Computational Methods and Function Theory

Ingram, 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}.

Zhang, 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.

Huang, 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.

Rainio, Oona; Vuorinen, Matti

2023 Computational Methods and Function Theory

doi: 10.1007/s40315-022-00447-3

The hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.

Beardon, 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.

Luo, Xi; Liu, Ming-Sheng

2023 Computational Methods and Function Theory

doi: 10.1007/s40315-022-00441-9

In this paper, we first establish a Landau–Bloch type theorem for certain bounded polyharmonic mappings, which improves upon a result given by Bai et al. (Complex Anal Oper Theory, 13(2):321–340, 2019). Then, we establish three new versions of Landau–Bloch type theorems for polyharmonic mappings, and obtain several sharp results. Finally, we provide four bi-Lipschitz theorems for these subclasses of polyharmonic mappings.

Du, 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.

Yang, Zhiqiang; Zhou, Qingshan

2023 Computational Methods and Function Theory

doi: 10.1007/s40315-022-00445-5

In this paper, several characterizations for both quasisymmetric mappings and freely quasiconformal mappings in real Banach spaces are established. Also, we get a characterization for a freely quasiconformal mapping to be quasisymmetric. All these characterizations consist of inequalities in terms of the point measure and the inner measure of topological angles, which were introduced by Agard and Gehring (Proc Lond Math Soc 3(14a):1–21, 1965). Also, we construct two examples which show that certain conditions in the obtained characterizations can not be removed.

Bongers, 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}.

Hinkkanen, Aimo; Miles, Joseph

2023 Computational Methods and Function Theory

doi: 10.1007/s40315-022-00464-2

We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.