Two-point Distortion Theorems Research Articles

Two-Point Distortion Theorems for Harmonic Mappings

Two-Point Distortion Theorems for Harmonic Mappings

Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions

For a meromorphic function f in the unit disk U={z:|z|<1} and arbitrary points z1,z2 in U distinct from the poles of f, a sharp upper bound on the product |f′(z1)f′(z2)| is established. Further, we prove a sharp distortion theorem involving the derivatives f′(z1), f′(z2) and the Schwarzian derivatives Sf(z1), Sf(z2) for z1,z2∈U. Both estimates hold true under some geometric restrictions on the image f(U).

Bounded Schwarzian and Two-Point Distortion

The Schwarzian derivative of a locally injective holomorphic function $$f$$ is $$S_f= f'''/f' - (3/2)\left( f''/f'\right) ^2$$ . As is well-known, $$S_f=0$$ if and only if $$f$$ is a Mobius transformation. Intuitively, if a locally injective holomorphic function has a small Schwarzian derivative, then it should behave roughly like a Mobius transformation. Two quantitative results of this type are established. First, if $$|S_f(z)|\le 2t, z \in \Omega $$ , on a convex region $$\Omega $$ , then sharp upper and lower two-point distortion bounds on $$|f(a)-f(b)|$$ for $$a,b \in \Omega $$ are given. The upper bound is valid for all $$a,b \in \Omega $$ while the lower bound is valid for $$|a-b| < \pi /\sqrt{t}$$ . For $$t=0$$ the bounds are the familiar identity $$\vert f(a)-f(b) \vert = \vert a-b\vert \sqrt{|f'(a)||f'(b)|}$$ for Mobius transformations. These upper and lower two-point distortion theorems characterize locally injective holomorphic functions with bounded Schwarzian derivative. Second, if $$\Omega $$ is a convex region with diameter $$D$$ and $$|S_f(z)| \le 2t<\pi ^2/D^2$$ for $$z\in \Omega $$ , then $$f$$ is $$K_t(D)$$ -quasi-Mobius, where the constant depends only on $$t$$ and $$D$$ . This means that $$1/K_t(D) \le |f(a),f(b),f(c),f(d)|/|a,b,c,d|\le K_t(D)$$ for all distinct $$a,b,c,d\in \Omega $$ , where $$|a,b,c,d|$$ denotes the absolute cross-ratio.

Symmetrization of condensers and inequalities for functions multivalent in a disk

Using the circular symmetrization of sets and condensers on Riemann surfaces, we establish new inequalities for multivalent functions with conditions on the critical values of the functions or on the coverings of concentric circles. Two-point distortion theorems, an inequality for the initial coefficients, and a lower bound for the modulus of functions (of diverse classes) p-valent in a disk are proved.

Growth and distortion theorems for linearly invariant families on homogeneous unit balls in [formula omitted

Let B be a homogeneous unit ball in X=Cn. In this paper, we obtain growth and distortion theorems for linearly invariant families F of locally biholomorphic mappings on the unit ball B with finite norm-order ‖ord‖e,1F. We use the Euclidean norm for the target space instead of the norm of X, because we are able to obtain lower bounds in the two-point distortion theorems for linearly invariant families on any homogeneous unit ball in Cn. We also obtain similar results for affine and linearly invariant families (A.L.I.F.s) of pluriharmonic mappings of the unit ball B into Cn. Again, in most of these results, we use the Euclidean norm for the target space, to obtain lower bounds in the two-point distortion theorems for A.L.I.F.s on B. These results are generalizations to homogeneous unit balls of recent results due to Graham, Kohr and Pfaltzgraff, the authors of this paper, and Duren, Hamada and Kohr. In the last section, we consider two-point distortion theorems for L.I.F.s and A.L.I.F.s on the unit polydisc Un in Cn.

Two-Point Distortion for Nehari Functions

Let \(\mathcal N (t)\), \(t\ge 0\), be the Nehari class of locally injective holomorphic functions on the unit disk \(\mathbb D \) that satisfy $$\begin{aligned} \sup _{z\in \mathbb D }\big (1-|z|^2\big )^2|S_f(z)| \le 2t, \end{aligned}$$ where \(S_f = f^{\prime \prime \prime }/f^{\prime } - (3/2)\big (f^{\prime \prime }/f^{\prime }\big )^2\) is the Schwarzian derivative of \(f\). Sharp two-point upper and lower distortion theorems for these functions were recently established by Chuaqui, Duren, Ma, Mejia, Minda and Osgood. A classical result of Krauss shows that all univalent functions on \(\mathbb D \) lie in \(\mathcal N (3)\). There are two different two-point upper distortion theorems for univalent functions due to Jenkins, Ma and Minda, and Kraus and Roth. Two similar two-point upper distortion theorems hold for \(\mathcal N (t)\). These two-point upper distortion theorems for \(\mathcal N (3)\) are the known two-point upper distortion theorems for univalent functions, so the latter are actually valid for the larger class \(\mathcal N (3)\). Two-point distortion theorems for \(\mathcal N (t)\) imply local uniform control in the hyperbolic sense on absolute cross-ratio distortion for functions in \(\mathcal N (t)\).

