160 research outputs found
The ubiquitous quasidisk
No full textThis book focuses on gathering the numerous properties and many different connections with various topics in geometric function theory that quasidisks possess. A quasidisk is the image of a disk under a quasiconformal mapping of the Riemann sphere. In 1981 Frederick W. Gehring gave a short course of six lectures on this topic in Montreal and his lecture notes "Characteristic Properties of Quasidisks" were published by the University Press of the University of Montreal. The notes became quite popular and within the next decade the number of characterizing properties of quasidisks and their ramifications increased tremendously. In the late 1990s Gehring and Hag decided to write an expanded version of the Montreal notes. At three times the size of the original notes, it turned into much more than just an extended version. New topics include two-sided criteria. The text will be a valuable resource for current and future researchers in various branches of analysis and geometry, and with its clear and elegant exposition the book can also serve as a text for a graduate course on selected topics in function theory. Frederick W. Gehring (1925-2012) was a leading figure in the theory of quasiconformal mappings for over fifty years. He received numerous awards and shared his passion for mathematics generously by mentoring twenty-nine Ph.D. students and more than forty postdoctoral fellows. Kari Hag received her Ph.D. under Gehring's direction in 1972 and worked with him on the present text for more than a decade
A fractional Gehring lemma, with applications to nonlocal equations
No full textTo Carlo Sbordone on his 65th birthday.This paper reports the content of a talk given by the second-named author at the Accademia dei Lincei on November 26, 2013.International audienceWe describe a fractional version of the classical Gehring lemma. As a consequence, new self-improving regularity properties of solutions to integrodifferential equations emerge
Univalence criteria for analytic functions.
No full textThis thesis is devoted to the study of univalence criteria for analytic functions, in particular a domain constant known as the inner radius of D. Let D be a simply connected plane domain and let B be the unit disk. We define the inner radius of D, by\sigma(D) = \sup\{a : a \geq 0,\ \Vert S\sb{f}\Vert\sb{D}\ \leq\ a\ {\rm implies}\ {\it f\/}\ {\rm is\ univalent\ in}\ D\}.Here S\sb{f} is the Schwarzian derivative of f, \rho\sb{D} the hyperbolic density on D and\Vert S\sb{f}\Vert\sb{D} = {\sup\limits\sb{z\in D}}\vert S\sb{f}(z)\vert\rho\sbsp{D}{-2}(z).Domains for which the value of is known include disks, angular sectors and regular polygons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that = 2-\Vert S\sb{h}\Vert\sb{B} where h maps B conformally onto D. Because of the importance of this property, we say that D is a regular domain if = 2 -\Vert S\sb{h}\Vert\sb{B} is satisfied. First we use regularity to give a simple new proof of the result on regular n-sided polygons P\sb{n}. Next we study rectangles and equiangular hexagons. We prove that if R is a rectangle whose ratio of longer over shorter side is bounded from above by a specific constant ( 1.52346 then R is regular and = = \sigma(P\sb4). In a similar fashion, we prove that if H is an equiangular hexagon whose sides form the sequence baabaa with 1.67117 then H is regular and = = \sigma(P\sb6).. An interesting problem is to characterize regular domains. For domains of bounded boundary rotation with convex corners we show that{\limsup\limits\sb{\vert z \vert\to1}}\vert S\sb{h}(z)\vert(1 - \vert z\vert\sp2)\sp{\sp2} = \Vert S\sb{h}\Vert\sb{B}is a sufficient condition for regularity, where h maps B conformally onto D. Some results previously known for B can now be extended for all regular domains, in particular theorems of Gehring-Pommerenke, Ahlfors and Minda. The last part of the thesis is devoted to investigating an alternative domain constant, The only domains for which is known are disks. We demonstrate some bounds on for convex angular sectors.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104352/1/9513432.pdfDescription of 9513432.pdf : Restricted to UM users only
Quasiextremal distance domains and quasiconformal reflections.
