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1
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Estimations of the second coefficient of a univalent, bounded, symmetric and non-vanishing function by means of Loewner's parametric method

100%
Śladkowska J.
Annales Polonici Mathematici
|
1998
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tom 68
|
nr 2
119-123
EN
Let $𝓑₀^{(R)}(b)$ denote the class of functions F(z) = b + A₁z + A₂z² + ...$ analytic and univalent in the unit disk U which satisfy the conditions: F(U) ⊂ U, 0 ∉ F(U), $Im F^{(n)}(0) = 0$. Using Loewner's parametric method we obtain lower and upper bounds of A₂ in $𝓑₀^{(R)}(b)$ and functions for which these bounds are realized. The class $𝓑₀^{(R)}(b)$, introduced in [6], is a subclass of the class $𝓑_u$ of bounded, non-vanishing univalent functions in the unit disk. This last class and closely related ones have been studied by various authors in [1]-[4]. We mention in particular the paper of D. V. Prokhorov and J. Szynal [5], where a sharp upper bound for the second coefficient in $𝓑_u$ is given.
2
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On the estimate of the fourth-order homogeneous coefficient functional for univalent functions

100%
Gromova L., Vasil'ev A.
Annales Polonici Mathematici
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1996
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tom 63
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nr 1
7-12
EN
The functional |c₄ + pc₂c₃ + qc³₂| is considered in the class 𝕊 of all univalent holomorphic functions $f(z) = z + ∑^{∞}_{n=2} c_n z^n$ in the unit disk. For real values p and q in some regions of the (p,q)-plane the estimates of this functional are obtained by the area method for univalent functions. Some new regions are found where the Koebe function is extremal.
3
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On starlikeness of certain integral transforms

80%
Ponnusamy S.
Annales Polonici Mathematici
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1991-1992
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tom 56
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nr 3
227-232
EN
Let A denote the class of normalized analytic functions in the unit disc U = {z: |z| < 1}. The author obtains fixed values of δ and ϱ (δ ≈ 0.308390864..., ϱ ≈ 0.0903572...) such that the integral transforms F and G defined by $F(z) = ∫_0^z (f(t)/t)dt$ and $G(z) = (2/z) ∫_0^z g(t)dt$ are starlike (univalent) in U, whenever f ∈ A and g ∈ A satisfy Ref'(z) > -δ and Re g'(z) > -ϱ respectively in U.
4
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On functions satisfying more than one equation of Schiffer type

80%
Macura J., Śladkowska J.
Annales Polonici Mathematici
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1993
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tom 58
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nr 3
237-252
EN
The paper concerns properties of holomorphic functions satisfying more than one equation of Schiffer type ($D_n$-equation). Such equations are satisfied, in particular, by functions that are extremal (in various classes of univalent functions) with respect to functionals depending on a finite number of coefficients.
5
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On the univalent, bounded, non-vanishing and symmetric functions in the unit disk

80%
Śladkowska J.
Annales Polonici Mathematici
|
1996
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tom 64
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nr 3
291-299
EN
The paper is devoted to a class of functions analytic, univalent, bounded and non-vanishing in the unit disk and in addition, symmetric with respect to the real axis. Variational formulas are derived and, as applications, estimates are given of the first and second coefficients in the considered class of functions.
6
Content available remote

Even coefficient estimates for bounded univalent functions

80%
Prokhorov D.
Annales Polonici Mathematici
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1993
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tom 58
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nr 3
267-273
EN
Extremal coefficient properties of Pick functions are proved. Even coefficients of analytic univalent functions f with |f(z)| < M, |z| < 1, are bounded by the corresponding coefficients of the Pick functions for large M. This proves a conjecture of Jakubowski. Moreover, it is shown that the Pick functions are not extremal for a similar problem for odd coefficients.
7
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On some radius results for normalized analytic functions

80%
Kim Y., Kwon E.
Annales Polonici Mathematici
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1998
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tom 68
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nr 1
51-60
EN
We investigate some radius results for various geometric properties concerning some subclasses of the class 𝓢 of univalent functions.
8
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Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

51%
Krushkal S.
Open Mathematics
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2007
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tom 5
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nr 3
551-580
EN
The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to $$\widehat{\mathbb{C}}$$ are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.
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