Trevor J. Richards

activity

20132019

most citedLevel Curve Configurations and Conformal Equivalence of Meromorphic Functions

5 citations · 9 across the 10 of their papers we have counts for

collaborators

13 papers

math.CV2019

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

Trevor J. Richards

In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii…

math.CV2019

Rouché's Theorem and the Geometry of Rational Functions

Trevor J. Richards

In this note, we use Rouché's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results regarding the geometry of the zer…

math.CV2018

Leaky Roots and Stable Gauss-Lucas Theorems

Trevor J. Richards, Stefan Steinerberger

Let $p : C \to C$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p^{'} (z) = 0$ , are contained in the convex hull of its roots. A re…

math.CA2018

Luzin-type properties and the difference quotient of a real function

Yuri Andreev, Trevor J Richards

We introduce the notion of the difference quotient set of a real valued function $f$ on a set $E \subset [0, 1]$ , and compare this set to the range of $f$ on $E$ . We discuss the measu…

math.CV2018

Computing polynomial conformal models for low-degree Blaschke products

Trevor Richards, Malik Younsi

For any finite Blaschke product $B$ , there is an injective analytic map $φ : D \to C$ and a polynomial $p$ of the same degree as $B$ such that $B = p \circ φ$ on $\mathbb…

math.CV2017★ 4 cited

On approximate Gauss-Lucas theorems

Trevor Richards

The Gauss--Lucas theorem states that any convex set $K \subset C$ which contains all $n$ zeros of a degree $n$ polynomial $p \in C [z]$ must also contain all $n - 1$ cri…

math.CA2017

Recognizing difference quotients of real functions

Trevor Richards, Jimmy Yau

For a real function $f : [0, 1] \to R$ , the difference quotient of $f$ is the function of two real variables $DQ_{f} (a, b) = \frac{f ( b ) - f ( a )}{b - a}$ , which we view as…

math.CV2016

Boundary convergence and path divergence sets for bounded analytic functions in the disk

Trevor Richards

Let $f : D \to C$ be a bounded analytic function. A set $K \subset D$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ i…

math.CV2016

Characterizing meromorphic pseudo-lemniscates

Trevor Richards

Let $f$ be a meromorphic function with simply connected domain $G \subset C$ , and let $Γ \subset C$ be a smooth Jordan curve. We call a component of $f^{- 1} (Γ)$ in $G…

math.CV2015

Conformal equivalence of analytic functions on compact sets

Trevor Richards

In this paper we present a geometric proof of the following fact. Let $D$ be a Jordan domain in $C$ , and let $f$ be analytic on $c l (D)$ . Then there is an injective analyti…

math.CV2015

Conformal models and fingerprints of pseudo-lemniscates

Trevor Richards, Malik Younsi

We prove that every function that is meromorphic on the closure of an analytic Jordan domain and sufficiently well-behaved on the boundary is conformally equivalent to a rational m…

math.CV2013★ 5 cited

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Trevor Richards

Let $f = B_{1} / B_{2}$ be a ratio of finite Blaschke products having no critical points on $\partial D$ . Then $f$ has finitely many critical level curves (level curves containing…

math.CV2013

Notes on the Level Curves of a Meromorphic Function

Trevor Richards

The subject of this paper is the bounded level curves of a meromorphic function $f$ with domain $G$ such that each component of $\partial G$ consists of a level curve of $f$ . (A p…