6,540 research outputs found
Pointwise convergence on the boundary in the Denjoy-Wolff Theorem
If is an analytic selfmap of the disk (not an elliptic automorphism)
the Denjoy-Wolff Theorem predicts the existence of a point with
such that the iterates converge to uniformly on compact subsets
of the disk. Since these iterates are bounded analytic functions, there is a
subset of the unit circle of full linear measure where they all well-defined.
We address the question of whether convergence to still holds almost
everywhere on the unit circle. The answer depends on the location of and
the dynamical properties of . We show that when (elliptic case),
pointwise a.e. convergence holds if and only if is not an inner
function. When things are more delicate. We show that when is
hyperbolic or type I parabolic, then pointwise a.e. convergence holds always.
The last case, type II parabolic remains open at this moment, but we conjecture
the answer to be as in the elliptic case.Comment: 11 page
Finite interpolation with minimum uniform norm in C^n
Given a finite sequence in a domain , and complex scalars , consider the classical extremal
problem of finding the smallest uniform norm of a holomorphic function
verifying for all . We show that the modulus of the solutions
to this problem must approach its least upper bound along a subset of the
boundary of the domain large enough to contain the support of a measure whose
hull contains a subset of the original large enough to force the same
minimum norm. Furthermore, all the solutions must agree on a variety which also
contains this hull. An example is given to show that the inclusions can be
strict
Neighborhoods of univalent functions
The main result shows a small perturbation of a univalent function is again a
univalent function, hence a univalent function has a neighborhood consisting
entirely of univalent functions.
For the particular choice of a linear function in the hypothesis of the main
theorem, we obtain a corollary which is equivalent to the classical
Noshiro-Warschawski-Wolff univalence criterion.
We also present an application of the main result in terms of Taylor series,
and we show that the hypothesis of our main result is sharp.Comment: 9 pages, 1 figur
Julius and Julia: Mastering the art of the Schwarz lemma
This article discusses classical versions of the Schwarz lemma at the
boundary of the unit disk in the complex plane. The exposition includes
commentary on the history, the mathematics, and the applications.Comment: Expository article to appear in the American Mathematical Monthly. 19
pages, 4 figures, uses tik
The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings
We give the lower bound for the modulus of the radial derivatives and
Jacobian of harmonic injective mappings from the unit ball onto convex domain
in plane and space. As an application we show co-Lipschitz property of some
classes of qch mappings. We also review related results in planar case using
some novelty.Comment: We added two references and subsection 3.1. Also we made small
changes concerning the title and the abstrac
Conformal models and fingerprints of pseudo-lemniscates
We prove that every function that is meromorphic on the closure of an
analytic Jordan domain and sufficiently well-behaved on the boundary is
conformally equivalent to a rational map whose degree is smallest possible. We
also show that the minimality of the degree fails in general without the
boundary assumptions. As an application, we generalize a theorem of Ebenfelt,
Khavinson and Shapiro by characterizing fingerprints of polynomial
pseudo-lemniscates.Comment: 12 page
Projective Hulls and Characterizations of Meromorphic Functions
We give conditions characterizing holomorphic and meromorphic functions in
the unit disk of the complex plane in terms of certain weak forms of the
maximum principle. Our work is directly inspired by recent results of John
Wermer, and by the theory of the projective hull of a compact subset of complex
projective space developed by Reese Harvey and Blaine Lawson
Distinguished Varieties
A distinguished variety is a variety that exits the bidisk through the
distinguished boundary. We show that Ando's inequality for commuting matrix
contractions can be sharpened to looking at the maximum modulus on a
distinguished variety, not the whole bidisk. We show that uniqueness sets for
extremal Pick problems on the bidisk always contain a distinguished variety
The Projective Hull of Certain Curves in C^2
The projective hull X^ of a subset X in complex projective space P^n is an
analogue of the classical polynomial hull of a set in C^n. If X is contained in
an affine chart C^n on P^n, then the affine part of X^ is the set of points x
in C^n for which there exists a constant M=M_x so that
|p(x)| < M^d sup{|p(y)| : y in X} for all polynomials p of degree less than
or equal to d, and any d > 0. Let X^(M) be the set of points x where M_x can be
chosen < M. Using an argument of E. Bishop, we show the following. Let G be a
compact real analytic curve (not necessarily connected) in C^2. Then for any
linear projection p: C^2 --> C^1, the set of points in G^(M) lying above a
point z in C^1 is finite for almost all z. Using this, we prove the conjecture
that for any compact stable real-analytic curve G in P^n, the set G^-G is a
1-dimensional complex analytic subvariety of P^n-G.Comment: We have added a new section giving the details of boundary regularity
in this contex
Some extensive discussions of Liouville's theorem and Cauchy's integral theorem on structural holomorphic
Classic complex analysis is built on structural function only
associated with Cauchy-Riemann equations, subsequently various generalizations
of Cauchy-Riemann equations start to break this situation. The goal of this
article is to show that only structural function such that
Liouville's theorem is held, otherwise, it's not valid any more on complex
domain based on structural holomorphic, the correction should be , where . Those theories in complex analysis which keep
constant are unable to be held as constant in the framework of structural
holomorphic. Synchronously, it deals with the generalization of Cauchy's
integral theorem by using the new perspective of structural holomorphic, it is
also shown that some of theories in the complex analysis are special cases at
, which are narrow to be applied such as maximum modulus principle.Comment: 14 page
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