No full textQuasiextremal distance (or QED) domains, domains whose complements do not change the extremal distance between continua by more than a fixed factor, were introduced by Gehring and Martio in connection with the theory of quasiconformal mappings in the Euclidean n-space \IR\sp{\rm n}. The purpose of this thesis is to study the geometry of these domains. With each domain D we associate two QED constants M(D) and M*(D), which measure how far D is from being a Mobius ball. It is known that M(D) and M*(D) are Mobius invariant and that M(D)=M*(D) = 2 when D is a ball or half space. We characterize all domains D for which M*(D) = 2 and prove that M*(D) = 2 if and only if D is a Mobius ball minus an NED set, a set each compact subset of which is the complement of a QED domain and has Lebesgue measure zero. To prove this, we establish some results about the extremal functions for the capacities of condensers by using variational integral methods. These functions turn out to be p-harmonic functions which are weak solutions of the non-linear equation div(u\vert\sp{\rm p-2}\nablau) = 0. We estimate the constants M(D) and M*(D) for different kinds of domains. For any domain D M(D)1 implies M(D)2. We find a sharp lower bound for M*(D) when D is locally raylike and a sharp upper bound for M*(D) when D is a reflection domain, a domain whose boundary admits a quasiconformal reflection. Based on these estimates, we are able to obtain the exact values of M(D) and M*() and for wedges, cones and related domains. For example, M(D) and M*(D) are equal to 6 or 8 when D is a regular triangle in \IR\sp2 or a cube in \IR\sp3, respectively. We also study some properties of reflection domains. In particular, by using the results about the estimates of M(D) and M*(D) mentioned above, we are able to find the extremal reflections for some domains such as regular n-gons, wedges and circular cones. This provides a new way to attack the important and difficult problem of finding extremal quasiconformal mappings between domains.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105631/1/9135725.pdfDescription of 9135725.pdf : Restricted to UM users only
[[alternative]]An Analogy of a Theorem of Gehring and Pommerenke under Nehari's
No full text[[abstract]]我們定義一個局部單葉的亞純函數 f 其 Schwarz 導數在 f'(z)≠0
的點為S_f=(f'')(f')'-1/2(f''/f')^2, 而在單極處則訂為 S_f=S_(1/
f). 我們早就知道S_f=0 若且唯若 f 是一個 Mobius 變換. 直接從定義
可得知 S_f 是一個可析函數.相反地, 如果 φ 是一個可析函數, 則存在
一個亞純函數 f 使得 S_f=φ. 早在1949年, Z. Nehari 就證明了 ∣
S_f(z)∣≦ 2/(1-∣z∣^2)^2) 及∣S_f(z)∣≦(π^2)/2 可導出 f 在單
位圓盤上的單葉性.F. W. Gehring 及 C.Pommerenke 更針對第一種條件
作深入的研究, 得到可同胚及可擬保角延拓的充分條件.而我們將注意力
集中在滿足Nehari第二種條件的函數和這些函數的延拓性.而這篇論文主
要是在單位圓盤D中就∣S_f(z)∣≦(π^2)/2 的條件作一檢視. 整個論文
的架構平行於Gehring 及 Pommerenke 的架構. 其中部份的技巧則模仿自
M. Chuaqui 及 B. Osgood 的研究工作.首先, 我們討論了 f 的解析性,
並且對 ∣f'∣及 ∣f∣ 的範圍作一估計.其次, 我們討論了 f 的單葉
性. 接著證明了 f 可連續延拓到 D 的邊界上. 再就此延拓是否一對一分
別討論. 如果是一對一, 則 f 可以擬保角延拓到整個複數平面; 若不是
一對一, f 則與 2/π tan πz/2 Mobius 共軛. 最後, 我們知道如果 f
不是與 2/ π tan πz/2 Mobius 共軛, 則 f 在 D 上滿足 Lipcshitz
條件.
We define the Schwarzian derivative of a function f which is
meromorphic andlocally univalent as S_f=(f''/f')'-(1/2)(f''/f')
^2 at the points where f'(0) is not equal to 0. And we define
S_f(z)=S_(1/f)(z) at the points which are simple poles. It's
shown in early age that S_f = 0 if and only if f is a Mobius
transformation. Directly from the definition, we know that S_f
is analytic. Conversely, suppose that φ is analytic
function, there exists a meromorphic function f such that S_f=
φ. Early in 1949, Z.
Nehari showed that conditions∣S_f(z)∣≦2/(1-∣z∣^2 )^2 and ∣
S_f(z)∣≦(π^2)/2 both imply that f is univalent in the unit
disk D. F. W. Gehring and C. Pommerenke focused their research
on the first condition that Nehari's deduced, and they had the
sufficient conditions that thefunction f can be extended to a
homeomorphic function and a quasiconformalfunction on the plane.
