The Morse Formula for Curves Which are not Locally Simple

2000 ◽  
Vol 7 (4) ◽  
pp. 599-608
Author(s):  
R. Abdulaev
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Critical Points ◽  
Smooth Curve ◽  
Piecewise Smooth ◽  
Number Of Solutions

Abstract Let be an interior mapping of the unit disk, continuous in D2 and such that the restriction of f to the unit circle S 1 is a locally simple curve γ. Suppose that f(a) ≠ a on S 1 and denote by n(a) the number of solutions of the equation f(z) = a in D2 , by μ(f) the sum of multiplicities of the critical points of f in , by q(a) the angular order of γ with respect to a, and by τ(γ) the angular order of γ. It is proved that the Morse formula 2n(a) – μ(f) – 2q(a) + τ(γ) – 1 = 0 remains correct for a piecewise smooth curve which is not locally simple.

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

scholarly journals Some Results and Problems Concerning Chordal Principal Cluster Sets

1967 ◽  
Vol 29 ◽  
pp. 7-18 ◽  
Author(s):  
F. Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Continuous Curve ◽  
Cluster Set ◽  
Asymptotic Values ◽  
Principal Cluster

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. By an arc at a point ζ∈Γ we mean a continuous curve such that |z(t)| < 1 for 0 ≦ t < 1 and . A terminal subarc of an arc Λ at ζ is a subarc of the form z = z (t) (t0 ≦ t < 1), where 0 ≦ t0<1. Suppose that f(z) is a meromorphic function in D. Then A(f) denotes the set of asymptotic values of f; and if ζ∈Γ, then C(f, ζ) means the cluster set of f at ζ and is the outer angular cluster set of f at ζ (see [13]).

On a Problem Related to The Conjecture of Sendov About The Critical Points of a Polynomial

1987 ◽  
Vol 30 (4) ◽  
pp. 476-480
Author(s):  
Q. I. Rahman ◽  
Q. M. Tariq
Keyword(s):  
Unit Disk ◽  
Critical Points ◽  
Closed Unit Disk

AbstractLet P be a polynomial of degree n having all its zeros in the closed unit disk. Given that a is a zero (of P) of multiplicity k we seek to determine the radius ρ(n; k; a) of the smallest disk centred at a containing at least k zeros of the derivative P'. In the case k = 1 the answer has been conjectured to be 1 and is known to be true for n ≦ 5. We prove that ρ(n; k; a) ≦ 2k/(k + 1) for arbitrary k ∊ N and n ≦ k + 4.

scholarly journals On a problem of Bonar concerning Fatou points for annular functions

1975 ◽  
Vol 56 ◽  
pp. 163-170
Author(s):  
Akio Osada
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Open Unit ◽  
Open Unit Disk ◽  
Minimum Modulus ◽  
Jordan Curves ◽  
Fatou Points ◽  
Annular Functions

The purpose of this paper is to study the distribution of Fatou points of annular functions introduced by Bagemihl and Erdös [1]. Recall that a function f(z), regular in the open unit disk D: | z | < 1, is referred to as an annular function if there exists a sequence {Jn} of closed Jordan curves, converging out to the unit circle C: | z | = 1, such that the minimum modulus of f(z) on Jn increases to infinity. If the Jn can be taken as circles concentric with C, f(z) will be called strongly annular.

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 Continuations of Analytic Functions of Class S and Class U

1970 ◽  
Vol 40 ◽  
pp. 33-37
Author(s):  
Shinji Yamashita
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Analytic Functions ◽  
Inner Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Radial Limit

Let f be of class U in Seidel’s sense ([4, p. 32], = “inner function” in [3, p. 62]) in the open unit disk D. Then f has, by definition, the radial limit f(eiθ) of modulus one a.e. on the unit circle K. As a consequence of Smirnov’s theorem [5, p. 64] we know that the function

β-flatness condition in CR spheres multiplicity results

2020 ◽  
Vol 31 (03) ◽  
pp. 2050023
Author(s):  
Najoua Gamara ◽  
Boutheina Hafassa ◽  
Akrem Makni
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Critical Points ◽  
Number Of Solutions ◽  
Multiplicity Results ◽  
Type Formula ◽  
Critical Points At Infinity ◽  
Cauchy Riemann

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Riemann spheres under [Formula: see text]-flatness condition. To find a lower bound for the number of solutions, we use Bahri’s methods based on the theory of critical points at infinity and a Poincaré–Hopf-type formula.

scholarly journals Regular Tsuji Functions with Infinitely Many Julia Points

1967 ◽  
Vol 29 ◽  
pp. 185-196 ◽  
Author(s):  
W. K. Hayman
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Spherical Derivative

Let D denote the unit disk | z | < 1, and C the unit circle | z | = 1. Corresponding to any function f meromorphic in D we denote by f* the spherical derivative

scholarly journals On area integrals and radial variations of analytic functions in the unit disk

1976 ◽  
Vol 61 ◽  
pp. 135-159
Author(s):  
Takafumi Murai
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Analytic Functions ◽  
Area Integral

We are concerned with the behaviour of analytic functions near the boundary. Let T and D be the unit circle |z| = 1 and the unit disk |z| < 1, respectively. The element of T is denoted by θ (0 ≤ θ < 2π). Let be analytic in D. The area integral A(f, θ) of f at θ is defined by

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