Two-point distortion theorems for harmonic and pluriharmonic mappings

Two-point distortion theorems for harmonic and pluriharmonic mappings

Two-point distortion theorems for harmonic mappings

In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in ${\Bbb R}^n$ are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.

Growth and two-point distortion for biholomorphic mappings of the ball¶

In this article we obtain invariant two-point distortion theorems for univalent holomorphic mappings of the unit ball in one and in several variables. A new locally uniform minimum growth bound for biholomorphic mappings of the unit ball in dimension n>1 is obtained for those mappings with finite “norm order”. The results are framed in the context of linear invariant families and the Koebe transform. ¶Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday

Extremal Problems of Function Theory Connected with the n-Fold Symmetry

We consider some conventional problems of the theory of functions of a complex variable such that their extremal configurations have the n-fold symmetry. We discuss two-point distortion theorems corresponding to the two-fold symmetry. New estimates are obtained for the module of a doubly connected domain. These estimates generalize known results by Rengel, Grotzsch, and Teichmuller to the case of rings with the n-fold symmetry, where \( \geqslant 2\). New distortion theorems are proved for functions meromorphic and univalent in a disk or in a ring. In these theorems, the extremal function also has the corresponding symmetry. All of the problems mentioned above are unified by the method applied; this method is based on properties of the conformal capacity and on symmetrization. Bibliography: 27 titles.

Distortion theorems for hyperbolically convex functions

In the paper we study the class of univalent hyperbolically convex (h-convex) functions in the unit disc . Two types of normalization are considered. The class K h(α) is defined by the inner normalization . Another natural normalization is the so-called boundary normalization. We consider all h-convex functions f with fixed distance >0 from the boundary of to the origin. This class is denoted by H(c). The main result for both normalizations is the sharp upper estimate of . For the boundary normalization we use the extremal length method and derive some general two-point distortion theorems. For the class K h(α) we use the fact that the map f:2/z is starlike in .

A Coefficient Problem for Univalent Functions Related to Two-Point Distortion Theorems

A Coefficient Problem for Univalent Functions Related to Two-Point Distortion Theorems

Two-Point Distortion Theorems for Spherically Convex Functions

One-parameter families of sharp two-point distortion theorems are established for spherically convex functions f , that is, meromorphic univalent functions f defined on the unit disk D such that f(D) is a spherically convex subset of the Riemann sphere P. These theorems provide for a, b ∈ D sharp lower bounds on dP(f(a), f(b)), the spherical distance between f(a) and f(b), in terms of dD(a, b), the hyperbolic distance between a and b, and the quantities (1 − |a|2)f (a), (1 − |b|2)f (b), where f = |f ′|/(1 + |f |2) is the spherical derivative. The weakest lower bound obtained is an invariant form of a known growth theorem for spherically convex functions. Each of the two-point distortion theorems is necessary and sufficient for spherical convexity. These twopoint distortion theorems are equivalent to sharp two-point comparison theorems between hyperbolic and spherical geometry on a spherically convex region Ω on P. Each of these two-point comparison theorems characterize spherically convex regions.

Distortion theorems for higher order Schwarzian derivatives of univalent functions

Let S ~ \tilde {\mathcal {S}} denote the class of functions which are univalent and holomorphic on the unit disc. We derive a simple differential equation for the Loewner flow of the Schwarzian derivative of a given f ∈ S ~ f \in \tilde {\mathcal {S}} . This is used to prove bounds on higher order Schwarzian derivatives which are sharp for the Koebe function. As well we prove some two-point distortion theorems for the higher order Schwarzians in terms of the hyperbolic metric.

Two-Point Distortion Theorems for Strongly Close-to-Convex Functions

A holomorphic function f defined on the unit disk is called strongly close-to-convex of order α > 0 if there is a convex univalent function φ defined on such that For α ε (0.1) this condition f is univalent in . By convention a convex univalent function is called close-to-convex of order 0. If f is close-to-convex of order α ε [0, 1 ] and , then where and is the hyperbolic distance between a and b. Equality holds for distinct if and only if where S is a conformal automorphism of automorphism , T is a conformal automorphism of ; and .

Two-point distortion theorems for univalent functions

Two-point distortion theorems for univalent functions