can be chosen in a power form. Applying this observation and a deformed cross–ratio introduced by Bonk and Kleiner, we next show that relatively quasimöbius mappings are coarsely bilipschitz in the distance ratio metric. Combined with the assumption that the mapping is coarsely bilipschitz in the quasihyperbolic metric, we establish relatively quasimöbius invariance of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi">
<mml:semantics>
<mml:mi>φ<!-- φ --></mml:mi>
<mml:annotation encoding="application/x-tex">\varphi</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-uniform domains and natural domains. As a by-product, we also obtain a similar result for uniform domains which provides a new method to answer a question posed by Väisälä.</p>
作者机构：
[Li, Liulan] College of Mathematics and Statistics (Hunan Provincial Key Laboratory of Intelligent Information Processing and Application), Hengyang Normal University, Hunan, Hengyang, 421002, China;Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India;Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics, Moscow, Russian Federation;[Ponnusamy, Saminathan] Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India, Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics, Moscow, Russian Federation
关键词：
and convolution;convex in a direction;convex mappings;Harmonic;slanted half-plane mappings;univalent
作者机构：
[周青山] School of Mathematics and Big Data, Foshan University, Foshan, 528000, China;[李浏兰] College of Mathematics and Statistics, Hengyang Normal University, Hengyang, 421001, China;[李希宁] Sun Yat-sen University, School of Mathematics(Zhuhai), Zhuhai, 519082, China
期刊：
Journal of Mathematical Analysis and Applications,2020年488(2):124083 ISSN：0022-247X
通讯作者：
Kovalev, Leonid, V
作者机构：
[Li, Liulan] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Kovalev, Leonid, V] Syracuse Univ, Dept Math, 215 Carnegie, Syracuse, NY 13244 USA.
通讯机构：
[Kovalev, Leonid, V] S;Syracuse Univ, Dept Math, 215 Carnegie, Syracuse, NY 13244 USA.
关键词：
Circle embeddings;Circle homeomorphisms;Blaschke products;Rational functions
摘要：
This paper continues the investigation of the relation between the geometry of a circle embedding and the values of its Fourier coefficients. First, we answer a question of Kovalev and Yang concerning the support of the Fourier transform of a starlike embedding. An important special case of circle embeddings are homeomorphisms of the circle onto itself. Under a one-sided bound on the Fourier support, such homeomorphisms are rational functions related to Blaschke products. We study the structure of rational circle homeomorphisms and show that they form a connected set in the uniform topology. (C) 2020 Elsevier Inc. All rights reserved.
期刊：
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY,2019年99(3):421-431 ISSN：0004-9727
通讯作者：
Ponnusamy, Saminathan
作者机构：
[Li, Liulan] Henyang Normal Univ, Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
关键词：
convex in a direction;convex mapping;convolution;harmonic;slanted half-plane mapping;univalent
摘要：
Dorff et al. \cite{DN} formulated an open problem concerning the convolution of two right half-plane mappings, where the normalization of the harmonic mappings has been considered incorrectly. Without realizing the error, the present authors considered the open problem (see \cite[Theorem 2.2]{LiPo1} and \cite[Theorem 1.3]{LiPo2}). In this paper, we have reformulated the open problem in correct form and provided solution to it in a more general form. In addition, we also obtain two new results which correct and improve some other results.
摘要：
<jats:p>While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain <jats:italic>Ω*</jats:italic> there exists a <jats:italic>ε ></jats:italic> 0 such that for every doubly connected domain <jats:italic>Ω</jats:italic> with Mod<jats:italic>Ω* <</jats:italic> Mod<jats:italic>Ω <</jats:italic> Mod<jats:italic>Ω*</jats:italic> + <jats:italic>ε</jats:italic> there exists a harmonic diffeomorphism from <jats:italic>Ω</jats:italic> onto <jats:italic>Ω*</jats:italic>.</jats:p>
期刊：
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY,2017年145(2):833-846 ISSN：0002-9939
通讯作者：
Li, Liulan
作者机构：
[Li, Liulan] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] ISI, Chennai Ctr, SETS, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Li, Liulan] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.
关键词：
Univalent;convex;starlike and close-to-convex functions;Fekete-Szegö inequality;Zalcman and generalized Zalcman functionals
摘要：
Let S denote the class of all functions f(z) = z + Sigma(infinity)(n=2) a(n)az(n) 2 anzn analytic and univalent in the unit disk D. For f. S, Zalcman conjectured that vertical bar a(n)(2) - a(2n-1)vertical bar <= (n -1)(2) for n >= 3. This conjecture has been verified for only certain values of n for f is an element of S and for all n >= 4 for the class C of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional vertical bar lambda a(n)(2) n-a2(n-1)vertical bar for functions in C and for all n >= 3, where. is a positive constant.
作者机构：
[Li, Liulan] Hengyang Normal Univ, Coll Math & Stat, E Ring Rd, Hengyang 421002, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Stat Inst, Chennai Ctr, Soc Elect Transact & Secur, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Li, Liulan] H;Hengyang Normal Univ, Coll Math & Stat, E Ring Rd, Hengyang 421002, Hunan, Peoples R China.
摘要：
We consider the class H
0 of sense-preserving harmonic functions
$$f = h + \bar g$$
defined in the unit disk |z| < 1 and normalized so that h(0) = 0 = h′(0) − 1 and g(0) = 0 = g′(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes P
H
0(α) and G
H
0(β) of functions from H
0 and show that if f ∈ P
H
0(α) and F ∈ G
H
0(β), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums)
$${s_{n,n}}\left( f \right)\left( z \right) = {s_n}\left( h \right)\left( z \right) + \overline {{s_n}\left( g \right)\left( z \right)} ,$$
where
$$f = h + \bar g$$
∈ H
0, s
n
(h) and s
n
(g) denote the n-th partial sums of h and g, respectively. We prove, among others, that if
$$f = h + \bar g$$
∈ H
0 is a univalent harmonic convex mapping, then s
n
,n(f) is univalent and close-to-convex in the disk |z| < 1/4 for n ≥ 2, and s
n
,n(f) is also convex in the disk |z| < 1/4 for n ≥ 2 and n ≠ 3. Moreover, we show that the section s
3,3(f) of f ∈ C
H
0 is not convex in the disk |z| < 1/4 but it is convex in a smaller disk.
摘要：
In this paper, which is sequel to [10], we give a generalisation of the second Klein-Maskit combination theorem, the one dealing with HNN extensions, to higher dimension. We give some examples constructed as an application of the main theorem.
