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.
作者机构：
[Li, Liulan] Hengyang Normal Univ, Dept Math & Computat Sci, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] MGR Knowledge City, SETS, Chennai Ctr, ISI, Madras 600113, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] M;MGR Knowledge City, SETS, Chennai Ctr, ISI, Madras 600113, Tamil Nadu, India.
期刊：
Complex Analysis and Operator Theory,2015年9(1):183-199 ISSN：1661-8254
通讯作者：
Ponnusamy, Saminathan
作者机构：
[Li, Liulan] Hengyang Normal Univ, Dept Math & Computat Sci, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Stat Inst, SETS, MGR Knowledge City, Chennai Ctr, Madras 600113, Tamil Nadu, India.;[Ponnusamy, Saminathan] Indian Stat Inst, SETS, MGR Knowledge City, Chennai Ctr, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Stat Inst, SETS, MGR Knowledge City, Chennai Ctr, CIT Campus, Madras 600113, Tamil Nadu, India.
关键词：
Harmonic;Univalent;Close-to-convex;Starlike;Convex mappings;Convex in a direction;Convolution
摘要：
Let be a fixed univalent and sense-preserving harmonic mapping which is convex in the real direction but is not starlike in the unit disk . The convolution of with other harmonic mappings, eg. half-plane mappings, is not necessarily univalent in . In this paper, under suitable restriction on the dilatation of , we show that the convolutions of with certain slanted half-plane harmonic mappings are necessarily convex in a direction. In addition, we consider a fixed harmonic mapping and with the dilatation , where denotes the class of asymmetric vertical strip mappings. We find the relationship between and such that is a sense-preserving univalent harmonic mapping and is convex in some direction. These results are a generalization of the corresponding recent result of Dorff et al.. The contents of this paper enhance the interest in univalent harmonic mappings, especially when much is not known on the harmonic convolution.
作者：
Abu Muhanna, Yusuf*;Li, Liulan;Ponnusamy, Saminathan
期刊：
Archiv der Mathematik,2014年103(6):461-471 ISSN：0003-889X
通讯作者：
Abu Muhanna, Yusuf
作者机构：
[Abu Muhanna, Yusuf] Amer Univ Sharjah, Dept Math, Sharjah 26666, U Arab Emirates.;[Li, Liulan] Hengyang Normal Univ, Dept Math & Computat Sci, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] SETS, Indian Stat Inst, Chennai Ctr, Madras 600113, Tamil Nadu, India.
通讯机构：
[Abu Muhanna, Yusuf] A;Amer Univ Sharjah, Dept Math, Sharjah 26666, U Arab Emirates.
关键词：
Univalent;Convex;Starlike;Close-to-convex;Extreme points;Hayman index;Arclength;Zalcman functional;and Harmonic function
摘要：
Lawrence Zalcman's conjecture states that if is analytic and univalent in the unit disk , then for each , with equality only for the Koebe function and its rotations. This conjecture remains open although it has been verified for a few geometric subclasses of the class of univalent analytic functions. In this paper, we consider this problem for the family of normalized functions f analytic and univalent in the unit disk |z| < 1 satisfying the condition Functions satisfying this condition are known to be convex in some direction (and hence close-to-convex and univalent) in |z| < 1. A few other related basic results and remarks about the Hayman index of functions in this family are also presented.
