期刊：
Journal of Mathematical Analysis and Applications,2013年408(2):589-596 ISSN：0022-247X
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, Chennai Ctr, Soc Elect Transact & Secur, MGR Knowledge City, Madras 600113, Tamil Nadu, India.;[Ponnusamy, Saminathan] Indian Stat Inst, Chennai Ctr, Soc Elect Transact & Secur, MGR Knowledge City, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Stat Inst, Chennai Ctr, Soc Elect Transact & Secur, MGR Knowledge City, CIT Campus, Madras 600113, Tamil Nadu, India.
关键词：
Harmonic mappings;Partial sum;Univalent;Convex;Starlike and close-to-convex mappings
摘要：
One of the classical results from Szego shows that if h(z) = z + Sigma(infinity)(n=2) a(n)z(n) is analytic and univalent in the unit disk D := {z is an element of C: vertical bar Z vertical bar < 1), then the section s(n) (h)(z) = Sigma(n)(k=1) a(k)z(k) of h is univalent in vertical bar z vertical bar < 1/4. The exact (largest) radius of the univalence r(n) of s(n) (h) remains an open problem. On the other hand, not much is known in the case of harmonic univalent functions. It is then natural to consider the class P-H(0) of normalized harmonic mappings f = h + (g) over bar in the unit disk D satisfying the condition Re h' (z) > vertical bar g'(z)vertical bar for z is an element of D, where g'(0) = 0. Functions in P-H(0) are known to be univalent and close-to-convex in D. In this paper, we first show that each f is an element of P-H(0) is convex in the disk vertical bar z vertical bar < root 2 - 1, and then determine the value of r so that the partial sums off f is an element of P-H(0) are convex in vertical bar z vertical bar < r. (C) 2013 Elsevier Inc. All rights reserved.
作者机构：
[Liulan Li] Department of Mathematics and Computational, Science Hengyang Normal University, Hengyang, Hunan 421008, China;[Ponnusamy S.] Indian Statistical Institute (ISI), Chennai Centre, SETS, Society for Electronic Transactions and Security, MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India
通讯机构：
[Ponnusamy, S.] D;Department of Mathematics, Indian Institute of Technology Madras, India
关键词：
convex functions;convolution;half-plane and slanted half-plane mappings;Harmonic mapping;univalent
摘要：
Let S0(Hγ) denote the class of all univalent, harmonic, sense-preserving and normalized mappings f of the unit disk D onto the slanted half-plane Hγ: = {w:Re(eiγw) > −1/2} with an additional condition fz(0) = 0. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in D is not necessarily even univalent in D. In this paper, we fix f0 ∈ S0(H0) and show that the convolutions of f0 and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solve a recent open problem of Dorff et al. in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.
作者机构：
[Li, Liulan] Hengyang Normal Univ, Dept Math & Computat Sci, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol, Dept Math, Madras 600036, Tamil Nadu, India.;[Ponnusamy, Saminathan] SETS, Chennai Ctr, Indian Stat Inst, CIT Campus, Madras 600113, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] S;SETS, Chennai Ctr, Indian Stat Inst, CIT Campus, Madras 600113, Tamil Nadu, India.
关键词：
Close-to-convex;Convex;Partial sum;Starlike and univalent harmonic mappings
摘要：
If S denotes the class of functions h(z) = z + Sigma(infinity)(n=2) a(n)Z(n) which are analytic and univalent in the unit disk vertical bar z vertical bar < 1, then the classical result of Szego shows that every section s(n)(h)(z) = Sigma(n)(k=1) a(k)z(k) of h is univalent in vertical bar z vertical bar < 1/4. Exact (largest) radius of univalence r(n) of s(n)(h) remains an open problem, although the corresponding results for sections of various geometric subclasses of S have been obtained which include those h is an element of S for which Re h'(z) > 0 holds in the unit disk. However, no attempt has been made to derive harmonic analog of these results. The central object in this case is the class S-H(0) of sense-preserving harmonic univalent mappings f = h+(g) over bar defined on the unit disk, normalized so that h(0) = g(0) = h'(0) - 1 = g'(0) = 0. Our primary objective in this paper is to solve the univalency of every section of a harmonic function in the class P-H(0)(alpha), where P-H(0)(alpha) denotes the class of normalized univalent harmonic mapping f = h+(g) over bar in the unit disk vertical bar z vertical bar < 1 satisfying the condition Re(h'(z) - alpha) > vertical bar g'(z)vertical bar for vertical bar z vertical bar < 1, where g'(0) = 0 and 0 <= alpha < 1. In this paper, we first present sharp bounds for the moduli of the coefficients a(n), b(n) of f is an element of P-H(0)(alpha) and then determine the value of r so that the partial sums of f is an element of P-H(0) are close-to-convex in vertical bar z vertical bar < r. (C) 2013 Elsevier Ltd. All rights reserved.
