scholarly journals Quasiconformal extensions for some geometric subclasses of univalent functions

1984 ◽  
Vol 7 (1) ◽  
pp. 187-195 ◽  
Author(s):  
Johnny E. Brown
Keyword(s):  
Recent Result ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Plane ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Sufficiency Condition ◽  
Extended Complex ◽  
Extended Complex Plane

LetSdenote the set of all functionsfwhich are analytic and univalent in the unit diskDnormalized so thatf(z)=z+a2z2+…. LetS∗andCbe those functionsfinSfor whichf(D)is starlike and convex, respectively. For0≤k<1, letSkdenote the subclass of functions inSwhich admit(1+k)/(1−k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a functionfbelongs toSk⋂S∗orSk⋂C. Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong toSk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.

scholarly journals On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

scholarly journals Subgroups of infinite index in the modular group

1978 ◽  
Vol 19 (1) ◽  
pp. 33-43 ◽  
Author(s):  
W. W. Stothers
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Modular Group ◽  
Infinite Index ◽  
Upper Half Plane ◽  
Extended Complex ◽  
Extended Complex Plane

The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.

Covering Theorems for Univalent Functions Mapping onto Domains Bounded by Quasiconformal Circles

1976 ◽  
Vol 28 (3) ◽  
pp. 627-631 ◽  
Author(s):  
Donald K. Blevins
Keyword(s):  
Complex Plane ◽  
Jordan Curve ◽  
Univalent Functions ◽  
Cross Ratio ◽  
Covering Theorems ◽  
Extended Complex ◽  
Chordal Distance ◽  
Image Position ◽  
Extended Complex Plane

Let Γ be a Jordan curve in the extended complex plane C. Γ is called a quasiconformal circle if it is the image of a circle by a homeomorphism ƒ which is quasiconformal in a neighborhood of that circle. If q(zi, z2) is the chordal distance from z1 to z2, the chordal cross ratio of a quadruple z1, z2, z3, z4 in C is

scholarly journals Extended Complex Plane and Riemann Sphere

2020 ◽  
Vol 08 (04) ◽  
pp. 44-51
Author(s):  
Egahi M. ◽  
Agbata B.C. ◽  
Ogwuche O.I. ◽  
Soomiyol M. C
Keyword(s):  
Complex Plane ◽  
Riemann Sphere ◽  
Extended Complex ◽  
Extended Complex Plane

scholarly journals On Univalence Criteria for a General Integral Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Vasile Marius Macarie ◽  
Daniel Breaz
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Complex Plane ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
General Integral ◽  
General Integral Operator ◽  
Univalence Criteria

We consider a new general integral operator, and we give sufficient conditions for the univalence of this integral operator in the open unit disk of the complex plane. Several consequences of the main results are also shown.

scholarly journals Discrete free products of two complex cyclic matrix groups

1979 ◽  
Vol 20 (1) ◽  
pp. 69-80 ◽  
Author(s):  
Ronald J. Evans
Keyword(s):  
Complex Plane ◽  
Imaginary Axis ◽  
Free Products ◽  
Matrix Groups ◽  
Linear Fractional Transformations ◽  
Real Trace ◽  
Extended Complex ◽  
Image Position ◽  
The Right ◽  
Extended Complex Plane

All 2-by-2 matrices in this paper are to be viewed as linear fractional transformations on the extended complex plane ℂ*. Let L+ and L− be the open half-planes to the right and left, respectively, of the extended imaginary axis L. Let Λ be the set of complex 2-by-2 matrices A with real trace and determinant ±1 such that A(L+) ⊂L−. Let Ω = Ω1 ∪ Ω2 ∪ Ω3 ∪ Ω4, Whereand

scholarly journals Types of complete Infinitely sheeted Planes

2004 ◽  
Vol 176 ◽  
pp. 181-195 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Complex Plane ◽  
Parameter Family ◽  
Real Numbers ◽  
Extended Complex ◽  
Extended Complex Plane

AbstractWe will answer negatively to the question whether the completeness of infinitely sheeted covering surfaces of the extended complex plane have anything to do with their types being parabolic or hyperbolic. This will be accomplished by giving a one parameter family {W[α]: α ∈ A} of complete infinitely sheeted planes W[α] depending on the parameter set A of sequences α = (an)n>1 of real numbers 0 < an ≤ 1/2 (n ≥ 1) such that W[α] is parabolic for ‘small’ α’s and hyperbolic for ‘large’ α’s.

scholarly journals Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Abhijit Banerjee ◽  
Saikat Bhattacharyya
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Shift Operator ◽  
Future Research ◽  
Weighted Sharing ◽  
Shift Operators ◽  
Extended Complex ◽  
Open Question ◽  
Extended Complex Plane

AbstractIn the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$Lcrf, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$f(z) and $L_{c}^{r}\,f$Lcrf sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.

scholarly journals Radius of starlikeness for some classes containing non-univalent functions

2021 ◽  
pp. 2250009
Author(s):  
Shalu Yadav ◽  
Kanika Sharma ◽  
V. Ravichandran
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Univalent Function ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Radius Of Starlikeness ◽  
The Past ◽  
The Right ◽  
Starlike Univalent Function

A starlike univalent function [Formula: see text] is characterized by [Formula: see text]; several subclasses of starlike functions were studied in the past by restricting [Formula: see text] to take values in a region [Formula: see text] on the right-half plane, or, equivalently, by requiring [Formula: see text] to be subordinate to the corresponding mapping of the unit disk [Formula: see text] to the region [Formula: see text]. The mappings [Formula: see text], [Formula: see text], defined by [Formula: see text] and [Formula: see text] map the unit disk [Formula: see text] to certain nice regions in the right-half plane. For normalized analytic functions [Formula: see text] with [Formula: see text] and [Formula: see text] are subordinate to the function [Formula: see text] for some analytic functions [Formula: see text] and [Formula: see text], we determine the sharp radius for them to belong to various subclasses of starlike functions.

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