Univalent Polynomials Research Articles

Asymptotic Conformality and Polygonal Approximation

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given.

On the Koebe quarter theorem for trinomials with fold symmetry

The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and T T -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials z + a z T + 1 + b z 2 T + 1 z + az^{T + 1} + bz^{2T + 1} . Namely, the Koebe radius is r = 4 cos 2 ⁡ ( π ( 1 + T ) 2 + 3 T ) r=4\cos ^2\left (\frac {\pi (1+T)}{2+3T}\right ) , and the only extremizer of the Koebe problem is the trinomial B ( T ) ( z ) = z + 2 2 + 3 T ( − T + ( 2 + 2 T ) cos ⁡ ( π T 2 + 3 T ) ) z 1 + T + + 1 2 + 3 T ( 2 + T − 2 T cos ⁡ ( π T 2 + 3 T ) ) z 1 + 2 T . \begin{gather*} B^{(T)}(z)=z+\frac 2{2+3T}\left (-T+(2+2T)\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+T}+\\ +\frac 1{2+3T}\left (2+T-2T\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+2T}. \end{gather*}

An Extremal Problem for Odd Univalent Polynomials

An Extremal Problem for Odd Univalent Polynomials

An extremal problem for polynomials

For the polynomials F ( z ) = ∑ j = 1 N a j z j with real coefficients and normalization a 1 = 1 we solve the extremal problem sup a 2 , … , a N ⁡ ( inf z ∈ D ⁡ { Re ( F ( z ) ) : Im ( F ( z ) ) = 0 } ) . We show that the solution is − 1 4 sec 2 ⁡ π N + 2 , and the extremal polynomial 1 U N ′ ( cos ⁡ π N + 2 ) ∑ j = 1 N U N − j + 1 ′ ( cos ⁡ π N + 2 ) U j − 1 ( cos ⁡ π N + 2 ) z j is unique and univalent, where the U j ( x ) are the Chebyshev polynomials of the second kind, j = 1 , … , N . As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.

Peaking and interpolation by complex polynomials

AbstractClassical results about peaking from complex interpolation theory are extended to polynomials on a closed disk, and on the complement of its interior. New results are obtained concerning interpolation by univalent polynomials on a Jordan domain whose boundary satisfies certain smoothness conditions.

Univalent polynomials and Hubbard trees

We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such $f$ are in natural 1:1 correspondence with anti-holomorphic polynomials of degree $d$ with $d-1$ distinct, fixed critical points (classified by their Hubbard trees).

On a subclass of close-to-convex harmonic mappings

For $\alpha > -1$ and $\beta >0, $ let $\mathcal{B}_{\mathcal{H}}^0(\alpha, \beta)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+\alpha(h'(z)-1)|\leq \beta-|zg''(z)+\alpha g'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $\beta \in (0, 1+\alpha]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.

On a subclass of close-to-convex harmonic mappings

In this paper, we introduce a new class of sense preserving harmonic mappings [Formula: see text] in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.

On a subclass of close-to-convex harmonic mappings

For $\alpha > -1$ and $\beta >0, $ let $\mathcal{B}_{\mathcal{H}}^0(\alpha, \beta)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+\alpha(h'(z)-1)|\leq \beta-|zg''(z)+\alpha g'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $\beta \in (0, 1+\alpha]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.

Generalized Chudnovsky Algorithm for Real-Time PWM Selective Harmonic Elimination/Modulation: Two-Level VSI Example

This article presents a novel approach to significantly reducing the mathematical complexity associated with the selective harmonic elimination (SHE), which allows for the real-time implementation of SHE with a reasonable utilization of the computational power of the digital controller. First, the transcendental equations of the optimization problem are converted to a set of algebraic equations using Chebyshev polynomials. Second, the set of algebraic equations is reduced to a single univalent polynomial using generalized Newton's identities. The roots of the polynomial hold a unique solution to the optimization problem. We show that the roots of the univalent polynomial can be solved in real time by direct substitution with no initial guess or iteration. The proposed method is a generalization of a Chudnovsky algorithm; it allows the modulation of the selected harmonics rather than eliminating them. Furthermore, the technique by which the roots of the polynomial are obtained enables one to implement the algorithm in real time with determinate execution time. This article lays the groundwork for presenting the generalized algorithm through a two-level voltage-source inverter drive. More application will follow in the near future.

Univalent polynomials and Koebe’s one-quarter theorem

The famous Koebe $\frac14$ theorem deals with univalent (i.e., injective) analytic functions $f$ on the unit disk $\mathbb D$. It states that if $f$ is normalized so that $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disk of radius $\frac14$ about the origin, the value $\frac14$ being best possible. Now suppose $f$ is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.

On univalent polynomials with critical points on the unit circle

Brannan showed that a normalized univalent polynomial of the form (P(z)=z+a2 z2+...+ an-1zn-1+znn) is starlike if and only if (a2=...=an-1=0). We give a new and simple proof of his result, showing further that it is also equivalent to the membership of P in the Noshiro–Warschawski class of univalent functions whose derivative has positive real part in the disk. Both proofs are based on the Fejér lemma for trigonometric polynomials with positive real part.

Some basic properties of certain subclass of harmonic univalent functions

For , let denote the class of sense preserving harmonic mappings in the unit disk satisfying . The main aim of this paper is to study some basic properties such as coefficient bounds, growth estimates and convolution for functions in the class . We end the paper with an application, and construct harmonic univalent polynomials belonging to .

On the dynamics of roots and poles for solutions of the Polubarinova-Galin equation

We study the dynamics of roots of f'(z,t), where f(z,t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(z,t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.

A family of univalent polynomials in an angular domain

We study the properties of symmetric polynomials of a particular form and use them to find a maximal angular domain of univalence for some family of polynomials.

Univalent polynomials and fractional order differences of their coefficients

Using fractional order differences, we find sufficient conditions on the coefficients of certain polynomials defined on the unit disk D : { z , | z | < 1 } such that the zeros of these polynomials does not lie in D .

The univalent polynomial of Suffridge as a summability kernel

A positive summability trigonometric kernel is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K n *f } approximates every continuous 2π-periodic function f with the rate ω( f,1 / n), where ω( f,δ) denotes the modulus of continuity, and this provides a new proof of the classical Jackson's theorem. Despite that it turns out that Kn (θ) coincide with positive cosine polynomials generated by Fejér, our proof differs from others known in the literature.

Relations between univalent and orthogonal polynomials

Two sequences of polynomials are studied. One satisfies a three term recurrence relation for specific parameters and another a para-orthogonality property. Using the fact that these polynomials have their zeros lying on the unit circle and some other properties, we establish a criterion in order that the polynomials be univalent in the open unit disk.

Ullemar's formula for the jacobian of the complex moment mapping

The complex moment sequence μ ( P ) is assigned to a univalent polynomial P ( z ) by the Cauchy transform of the domain , where is the unit disk. We establish the representation of the Jacobian in terms of roots of the derivative . Combining this result with the special decomposition for the Hurwitz determinants, we prove a formula for , which was previously conjectured by Ullemar. As a consequence, we show that the boundary of the class of all locally univalent polynomials in is contained in the union of three irreducible algebraic surfaces.

Polynomial Solutions to the Hele–Shaw Problem

We introduce a new class Hn of univalent polynomials and establish that for every polynomial in Hn the Hele–Shaw problem has a polynomial solution w(z;t) for all values t>0. We also demonstrate that the members of Hn are starlike.