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1
Content available remote

On the coefficient bodies of meromorphic univalent functions omitting a disc

100%
Tammi O.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 1
47-58
EN
Let S(b) be the class of bounded normalized univalent functions and Σ(b) the class of normalized univalent meromorphic functions omitting a disc with radius b. The close connection between these classes allows shifting the coefficient body information from the former to the latter. The first non-trivial body can be determined in Σ(b) as well as the next one in the real subclass $Σ_{R}(b)$.
2
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On bounded univalent functions that omit two given values

80%
Betsakos D.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 2
253-258
EN
Let a,b ∈ {z: 0<|z|<1} and let S(a,b) be the class of all univalent functions f that map the unit disk 𝔻 into 𝔻\{a,b} with f(0)=0. We study the problem of maximizing |f'(0)| among all f ∈ S(a,b). Using the method of extremal metric we show that there exists a unique extremal function which maps 𝔻 onto a simply connnected domain $D_0$ bounded by the union of the closures of the critical trajectories of a certain quadratic differential. If a<0
3
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Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions

80%
Girela D.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 1
19-17
EN
A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $Δ ={z ∈ ℂ : |z|<1} $ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{iθ}$ for which the radial variation $V(f,e^{iθ})=∫_{0}^{1}|f'(re^{iθ})|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^{iθ}$ such that $(1 - r)|f'(re^{iθ})| ≠ o(1)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.
4
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On uniformly convex functions

80%
Goodman A.
Annales Polonici Mathematici
|
1991-1992
|
tom 56
|
nr 1
87-92
EN
We introduce a new class of normalized functions regular and univalent in the unit disk. These functions, called uniformly convex functions, are defined by a purely geometric property. We obtain a few theorems about this new class and we point out a number of open problems.
5
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An extension of typically-real functions and associated orthogonal polynomials

61%
Naraniecka I., Szynal J., Tatarczak A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2011
|
tom 65
|
nr 2
EN
Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties of obtained orthogonal polynomials.
6
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Univalent functions with logarithmic restrictions

61%
Grinshpan A.
Annales Polonici Mathematici
|
1991
|
tom 55
|
nr 1
117-139
EN
It is known that univalence property of regular functions is better understood in terms of some restrictions of logarithmic type. Such restrictions are connected with natural stratifications of the studied classes of univalent functions. The stratification of the basic class S of functions regular and univalent in the unit disk by the Grunsky operator norm as well as the more general one of the class 𝔐 * of pairs of univalent functions without common values by the τ-norm (this concept is introduced here) are given in the paper. Moreover, some properties of univalent functions whose range has finite logarithmic area are considered. To apply the logarithmic restrictions a special exponentiation technique is used.
7
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Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions of complex order associated with the Hohlov operator

61%
Murugusundaramoorthy G., Bulboaca T.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2018
|
tom 17
8
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The generalized Neumann-Poincaré operator and its spectrum

33%
Partyka D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction..........................................................................................................................................................................5 Preliminaries. Complex harmonic functions..........................................................................................................................7 I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19  1.1. On a boundary integral..............................................................................................................................................20  1.2. The generalized Cauchy singular integral operator $C_𝕍$.......................................................................................23  1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28  1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31  1.5. The generalized Neumann-Poincaré operator $N_𝕍$...............................................................................................36 II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41  2.1. The Douady-Earle extension $E_γ$..........................................................................................................................42  2.2. On an approximation of the Hersch-Pfluger distortion function $Φ_K$......................................................................46  2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48  2.4. The Hilbert space H...................................................................................................................................................54  2.5. The linear operator $B_γ$.........................................................................................................................................60 III. The generalized harmonic conjugation operator............................................................................................................64  3.1. The generalized harmonic conjugation operator $A_γ$.............................................................................................64  3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73  3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80  3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84 IV. Spectral values of a quasicircle.....................................................................................................................................90  4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91  4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93  4.3. Plemelj's formula for a quasicircle..............................................................................................................................96  4.4. The main spectral theorem for quasicircles.............................................................................................................103  4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108 Appendix. The inner completion of pseudo-normed spaces............................................................................................114 References......................................................................................................................................................................117 List of symbols.................................................................................................................................................................122 Index................................................................................................................................................................................124
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