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2014 | Buch

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Current Topics in Pure and Computational Complex Analysis

herausgegeben von: Santosh Joshi, Michael Dorff, Indrajit Lahiri

Verlag: Springer India

Buchreihe : Trends in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The book contains 13 articles, some of which are survey articles and others research papers. Written by eminent mathematicians, these articles were presented at the International Workshop on Complex Analysis and Its Applications held at Walchand College of Engineering, Sangli. All the contributing authors are actively engaged in research fields related to the topic of the book. The workshop offered a comprehensive exposition of the recent developments in geometric functions theory, planar harmonic mappings, entire and meromorphic functions and their applications, both theoretical and computational. The recent developments in complex analysis and its applications play a crucial role in research in many disciplines.

Inhaltsverzeichnis

Frontmatter

1. Boundary Behavior of Univalent Harmonic Mappings

A Survey of Recent Boundary Behavior Results of Univalent Harmonic Mappings

Abstract

This chapter is a survey of the boundary behavior properties of univalent harmonic mappings of the unit disk $\mathbb{D}$ over the last two decades. Particular emphasis has been given to the boundary behavior of univalent harmonic mappings “onto” $\mathbb{D}$ in the sense of Hengartner and Schober.

Daoud Bshouty, Abdallah Lyzzaik

2. Harmonic Univalent Mappings and Minimal Graphs

Abstract

We survey results and open problems in harmonic maps and minimal surface theory at a level appropriate for graduate students and others interested in contributing to the existing research. After covering some basic results, several topics are covered in more detail, including the shearing technique, inner mapping radius, convolutions, the Weierstrass Representation, determining minimal surfaces via change of variables, curvature bounds, and conjugate minimal surfaces. A variety of new and standing conjectures is included throughout. Examples are worked in detail and presented visually using ComplexTool, Mathematica, and other software packages.

Zach Boyd, Michael Dorff

3. The Minimal Surfaces Over the Slanted Half-Planes, Vertical Strips and Single Slit

Abstract

In this chapter, we discuss the minimal surfaces over the slanted half-planes, vertical strips, and single slit whose slit lies on the negative real axis. The representation of these minimal surfaces and the corresponding harmonic mappings are obtained explicitly. Finally, we illustrate the harmonic mappings of each of these cases together with their minimal surfaces pictorially with the help of mathematica. The content of this chapter is a shorter version of an article of the author’s report of 2011 and published in arXiv (http://arxiv.org/pdf/1204.2890.pdf) in 2012.

Liulan Li, Saminathan Ponnusamy, Matti Vuorinen

4. A Survey On Some Special Classes of Bazilevič Functions and Related Function Classes

Abstract

This chapter is a survey in which we analyze recent developments of some interesting subclasses ${\mathcal U}(\lambda)$, ${\mathcal U}(\lambda,\mu)$, and ${\mathcal U}(\alpha,\lambda,\mu)$ of Bazilevič and non-Bazilevič functions. We discuss historic development of these classes and discuss numerous properties and interesting results pertaining to these classes.

Pravati Sahoo, R. N. Mohapatra

5. Uniqueness of Entire Functions Sharing Certain Values with Derivatives

Abstract

The problem of entire functions sharing values with their derivatives is a special case of uniqueness of entire functions. In 1977 L. A. Rubel and C. C. Yang initiated this kind of investigation. Afterwards, a number of researchers like Mues, Steinmetz, Zhang, Zheng, Wang, Yang, Yi etc. proceeded further with this problem. Now-a-days the uniqueness problem of entire functions sharing values with derivatives has become one of the most well explored branches of the value distribution theory. In the short survey we discuss the gradual development of this branch starting from the result of Rubel and Yang. For the purpose, sometimes we shall mention some results for meromorphic functions also.

Indrajit Lahiri

6. Differential Superordinations and Sandwich-Type Results

Abstract

Let $\Omega\subset\mathbb{C}$, let p be analytic in the unit disc $\mathrm{U}=\left\{z\in\mathbb{C}:|z|<1\right\}$, and let $\psi(r,s,t;z):\mathbb{C}^3\times\mathrm{U}\rightarrow\mathbb{C}$. In a series of articles, S. S. Miller, P. T. Mocanu and many others have determined properties of functions ψ that satisfy the differential subordination (i.e. the differential inclusion)

Teodor Bulboacӑ, Nak Eun Cho, Pranay Goswami

7. Starlikeness and Convexity of Certain Integral Transforms by using Duality Technique

Abstract

Duality technique is applied on the integral transform of the type $V_\lambda(f)(z)=\int_0^1\lambda(t)\frac{f(tz)}{t}{} dt$, where $\int_0^1 \lambda(t)dt =1$ and f is functional class that satisfies certain analytic characterization in the unit disk. This leads to the investigation between the equivalence of non-negativity of a linear functional and the given integral transform involving starlike and convex functions. Particular values of $\lambda(t)$ give rise to well-known integral operators. Investigation of the parameters for such values leads to interesting results in univalent function theory. This chapter outlines all the possible results available in the literature in this direction to provide the reader an overview.

