Solutions to Difference Equations Have Few DefectsIngram, Patrick
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-022-00437-5
We establish a strong form of Nevanlinna’s Second Main Theorem for solutions to difference equations f(z+1)=R(z,f(z)),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} f(z+1)=R(z, f(z)), \end{aligned}$$\end{document}with the coefficients of R growing slowly relative to f, and degw(R(z,w))≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\deg _w(R(z, w))\ge 2$$\end{document}.
On Entire Function ep(z)∫0zβ(t)e-p(t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{p(z)}\int _0^{z}\beta (t)e^{-p(t)}dt$$\end{document} with Applications to Tumura–Clunie Equations and Complex DynamicsZhang, Yueyang
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-022-00446-4
Let p(z) be a non-constant polynomial and β(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta (z)$$\end{document} be a small entire function of ep(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e^{p(z)}$$\end{document} in the sense of Nevanlinna. We describe the growth behavior of the entire function H(z):=ep(z)∫0zβ(t)e-p(t)dt\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(z):=e^{p(z)}\int _0^{z}\beta (t)e^{-p(t)}dt$$\end{document} in the complex plane C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}$$\end{document}. As an application, we find entire solutions of the Tumura–Clunie type differential equation f(z)n+P(z,f)=b1(z)ep1(z)+b2(z)ep2(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(z)^n+P(z,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$$\end{document}, where b1(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b_1(z)$$\end{document} and b2(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b_2(z)$$\end{document} are non-zero polynomials, p1(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_1(z)$$\end{document} and p2(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_2(z)$$\end{document} are two polynomials of the same degree k≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\ge 1$$\end{document} and P(z, f) is a differential polynomial in f of degree at most n-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n-1$$\end{document} with meromorphic functions of order less than k as coefficients. These results allow us to determine all solutions with relatively few zeros of the second-order differential equation f′′-[b1(z)ep1(z)+b2(z)ep2(z)+b3(z)]f=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}+b_3(z)]f=0$$\end{document}, where b3(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b_3(z)$$\end{document} is a polynomial. We also prove a theorem on certain first-order linear differential equations related to complex dynamics.
Quasisymmetry and Quasihyperbolicity of Mappings on John DomainsHuang, Manzi; Rasila, Antti; Wang, Xiantao; Zhou, Qingshan
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-022-00440-w
Suppose that G is a proper subdomain of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb R}^n$$\end{document}, f:G→Y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f:G\rightarrow Y$$\end{document} is a homeomorphism with a continuous extension to the inner boundary of G, i.e., the boundary of G with respect to the corresponding inner metric, where (Y,d′)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(Y,d')$$\end{document} stands for a locally compact, non-complete and rectifiably connected metric space, and that G′=f(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G'=f(G)$$\end{document} is uniform in Y. The purpose of this paper is to prove that G is a John domain if f is M-quasihyperbolic in G and the restriction of f on the inner boundary of G is η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\eta $$\end{document}-quasisymmetric with respect to the inner metric.
Groups of Rotations of Euclidean and Hyperbolic SpacesBeardon, A. F.; Minda, D.
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-022-00436-6
In both the Euclidean plane R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^2$$\end{document} and the hyperbolic plane H2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {H}}^2$$\end{document}, a non-trivial group of rotations has a unique fixed point. We compare groups of rotations of the three-dimensional spaces R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^3$$\end{document} and H3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {H}}^3$$\end{document}, and in each case we discuss the existence of a (possibly non-unique) common fixed point of the elements in such a group.
Painlevé III and V Types Differential Difference EquationsDu, Yishuo; Zhang, Jilong
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-022-00442-8
In this paper, we show that if the equations w(z+1)w(z-1)+a(z)w′(z)w(z)=P(z,w(z))Q(z,w(z)),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} w(z+1)w(z-1)+a(z)\frac{w'(z)}{w(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{aligned}$$\end{document}and (w(z)w(z+1)-1)(w(z)w(z-1)-1)+a(z)w′(z)w(z)=P(z,w(z))Q(z,w(z)),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} (w(z)w(z+1)-1)(w(z)w(z-1)-1)+a(z)\frac{w'(z)}{w(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{aligned}$$\end{document}where a(z) is rational, P(z, w) and Q(z, w) are coprime polynomials of w(z) with rational functions coefficients, have a non-rational meromorphic solution with hyper-order less than one, then the degrees of the numerator and denominator on the right sides of the equations have to meet certain conditions.
Existence of Quasiconformal Maps with Maximal Stretching on Any Given Countable SetBongers, Tyler; Gill, James T.
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-022-00453-5
Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative Hölder continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for Hölder continuity. In this work, given any bounded countable set in Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^d$$\end{document}, we will construct an example of a K-quasiconformal map which exhibits the maximum stretching at each point of the set. This will provide an example of a quasiconformal map that exhibits the worst-case regularity on a surprisingly large set, and generalizes constructions from the planar setting into Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^d$$\end{document}.