期刊：
International Journal of Applied Mathematics & Statistics,2013年47(17):408-415 ISSN：0973-1377
通讯作者：
Li, L.
作者机构：
[Rui Xiao] Department of Electronic Science, Huizhou University, Huizhou 516067, Guangdong, China;[Long Li] College of Computer, National University of Defense Technology, Changsha 410005, Hunan, China;[Guohui Zhang; Long Li] Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang 421008, Hunan, China
通讯机构：
[Li, L.] C;College of Computer, , Changsha 410005, Hunan, China
期刊：
Complex Variables and Elliptic Equations,2013年58(12):1647-1653 ISSN：1747-6933
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 Inst Technol, Dept Math, Madras 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol, Dept Math, Madras 600036, Tamil Nadu, India.
期刊：
Journal of Theoretical and Applied Information Technology,2012年46(1):114-119 ISSN：1817-3195
通讯作者：
Li, L.
作者机构：
[Xu, Tian; Li, Long] Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421008, China;[Yang, Jie] School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China;[Liu, Yan] Department of Applied Mathematics, Dalian Polytechnic University, Dalian, 116034, China
通讯机构：
Department of Mathematics and Computational Science, Hengyang Normal University, China
摘要：
In the training of feedforward neural networks, it is usually suggested that the initial weights should be small in magnitude in order to prevent premature saturation. The aim of this paper is to point out the other side of the story: In some cases, the gradient of the error functions is zero not only for infinitely large weights but also for zero weights. Slow convergence in the beginning of the training procedure is often the result of sufficiently small initial weights. Therefore, we suggest that, in these cases, the initial values of the weights should be neither too large, nor too small. For instance, a typical range of choices of the initial weights might be something like (−0.4,−0.1) ∪ (0.1, 0.4), rather than (−0.1, 0.1) as suggested by the usual strategy. Our theory that medium size weights should be used has also been extended to a few commonly used transfer functions and error functions. Numerical experiments are carried out to support our theoretical findings.
作者机构：
[刘燕; 杨洁] School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;[李龙] Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang 421008, China;[刘燕] School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China
通讯机构：
[Liu, Y.] S;School of Mathematical Sciences, Dalian University of Technology, China
期刊：
NEW YORK JOURNAL OF MATHEMATICS,2011年17:41-49 ISSN：1076-9803
通讯作者：
Li, L.
作者机构：
Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang Hunan 421008, China
通讯机构：
Department of Mathematics and Computational Science, Hengyang Normal University, China
关键词：
Clifford algebra;Jørgensen's inequality;Möbius transformation in infinite dimension
摘要：
In this paper, we give a generalization of Jørgensen's inequality to hyperbolic Möbius transformations in infinite dimension by using Clifford algebras. We also give an application.
期刊：
OSAKA JOURNAL OF MATHEMATICS,2009年46(4):1097-1141 ISSN：0030-6126
通讯作者：
Li, Liulan
作者机构：
[Li, Liulan] Hengyang Normal Univ, Dept Math, Hengyang 421008, Hunan, Peoples R China.;[Ohshika, Ken'Ichi] Osaka Univ, Grad Sch Sci, Dept Math, Osaka 5600043, Japan.;[Wang, Xiantao] Hunan Normal Univ, Dept Math, Changsha 410081, Hunan, Peoples R China.
通讯机构：
[Li, Liulan] H;Hengyang Normal Univ, Dept Math, Hengyang 421008, Hunan, Peoples R China.
关键词：
In this paper;we generalise the first Klein-Maskit combination theorem to discrete groups of Möbius transformations in higher dimensions. The application of the main theorem is discussed in the last section. Published: December 2009 First available in Project Euclid: 15 December 2009 zbMATH: 1196.30036 MathSciNet: MR2604923 Subjects: Primary: 30F40 Secondary: 20H10
摘要：
In this paper, we generalise the first Klein-Maskit combination theorem to discrete groups of Möbius transformations in higher dimensions. The application of the main theorem is discussed in the last section.