This paper mainly makes an investgation on which properties f
possessesunder the condition ∣S_f∣≦(π^2)/2 in the unit disc
D. The structure of this paper is parallel with the one of
Gehring and Pommerenke. And there are some skills imitated from
the research of M. Chuaqui and B. Osgood. First, we discuss the
analytic property of f under∣S_f∣≦(π^2)/2. We also estimate
the bound of ∣f'∣and ∣f∣. Next, we discuss the univalence of
fand prove that f has a continuous extension to the boundary of
D. Whether this extension is univalent or not give us two
directions. If the extension isone-to-one, f possess a
quasiconformal extension to the whole complex plane.If not, f is
a Mobius conjugate to 2/π tan πz/2. Finally, we prove that if
f is not a Mobius conjugate to f, then f satisfies Lipschitz
condition,which is the special case of Holder's continuity.
We define the Schwarzian derivative of a function f which is
Extension domains.
No full textThis research is concerned with a problem in higher dimensional quasiconformal mappings and a related problem in the complex plane. Specifically, we study domains D in \IR\sp{\rm n} for which each quasiconformal self map of D extends to a quasiconformal self map of \IR\sp{\rm n}. For n = 2, it was shown by Ahlfors and Rickman that a simply connected domain D has the above extension property if and only if D is a quasidisk. Suppose that G is a Jordan domain in \overline{\IR}\sp2. We consider the relationship between the geometry of G and the above extension property for D = G \IR. Vaisala showed that G \IR is quasiconformally equivalent B\sp3 if and only if G is an inner chordarc domain. For our extension problem we need to consider the geometry of such domains. We show that a Jordan domain \rm G\subset\IR\sb2 with locally rectifiable boundary is an inner chordarc domain if and only if for each straight cross-cut = (z, w) of G, where is the shorter arc in G joining z, w. Next G is an inner chordarc domain if and only if G is a John domain and G is regular. Finally, a characterization in terms of the corresponding Riemann map is given. Our extension results are the following. If G is a bounded inner chordarc Jordan domain in \IR\sp2, then G \IR does not have the extension property for the class of quasiconformal maps fixing . On the other hand, if G is an inner chordarc domain and if G* = \IR\sp2\\G, then G* \IR has the extension property for quasisymmetric maps. Next if D = G \IR is quasiconformally equivalent to B\sp{3}, then D has the quasiconformal extension property if and only if D is a quasiball. Finally, using the reflection properties of quasiconformal maps we show that if G is an unbounded quasidisk in \IR\sp2, then G \IR has the quasiconformal extension property in \overline{\IR}\sp3. We also construct a bounded quasidisk G such that G \IR has the above property.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105185/1/9116182.pdfDescription of 9116182.pdf : Restricted to UM users only
Quasiregular mappings and Royden algebras.
No full textLet be as domain in euclidean n-space and A() the Royden algebra of . Then A() is the Banach algebra of all functions \rm u \in C(\Omega) \cap L\sp\infty (\Omega) \cap L\sbsp{n}{1}(\Omega) with addition and multiplication defined pointwise and norm \rm \Vert u \Vert \sb\Omega = \Vert u \Vert \sb\infty + \Vert \nabla u \Vert \sb{L\sp n}\sb{(\Omega)}. It is known that domains and \Omega\sp\prime are quasiconformally equivalent if and only if there exists an algebra isomorphism T:A( \Omega \sp\prime)\to\rm A(\Omega). In this thesis, we characterize quasiregular mappings f:\Omega\to\Omega\sp\prime with finite multiplicity as exactly those mappings induced by algebra isomorphisms T:A(\Omega\sp\prime)\to\rm A where A is a subalgebra of A() which satisfies certain conditions concerning the separation of sets in by functions in A. We characterize closed quasiregular mappings in a similar fashion and show that the multiplicity of a quasiregular mapping induced by T:A(\Omega \sp\prime )\to \rm A \subset A(\Omega) is bounded above by T\Vert \sp{\rm n \sp2}. We also investigate the maximal ideal space * of A(), which is a compact, Hausdorff, topological space. We let denote the Royden boundary of which is the set *. If two domains are quasiconformally equivalent, then their Royden boundaries must be homeomorphic. Using the theory of nets, we are able to characterize elements of as certain types of nets for which no subnet is a sequence. We define fibers over points in and show that even in the special case that is the unit ball and all fibers over are homeomorphic, is not the natural topological product of with any particular fiber. Finally, we discuss two ways in which A() reflects the geometry of . First, we define a condition on sequences of level sets linked in which guarantees that is not quasiconformally equivalent to the unit ball. Second, we use Sario's and Nakai's definition of the harmonic boundary of to give a simple condition on the Royden algebra which is equivalent to the existence of a Green's function on .PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105449/1/9124109.pdfDescription of 9124109.pdf : Restricted to UM users only
The quasihyperbolic metric, growth, and John domains.
