scholarly journals The complex geometry of Blaschke products of degree 3 and associated ellipses

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 61-68
Author(s):  
Masayo Fujimuraa
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Geometry ◽  
Sufficient Condition ◽  
Blaschke Products ◽  
Necessary And Sufficient Condition ◽  
Geometrical Properties ◽  
Natural Extensions ◽  
Necessary And Sufficient

A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of bicentric triangle for these two circles is known as Chapple?s formula or Euler?s theorem. As one of natural extensions of this formula, we characterize the inscribed ellipses of a triangle which is inscribed in the unit circle. We also discuss the condition for the ?circumscribed? ellipse of a triangle which is circumscribed about the unit circle. For the proof of these results, we use some geometrical properties of Blaschke products on the unit disk.

scholarly journals On the asymptotic boundary behavior of functions analytic in the unit disk

1977 ◽  
Vol 67 ◽  
pp. 1-13
Author(s):  
James R. Choike
Keyword(s):  
Riemann Surface ◽  
Analytic Function ◽  
Unit Disk ◽  
Boundary Behavior ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Asymptotic Boundary ◽  
Necessary And Sufficient ◽  
Class Of Functions

In [8] a necessary and sufficient condition was given for determining the equivalence of two asymptotic boundary paths for an analytic function w = f(p) on a Riemann surface F. In this paper we give a necessary and sufficient condition for determining the nonequivalence of two asymptotic boundary paths for f(z) analytic in |z| < R, 0 < R ≤ + ∞. We shall, also, illustrate some applications of the main result and examine a class of functions introduced by Valiron.

scholarly journals QKSpaces on the Unit Circle

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jizhen Zhou
Keyword(s):  
Sobolev Space ◽  
Unit Circle ◽  
Measurable Function ◽  
Weight Functions ◽  
Sufficient Condition ◽  
General Criterion ◽  
Necessary And Sufficient Condition ◽  
New Space ◽  
Necessary And Sufficient

We introduce a new spaceQK(∂D)of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition onKsuch thatQK(∂D)=BMO(∂D), as well as a general criterion on weight functionsK1andK2,K1≤K2, such thatQK1(∂D)QK2(∂D). We also prove that a measurable function belongs toQK(∂D)if and only if it is Möbius bounded in the Sobolev spaceLK2(∂D). Finally, we obtain a dyadic characterization of functions inQK(∂D)spaces in terms of dyadic arcs on the unit circle.

scholarly journals Badly approximable functions and interpolation by Blaschke products

1976 ◽  
Vol 20 (2) ◽  
pp. 159-161 ◽  
Author(s):  
L. A. Rubel ◽  
A. L. Shields
Keyword(s):  
Continuous Function ◽  
Supremum Norm ◽  
Sufficient Condition ◽  
Blaschke Products ◽  
Winding Number ◽  
Necessary And Sufficient Condition ◽  
Uniform Limit ◽  
Disc Algebra ◽  
Necessary And Sufficient ◽  
Badly Approximable

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.

scholarly journals Composition operators fromℬαtoQKtype spaces

2008 ◽  
Vol 6 (1) ◽  
pp. 88-104 ◽  
Author(s):  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Analytic Functions ◽  
Composition Operators ◽  
Nondecreasing Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Analytic Functions ◽  
Type Spaces ◽  
Necessary And Sufficient

Suppose thatϕis an analytic self-map of the unit diskΔ. Necessary and sufficient condition are given for the composition operatorCϕf=fοϕto be bounded and compact fromα-Bloch spaces toQKtype spaces which are defined by a nonnegative, nondecreasing functionk(r)for0≤r<∞. Moreover, the compactness of composition operatorCϕfromℬ0toQKtype spaces are studied, whereℬ0is the space of analytic functions offwithf′∈H∞and‖f‖ℬ0=|f(0)|+‖f′‖∞.

