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Univalent Functions Volumes 1 And 2 9780936166124 A W Goodman
in this paper we consider two classical objects of univalent functions: starlike and convex functions. we show that if the boundary of a polygon is convex and of positive curvature, then the starlike extension of the boundary of this polygon is a convex function.the converse is not true, but this property, i.e. the fact that the starlike extension of the boundary of a convex polygon is a convex function, has a deep geometric meaning. this fact and some applications are proved in this paper.
in this paper we introduce and study the class of harmonic univalent functions of order k between a circle and a jordan arc in the complex plane. the class of harmonic univalent functions of order k of one-to-one holomorphic mappings between two regions of the plane has been introduced by the author in [1]. in this article we study the class of functions introduced by the author in [1]. we study some of the properties of this class, and give a representation theorem for the harmonic univalent mappings of this class. in particular, we show that the harmonic univalent mappings of this class have the so-called wright's asymptotic function as their boundary values at the boundary of the circle, when the function is extended to the interior of the circle by means of the harmonic mean. we also show that the class of harmonic univalent mappings of this class is closed with respect to the generalized composition. some interesting applications of these results to the study of univalent functions are also discussed.
in this paper we study the problem of matching generalized polygons under affine transformations. our approach is based on invariants. firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. these invariant functions are defined as quotients of the so-called fourier descriptors, also known as discrete fourier transforms.each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.more formally, given a collection { z 1, z 2,, z m } of n-sided polygons in the plane and a query polygon w, we give algorithms to find all z such that f ( z ) = w + w, where f is an unknown affine transformation and w = ( w 1,, w n ) with w k, where is certain tolerance.
article{a1991, abstract = {we introduce a new class of normalized functions regular and univalent in the unit disk. these functions, called uniformly convex functions, are defined by a purely geometric property. we obtain a few theorems about this new class and we point out a number of open problems.}, author = {a. w. goodman}, journal = {annales polonici mathematici}, keywords = {univalent functions; convex functions; coefficient bounds; circular arc; convex arc; uniformly convex functions}, language = {eng}, number = {1}, pages = {87-92}, title = {on uniformly convex functions}, url = { volume = {56}, year = {1991},} we consider the classical subclasses of univalent functions in the upper half-plane and show how they can be constructed from the classical subclasses of analytic and univalent functions in the unit disk. in addition, we construct several subclasses of univalent functions in the unit disk from certain classical subclasses of harmonic and univalent functions in the unit disk. the composition of two such functions is also studied. we introduce and study several well-known subclasses of analytic and univalent functions in the upper half-plane. these are the functions of b, b_m, and b_u. here the function of b is the hyperbolic analog of the blaschke products, the function of b_m is a hyperbolic analog of b_u, and finally the function of b_u is a hyperbolic analog of the well-known univalent functions volumes 1 and 2 9780936166124 a w goodman. we introduce the class of harmonic-convex mappings, which extends the class of harmonic mappings of positive real part by adding a condition of bounded real part of the square of the derivative. we give some properties of this class of functions. 5ec8ef588b
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