No full textThe research in this thesis stems from three elements of classical function theory. The first is a criterion due to Hardy and Littlewood for a function to be Holder continuous in the unit disk \rm I\!B\subset\doubc. The second is a famous inequality due to Bernstein which bounds the modulus of the derivative of a polynomial in terms of the degree and the L\sp\infty-norm of the polynomial in . The third is a class of domains first considered by Fritz John in his studies of plane elasticity and rigidity of local quasi-isometries. Suppose that f is a function analytic in and that Then the theorem of Hardy and Littlewood mentioned above asserts that\vert f\sp\prime(z)\vert\le m\ {\rm dist}(z, \partial {\rm I\!B})\sp{\alpha-1}for all if and only if\vert f(z\sb1)-f(z\sb2)\vert\le{M\over\alpha}\vert z\sb1-z\sb2\vert\sp\alphafor all z\sb1,\ z\sb2\in\rm I\!B, where m and M depend only on each other. Bernstein's inequality states that if p(z) is a polynomial of degree n, then\sup\sb{\rm I\!B}\vert p\sp\prime(z)\vert\le n\ \sup\sb{\rm I\!B}\vert p(z)\vert.\eqno(1) A domain D\subset\IR\sp{n} is a b-John domain if each pair of points x\sb1,\ x\sb2\in D can be joined by an arc for which\min\sb{j=1,2}l(\gamma(x\sb{j},y))\le b\ {\rm dist}(y, \partial D)for all , where \gamma(x\sb{j},y) is the subarc of with endpoints x\sb{j} and y. A domain is John if it is b-John for some constant b, and a simply-connected John domain in the plane is a John disk. John domains appear naturally in many areas of analysis, including complex dynamics, approximation theory, and elasticity. My research concerns the following four questions. (1) What analogues of the Hardy-Littlewood result hold when ? (2) What analogues of this result hold when is replaced by a domain D\subset\doubc and ? (3) What analogues of this result hold when f is an arbitrary function defined in a domain D \subset \IR\sp{n} and ? (4) For which continua E\subset\doubc does an analogue of inequality (1) hold? John domains arise in examining the second, third, and fourth questions listed above.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104976/1/9624658.pdfDescription of 9624658.pdf : Restricted to UM users only
Properties of John disks.
No full textA domain D in euclidean n-space R\sp n is said to be a John Domain if there exist a point x\sb0 in D and a constant such that point x\sb1 in D can be joined in D to x\sb0 by an arc such that the length of the subarc from x\sb1 to x in is bounded above by c times the distance from x to the boundary of D. This class was first studied by Fritz John in 1961 in connection with his work on elasticity and local quasi-isometries. In this thesis we study John disks, the John domains D in R\sp2 for which is a Jordan curve, and we give new conformally invariant, geometric and function theoretic properties and characterizations for this class. The conformally invariant properties involve harmonic and hyperbolic measure, the geometric conditions concern quasidisks and a quasiextremal distance property, and the function theoretic properties consist of an analogue of the Bernstein inequality and the conformal mapping of the exterior of the unit disk onto the exterior of D. In the final chapter we characterize the compact sets on which the above analogue of the Bernstein inequality holds in terms of the exterior mapping function.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105598/1/9135685.pdfDescription of 9135685.pdf : Restricted to UM users only
The apollonian metric, sets of constant width and Moebius modulus of ring domains.
No full textIn this thesis we study the concepts of apollonian metric, sets of constant width and Mobius modulus of a ring domain. The apollonian metric, introduced in 1995 by A. Beardon, can be thought of as a generalization of the hyperbolic metric to more general domains. We give a positive answer to a question of Beardon, namely, that the apollonian isometries are the restrictions of Mobius transformations, in two instances. Sets of constant width have been an object of study by geometers for several centuries; some non-trivial examples of such sets were already known to Euler. We give a new characterization of these sets in terms of the apollonian metric. In particular, we give characterize balls in terms of this metric. We introduce the Mobius modulus of a ring domain in the context of quasiconformal mappings. It shares many properties with the classical conformal modulus of such domains which is one of the main tools in the study of quasiconformal mappings. We solve Teichmuller's problem for a Mobius modulus, and as a result we settle a conjecture of Vuorinen.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132111/2/3057971.pd
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