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1.
We apply the capacity and symmetrization methods to distortion theorems for analytic functions in an annulus. We show that the classical Teichmüller estimate for the capacity of a doubly-connected domain yields a series of the already known and new inequalities for univalent functions. In particular, we supplement the results of Grötzsch, Duren, and Huckemann. Using the dissymmetrization of condensers we establish sharp estimates for local distortion and the distortion of level curves in n ≥ 2 symmetric directions. In terms of Robin functions we give an analog of the Nehari inequality: some general distortion theorem for several points taking into account the boundary behavior of the function and describing the cases of equalities. As a corollary we give analogs of some inequalities of Solynin, Pommerenke, and Vasil’ev that were obtained previously for univalent and bounded functions in a disk. We prove a distortion theorem that involves the Schwarzian derivatives at symmetric points on the unit circle. 相似文献
2.
借用著名的Ruscheweyh导数,引入了一类单叶保向调和函数族. 通过建立极值理论,得到了关联该族的最优系数边界、最优增长定理和最优偏差定理. 同时,给出了该族与先前已有调和函数族之间的转换半径. 最后,讨论了基于该族的修正哈达玛乘积结果. 相似文献
3.
New applications of generalized condensers with two or more plates are considered. Based on one and the same approach and
simplest properties of such condensers, boundary distortion theorems for analytic univalent functions bounded in a disk are
established, as well as bounds for certain combinations of coefficients in expansions of such functions, inequalities for
polynomials, and theorems on extremal decompositions of the complex sphere. Part of the results obtained contains new information
on the Schwarzian derivative of a regular univalent function. Bibliography: 21 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 52–75. 相似文献
4.
A holomorphic functionf defined on the unit disk d is called a Bloch function provided {fx73-02} For α ∃ (0,1] letB∞(α)denote the class of locally univalent Bloch functionsf normalized by ∥f∥B ≤1f(0) = 0 andf’(0) = α. A type of subordination theorem is established for B∞(α). This subordination theorem is used to derive sharp growth,
distortion, curvature and covering theorems for B∞(α).
Supported as a Feodor Lynen Fellow of the Alexander von Humboldt Foundation.
Research supported in part by a National Science Foundation grant. 相似文献
5.
Kai Rajala 《Mathematische Annalen》2007,339(2):445-460
We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend
the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially
integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural
form.
Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author
was visiting at the University of Michigan. He wishes to thank the department for hospitality. 相似文献
6.
Mengkun Zhu & Xinzhong Huang 《数学研究》2015,48(3):265-274
Some sharp estimates for coefficients, distortion and the growth order are
obtained for harmonic mappings $f ∈ TL^α_H$which are locally univalent harmonic mappings
in the unit disk $\mathbb{D}:=\{z:|z| < 1\}$. Moreover, denoting the subclass $TS^α_H$ of the
normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f ∈ TS^α_H,$ and their covering theorems. 相似文献
7.
Mengkun Zhu & Xinzhong Huang 《数学研究》2016,49(1):23-32
Some sharp estimates for coefficients, distortion and the growth order are
obtained for harmonic mappings $f \in TL^{\alpha}_H,$ which are locally univalent harmonic mappings
in the unit disk $\mathbb{D}:=\{z:|z|<1\}.$ Moreover, denoting the subclass $TS^{\alpha}_H$ of the
normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f \in TS^α_H,$ and their covering theorems. 相似文献
8.
A. Khatamov 《Mathematical Notes》2000,67(4):516-522
We obtain an instance of the multidimensional analogs of the classical direct and converse Jackson and Bernshtein-Vallée-Poussin
theorems (a version stronger than those obtained by S. M. Nikol’skii) and significant generalizations of these theorems to
polynomial approximations of functions of real variables from Nikol’skii and Besov spaces on bounded domains with Lipschitzian
boundary. The direct theorem concerning the Nikol’skii spaces was previously obtained by Yu. A. Brudnyi and generalized to
joint polynomial approximations of functions and their derivatives by B. N. Konovalov.
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 608–615, April, 2000. 相似文献
9.
In this paper we introduce the notion of tolerance in connection with Helly-type theorems and prove, using the Erdős–Gallai
theorem, that any Helly-type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number
of exceptional sets or points. In particular, we analyze some of the classical Helly-type theorems, such as Caratheodory’s
and Tverberg’s theorems, as well as some other interesting ones. 相似文献
10.
