Sebastian Schleißinger
We consider support points of the class $S^0(\mathbb{D}^n)$ of normalized univalent mappings on the polydisc $\mathbb{D}^n$ with parametric representation and we prove sharp estimates for coefficients of degree 2.
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In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some natural assumptions. We obtain sharp coefficient bounds for the family $S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded symmetric domains…
Given a set $\mathfrak S$ of conformal maps of the unit disk $\mathbb D$ into itself that is closed under composition, we address the question whether $\mathfrak S$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$, $w_0=\mathsf{id}_{\mathbb D}$, where the control functions $t\mapsto G_t$ form a convex cone $\mathcal M_{\mathfrak S}$. For the set of all conformal $φ:\mathbb D\to\mathbb D$ with $φ(0)=0$, $φ'(0)>0$, an affirmative answer to this question is the essence of Loewner's well-known Parametric Representation of univalent …
Toeplitz Determinants for a Class of Holomorphic Mappings in Higher Dimensions
本文针对高维全纯映射，建立了由系数构成的Toeplitz行列式的锐界，为单叶函数子类提供了新结果，并推广了已知结论。
In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk $\mathbb{U}$. Further, these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc in $\mathbb{C}^n$. The obtained results provide the bounds of Toeplitz determinants for various subclasses of normalized univalent functions in higher dimensions.
Let $n\geq 2$. In this paper, we obtain approximation properties of various families of normalized univalent mappings $f$ on the Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$. First, we obtain approximation properties of spirallike, convex and $g$-starlike mappings $f$ on $\mathbb{B}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$, respectively. Next, for a nonresonant operator $A$ with $m(A)>0$, we obtain an a…
We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unit disk $\mathbb D$ extendible to $k$-q.c.(=quasiconformal) automorphisms of $\mathbb C$. In particular, we answer a question on estimation of $|a_3|$ raised by Kühnau and Niske [Math. Nachr. 78 (1977) 185-192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner's parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972) 23-43]. Another problem we consider is to find the maximal $k_*\in(0,1]$ such that every univalent f…
A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $ϕ$ of the unit disk $\mathbb{D}$ normalized by $ϕ(0) = 0$ and $ϕ'(0) > 0$ can be obtained as the reachable set of the Loewner - Kufarev control system $$ \frac{\mathrm{d} w_t}{\mathrm{d} t}=G_t\circ w_t,\quad t\geqslant0,\qquad w_0=\mathsf{id}_{\mathbb{D}}, $$ where the control functions $t\mapsto G_t\in\mathsf{Hol}(\mathbb{D},\mathbb{C})$ form a certain convex cone. Here we extend this result to the semigroup $\mathfrak U[F]$ consisting of all conformal…
We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system and calculate regular geodesics which turn t…
We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichl…
We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejer and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent.
Let $S$ stand for the usual class of univalent regular functions in the unit disk $U=\{z: |z|<1\}$ normalized by $f(z)=z+a_2z^2+...$ in $U$, and let $S^M$ be its subclass defined by restricting $|f(z)|<M$ in $U$, $M\geq 1$. We consider two classical problems: Bombieri's coefficient problem for the class $S$ and the sharp estimate of the fourth coefficient of a function from $S^M$. Using Löwner's parametric representation and the optimal control method we give exact initial Bombieri's numbers and derive a sharp constant $M_0$, such that for all $M\geq M_0$ the Pick function gives the local maxi…
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