scholarly journals On minimal thinness, reduced functions and Green potentials

1993 ◽  
Vol 36 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Matts Essén
Keyword(s):  
Unit Circle ◽  
Recent Work ◽  
Unit Disc ◽  
Minimum Principle ◽  
Higher Dimensions ◽  
Connected Subset ◽  
Whitney Decomposition ◽  
Minimal Thinness ◽  
Image Position ◽  
Green Potentials

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we considerwhere Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.

Interpolation by Linear Sums of Harmonic Measures

1975 ◽  
Vol 18 (2) ◽  
pp. 249-253
Author(s):  
Marvin Ortel
Keyword(s):  
Unit Circle ◽  
Harmonic Measure ◽  
Unit Disc ◽  
Boundary Values ◽  
Harmonic Measures ◽  
Image Position ◽  
Interior Points

Let α be an open arc on the unit circleand for z=reiθ, 0 ≤ r < 1, let(1)The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

On the Zeros of Functions with Derivatives in H1 and H∞

1970 ◽  
Vol 22 (2) ◽  
pp. 342-347 ◽  
Author(s):  
James Wells
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Open Unit ◽  
Hardy Class ◽  
Open Unit Disc ◽  
Zeros Of Functions ◽  
Limit Points ◽  
Image Position ◽  
Derivatives Of

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying(A)and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.THEOREM 1. If(1)(2)and(3)then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

scholarly journals A Boundary Theorem for Tsuji Functions

1967 ◽  
Vol 29 ◽  
pp. 197-200 ◽  
Author(s):  
E. F. Collingwood
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Spherical Derivative

Let D denote the unit disc | z | <1, C the unit circle | z | = 1 and Cr the circle | z| = r. Corresponding to any function w(z) meromorphic in D we denote by w*(z) the spherical derivative

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

scholarly journals A note on the phase retrieval of holomorphic functions

2020 ◽  
pp. 1-8
Author(s):  
Rolando Perez
Keyword(s):  
Unit Circle ◽  
Connected Domain ◽  
Unit Disc ◽  
Phase Retrieval ◽  
Holomorphic Functions ◽  
Nevanlinna Class

Abstract We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

p-Radial Exceptional Sets and Conformal Mappings

2007 ◽  
Vol 50 (4) ◽  
pp. 579-587
Author(s):  
Piotr Kot
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Conformal Mappings ◽  
Exceptional Sets ◽  
Image Position

AbstractFor p > 0 and for a given set E of type Gδ in the boundary of the unit disc ∂ we construct a holomorphic function f ∈ such thatIn particular if a set E has a measure equal to zero, then a function f is constructed as integrable with power p on the unit disc .

Coefficients of Symmetric Functions of Bounded Boundary Rotation

1974 ◽  
Vol 26 (6) ◽  
pp. 1351-1355 ◽  
Author(s):  
Ronald J. Leach
Keyword(s):  
Unit Disc ◽  
Symmetric Functions ◽  
The Family ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Image Position

Let denote the family of all functions of the formthat are analytic in the unit disc U, f′(z) ≠ 0 in U and f maps U onto a domain of boundary rotation at most . Recently Brannan, Clunie and Kirwan [2] and Aharonov and Friedland [1] have solved the problem of estimating |amp+1| for all , provided m = 1.

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

Sign in / Sign up

Export Citation Format

Share Document