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Register a new userA Schwarz lemma for a domain related to mu-synthesis
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We prove a Schwarz lemma for a domain E in 3-dimensional complex space that arises in connection with a problem in H infinity control theory. We describe a class of automorphisms of E and determine the distinguished boundary of E. We obtain a type of Schwarz-Pick lemma for a two by two mu-synthesis problem.
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In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Caratheodory theorem for univalent holomorphic self-mappings of the open unit disk $mathbb Dsubset mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma.
In this paper we establish several invariant bounda
In this paper, we generalize a recent work of Liu et al. from the open unit ball $mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Caratheodory metric near the boundary of $C^2$ domains and the well-known Grahams estimate on the boundary behavior of the Caratheodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.
La transformoj de Schwarz-Christoffel mapas, konforme, la superan kompleksan duon-ebenon al regiono limigita per rektaj segmentoj. Cxi tie ni priskribas kiel konvene kunigi mapon de la suba duon-ebeno al mapo de la supera duon-ebeno. Ni emfazas la bezonon de klara difino de angulo de kompleksa nombro, por tiu kunigo. Ni diskutas kelkajn ekzemplojn kaj donas interesan aplikon pri movado de fluido. ------- Schwarz-Christoffel transformations map, conformally, the complex upper half plane into a region bounded by right segments. Here we describe how to couple conveniently a map of the lower half plane to the map of the upper half plane. We emphasize the need of a clear definition of angle of a complex, to that coupling. We discuss some examples and give an interesting application for motion of fluid.
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f , (, =(f_1,f_2,cdots,f_n), )$ is a $mathbb C^n$-valued function on an open set $U$ of $mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$, $f$ is said to be $mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i ot=j$ implies $frac{partial f_i}{partial z_j}=0$. Here, $(z_1,z_2,cdots,z_n) in mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$ holds. We note that a $mathbb C^n$-analytic mapping and a $mathbb C^n$-analytic manifold can be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for a $mathbb C^n$-analytic function. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following partial result. If $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $frac{partial f_i}{partial z_j}=0$. Kasuya suggested that a $mathbb C^n$-analytic manifold and a $mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.
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