关于某类解析函数的星象性和Ruscheweyh的一个问题 总被引:2,自引:0,他引:2
本文引入了一个涉及Ruscheweyh导数的解析函数子类,应用微分从属方法和Carlson-Shaffer算子讨论了它的从属关系和偏差定理;其次,应用单叶函数的性质和一个微分不等式研究了它的星象性条件和覆盖定理,最后,部分地解决了Ruscheweyh的一个问题。 相似文献
11.
We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L
p
-spaces (Sobolev case) and spaces of H?lder continuous functions (Morrey case). In the case of bounded domains the generalized
embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be
“almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”,
where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives,
and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability
requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of
exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes
the embeddings indeed do not hold. 相似文献
12.
V. N. Dubinin 《Journal of Mathematical Sciences》2007,143(3):3069-3076
It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and
its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose
zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis,
are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by
Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of
a polynomial is proved. Bibliography: 16 titles.
Dedicated to the 100th anniversary of G. M. Goluzin’s birthday
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 101–112. 相似文献
13.
V. N. Dubinin 《Journal of Mathematical Sciences》1998,89(1):967-975
Let h(zξ)−log|z−ξ| be the Green function of a planar domain D. The behavior of the linear combination h(z,z)+h(ξ,ξ)−2h(z,ξ)
under certain symmetrization transformations of D is studied. Covering and distortion theorems in the theory of univalent
functions are proved as applications. Bibliography: 9 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 80–92. 相似文献
14.
A. I. Perov 《Differential Equations》2012,48(5):670-680
We obtain new conditions for the existence of bounded solutions of higher-order nonlinear differential equations. In addition to the classical contraction mapping principle, A.N. Tikhonov’s fixed-point principle is used in the proof of existence theorems. Assertions dealing with the stability of a bounded solution are derived directly from the corresponding results obtained by M.A. Krasnosel’skii and A.V. Pokrovskii. 相似文献
15.
Vector-valued laplace transforms and cauchy problems 总被引:43,自引:0,他引:43
Wolfgang Arendt 《Israel Journal of Mathematics》1987,59(3):327-352
Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central
tool is an “integrated version” of Widder’s theorem (characterizing Laplace transforms of bounded functions). It holds in
any Banach space (whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the
Radon-Nikodym property). The Hille-Yosida theorem and other generation theorems are immediate consequences. The method presented
here can be applied to operators whose domains are not dense. 相似文献
16.
In this paper, we establish distortion theorems for both normalized p-Bloch functions with branch points and normalized locally univalent p-Bloch functions defined on the unit disk, respectively. These distortion theorems give lower bounds on |f′(z)| and ■f′(z). As applications of these distortion theorems, the lower bounds of the radius of the largest schlicht disk on these Bloch functions are given, respectively. Notice that when p = 1, our results reduce to that of Liu and Minda. 相似文献
17.
Frederick Bagemihl 《Israel Journal of Mathematics》1966,4(4):230-232
Meier’s topological analogue of Fatou’s theorem is shown to be sharp by exhibiting a bounded holomorphic function in the unit
disk for which no point of a prescribed set of first category on the unit circle is a Meier point.
Supported by the U. S. Army Research Office, Durham. 相似文献
18.
Given scattered data on the real line, Favard [4] constructed an interpolant which depends linearly and locally on the data
and whose nth derivative is locally bounded by the nth divided differences of the data times a constant depending only on
n. It is shown that the (n —1)th derivative of Favard’s interpolant can be likewise bounded by divided differences, and that
one can bound at best two consecutive derivatives of any interpolant by the corresponding divided differences. In this sense,
Favard’s univariate interpolant is the best possible. Favard’s result has been extended [8] to a special case in several variables,
and here the extent to which this can be repeated in a more general setting is proven exactly. 相似文献
19.
Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the
Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of
the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues.
1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics.
Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption
is unnecessary and irrelevant.
2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently,
this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on
Riemann’s Mapping Theorem.)
3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While
previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence
assumption is unnecessary.
Oblatum 15-V-1995 & 13-XI-1995 相似文献
20.
Morteza Moniri 《Archive for Mathematical Logic》2010,49(3):291-300
In this paper we prove conservation theorems for theories of classical first-order arithmetic over their intuitionistic version.
We also prove generalized conservation results for intuitionistic theories when certain weak forms of the principle of excluded
middle are added to them. Members of two families of subsystems of Heyting arithmetic and Buss-Harnik’s theories of intuitionistic
bounded arithmetic are the intuitionistic theories we consider. For the first group, we use a method described by Leivant
based on the negative translation combined with a variant of Friedman’s translation. For the second group, we use Avigad’s
forcing method. 相似文献