On Some Classes of Univalent Polynomials

1978 ◽  
Vol 30 (02) ◽  
pp. 332-349 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Szynal
Keyword(s):  
Unit Disk ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient

It was in the year 1931 that Dieudonné [4] proved the following necessary and sufficient condition for a polynomial to be univalent in the unit disk. THEOREM A (Dieudonné criterion). The polynomial is univalent in |z| &lt; 1 if and only if for every θ in [0, π/2] the associated polynomial does not vanish in |z| &lt; 1. For θ = 0, ϕ(z, θ) is to be interpreted as Pn'(z).

scholarly journals Generalized typically real functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1697-1710 ◽  
Author(s):  
Stanisława Kanas ◽  
Anna Tatarczak
Keyword(s):  
Unit Disk ◽  
Real Function ◽  
Chebyshev Polynomials ◽  
Regular Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Functions ◽  
Typically Real ◽  
Necessary And Sufficient ◽  
Typically Real Functions

Let f(z)=z+a2z2+... be regular in the unit disk and real valued if and only if z is real and |z| < 1. Then f(z) is said to be typically real function. Rogosinski found the necessary and sufficient condition for a regular function to be typically-real. The main purpose of the paper is a consideration of the generalized typically-real functions defined via the generating function of the generalized Chebyshev polynomials of the second kind ?p,q(ei?;z)=1 /(1-pzei?)(1-qze-i?) = ??,n=0 Un(p,q; ei?)zn, where -1 ? p,q ? 1; ?? ?0,2??i, |z|<1.

scholarly journals Compressions of Multiplication Operators and Their Characterizations

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
M. Cristina Câmara ◽  
Kamila Kliś–Garlicka ◽  
Bartosz Łanucha ◽  
Marek Ptak
Keyword(s):  
Unit Circle ◽  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Multiplication Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Truncated Toeplitz Operator ◽  
Independent Variable ◽  
Necessary And Sufficient

AbstractDual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable $$M_z$$ M z on the classical $$L^2$$ L 2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated.

A NECESSARY AND SUFFICIENT CONDITION TO HAVE r PAIRS OF COMPLEX MULTIPLIERS WITH MODULUS EQUAL TO UNITY IN THE CASE OF AN n-DIMENSIONAL MAP

1995 ◽  
Vol 05 (04) ◽  
pp. 1193-1204 ◽  
Author(s):  
JEAN-PIERRE CARCASSES
Keyword(s):  
Unit Circle ◽  
Periodic Point ◽  
Parameter Plane ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Considering a cycle (or periodic point) of an n-dimensional map, a bifurcation occurs when at least one of its multipliers crosses the unit circle in the complex.plane. This paper presents a necessary and sufficient condition for the bifurcations occurring when r pairs of complex multipliers cross the unit circle. Among this kind of bifurcations, the most known is the Neimark's bifurcation corresponding to the case when r is equal to one. From the established condition, an algorithm to trace out these bifurcation curves in a parameter plane of the considered map is displayed.

Higher numerical ranges of quaternion matrices

2015 ◽  
Vol 30 ◽  
pp. 889-904 ◽  
Author(s):  
Narjes Haj Aboutalebi ◽  
Gholamreza Aghamollaei ◽  
Hossein Momenaee Kermani
Keyword(s):  
Numerical Range ◽  
Sufficient Condition ◽  
Quaternion Matrix ◽  
Necessary And Sufficient Condition ◽  
Geometrical Properties ◽  
Quaternion Matrices ◽  
Necessary And Sufficient ◽  
Positive Integers

Let n and k be two positive integers and k n. In this paper, the notion of k−numerical range of n−square quaternion matrices is introduced. Some algebraic and geometrical properties are investigated. In particular, a necessary and sufficient condition for the convexity of the k−numerical range of a quaternion matrix is given. Moreover, a new description of 1−numerical range of normal quaternion matrices is also stated.

scholarly journals On a generalization of the Levin-May Theorem

2014 ◽  
Vol 30 (1) ◽  
pp. 55-62
Author(s):  
JAN CERMAK ◽  
◽  
JIRI JANSKY ◽  
Keyword(s):  
Unit Circle ◽  
Practical Importance ◽  
Complete List ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Alternate Proof ◽  
Necessary And Sufficient

The paper discusses a distribution of the zeros of the polynomial ...with respect to the unit circle. This problem is of theoretic as well as practical importance which motivated S.A. Levin and R. May to formulate a necessary and sufficient condition guaranteeing the location of all the zeros of p(λ) inside the unit circle. We give a simple alternate proof of their criterion and, as the main result, present a complete list of all possible zero distributions of p(λ) with respect to this circle.

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