Satwanti Devi, A. Swaminathan

8. Eneström–Kakeya Theorem and Some of Its Generalizations

Abstract

The study of the zeros of polynomials has a very rich history. In addition to having numerous applications, this study has been the inspiration for much theoretical research (including being the initial motivation for modern algebra). The earliest contributors to this subject were Gauss and Cauchy. Algebraic and analytic methods for finding zeros of a polynomial, in general, can be quite complicated. So it is desirable to put some restrictions on polynomials. Eneström–Kakeya theorem, which is a result in this direction, states that if $p(z)=\sum_{j=0}^n a_j z^j$ is a polynomial of degree n with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots \leq a_n$, then all the zeros of p lie in $|z|\leq 1$. Eneström–Kakeya theorem has been the starting point for considerable literature in Mathematics, concerning the location of the zeros of polynomials. In this article we begin with the earliest results of Eneström and Kakeya and conclude this by presenting some of the recent results on this subject. Our article is expository in nature.

Robert B. Gardner, N. K. Govil

9. Starlikeness, Convexity and Close-to-convexity of Harmonic Mappings

Abstract

In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this chapter, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investigate the geometric properties of functions in the class ${\cal M}(\alpha)$ $(|\alpha|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=\alpha zh'(z)$, Re $(1+{zh''(z)}/{h'(z)})>-{1}/{2}$ for $|z|<1$. The coefficient estimates, growth results, area theorem and bounds for the radius of starlikeness, and convexity of the class ${\cal M}(\alpha)$ are determined. In particular, the bound for the radius of convexity is sharp for the class ${\cal M}(1)$.

Sumit Nagpal, V. Ravichandran

10. On Generalized p-valent Non-Bazilevic Type Functions

Abstract

In this chapter, we introduce and study a subclass ${\cal N}_{p}[k,\mu,\alpha;A,B]$ of p-valent analytic functions of the type

$$f(z)= z^p+ \sum ^{\infty}_{n=m}a_{n+p}z^{n+p}.$$

This class includes the class of non-Bazilevic functions. We use differential subordination to derive certain inclusion relations. Distortion theorems, radius problems, and coefficient result, are also discussed.

Khalida Inayat Noor

11. Integral Mean Estimates for a Polynomial with Restricted Zeros

Abstract

In this paper, we prove some $L^{q},~q\ge 0$ mean inequalities for a class of polynomials

$$\begin{aligned} \textbf{P}_{n,\mu}:=\bigg\{P(z)=a_{n}z^{n}+\sum\limits_{j=\mu}^{n}a_{n-j}z^{n-j},~~1\le\mu\le n\bigg\}\end{aligned}$$

having all zeros in $|z|\le k,~k\le 1.$

A. Liman, W. M. Shah

12. Uniqueness Results of Meromorphic Functions Concerning Small Functions

Abstract

We study the uniqueness question of meromorphic functions sharing four small functions. The results in this paper improve the corresponding results given by Nevalinna [8], Shirosaki [9], Toda [10], Li-Qiao [7], Li[6], Ishizaki-Toda [3] and Yi[13], etc.

Xiao-Min Li, Kai-Mei Wang, Hong-Xun Yi

13. Maximal Polynomial Ranges for a Domain of Intersection of Two Circular Disks

Abstract

Let Ω be a domain in the complex plane $\mathbb{C}$ containing the origin and let ${\cal P}_{n}^{0}(\Omega)$ denote the set of all polynomials P of degree ≤ n satisfying the conditions $P(0)=0$ and $P(\mathbb{D})\subset \Omega$, where $\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}$ is the unit disk in the complex plane. The maximal range, denoted by Ω _n, is then defined as the union of all sets $P(\mathbb{D}), P \in {\cal P}_{n}^{0}$ .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.

Chinta Mani Pokhrel

Titel: Current Topics in Pure and Computational Complex Analysis
herausgegeben von: Santosh Joshi
Michael Dorff
Indrajit Lahiri
Verlag: Springer India
Electronic ISBN: 978-81-322-2113-5
Print ISBN: 978-81-322-2112-8
DOI: https://doi.org/10.1007/978-81-322-2113-5