Risto Juhana Korhonen

University of Eastern Finland | UEF · Department of Physics and Mathematics

Tropical Nevanlinna theory, introduced by Halburd and Southall as a tool to analyze integrability of ultra-discrete equations, studies the growth and complexity of continuous piecewise linear real functions. The purpose of this paper is to extend tropical Nevanlinna theory to finite dimensional tropical projective spaces by introducing a natural ch...

Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations solved by elliptic functions.

It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag{\dag} \end{equation} where $R(z,f)$ is rational in both arguments, has a transcendental meromorphic solution $f$ of hyper-order $<1$, then either $f$ satisfies a difference linear or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be transformed int...

According to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function...

This is the first textbook-type presentation of tropical value distribution theory. It provides a detailed introduction of the tropical version of the Nevanlinna theory, describing growth and value distribution analysis of continuous, piecewise linear functions on the real axis. The book also includes applications of this theory to ultra-discrete e...

This paper is devoted to considering the quasiperiodicity of complex differential polynomials, complex difference polynomials and complex delay-differential polynomials of certain types, and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form \(f(z+1)^n=R(z,f)\) \((\dagger)\) that have transcendental meromorphic solutions, where \(R(z,f)\) is rational in both arguments. In this paper, we first complete the classification for the case \(\deg_{f}(R(z,f))...

It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{\Delta_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $\Delta_c f=f(z+c)-f(z)$ and $a,c\in\mathbb{C}$. This estimate implies a lower bound for the Nevanlinna ramification term in terms of the di...

A generalization of the second main theorem of tropical Nevanlinna theory is presented for noncontinuous piecewise linear functions and for tropical hypersurfaces without requiring a growth condition. The method of proof is novel and significantly more straightforward than previously known proofs. The tropical analogue of the Nevanlinna inverse pro...

It is shown that if the equation where R(z, f) is rational in both arguments and \(\deg _f(R(z,f))\not =n\), has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of W...

Recently, the present authors classified the Malmquist type difference equations $f(z+1)^n=R(z,f)$ $(\dag)$ that have a transcendental meromorphic solution, where $R(z,f)$ is rational in both arguments. In this paper, we first simplify the derivations of those equations obtained in the case $\deg_{f}(R(z,f))=n$ based on some recent observations on...

We study the higher order delay differential equationsw(z+1)−w(z−1)+a(z)w(k)(z)w(z)=R(z,w(z)), andw(z+1)+a(z)w(k)(z)w(z)=R(z,w(z)), where k is a positive integer, a(z) is a rational function and R(z,w) is rational in w with rational coefficients. We obtain necessary conditions on the degree of R(z,w) for these delay differential equations to admit...

The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, f(z)nf(k)(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document...

By observing the periodicity of transcendental entire solutions of the complex differential equation f(z)f″(z)=p(z)sin2⁡z, where p(z) is a non-zero polynomial with real coefficients and real zeros, Yang's Conjecture has been proposed and considered by many authors recently. In this paper, we consider the parity of transcendental entire solutions of...

It is shown that if equation \begin{equation*} f(z+1)^n=R(z,f), \end{equation*} where $R(z,f)$ is rational in both arguments and $\deg_f(R(z,f))\not=n$, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term $R(z,f)$ takes particular forms. Solutions o...

Differential calculus is not a unique way to observe polynomial equations such as $$a+b=c$$ a + b = c . We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a , b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a po...

In this paper, we study q-difference analogues of several central results in value distribution theory of several complex variables such as q-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura–Clunie theorem concerning partial q-differen...

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity...

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a δ \delta -subharmonic function is established allowing the case of hyper-order equal to one a...

First, we are concerned with a lemma on the difference quotients due to Halburd, Korhonen and Tohge. We show that for meromorphic functions whose deficiency is origin dependent the exceptional set associated with this lemma is of infinite linear measure. In particular, for such entire functions in this set there is an infinite sequence {r_n} such t...

We investigate transcendental entire solutions of complex differential equations f″+A(z)f=H(z), where the entire function A(z) has a growth property similar to the exponential functions, and H(z) is an entire function of order less than that of A(z). We first prove that the lower order of the entire solution to the equation is infinity. By using ou...

A class of discrete equations is considered from three perspectives corresponding to three measures of the complexity of solutions: the (hyper-) order of meromorphic solutions in the sense of Nevanlinna, the degree growth of iterates over a function field and the height growth of iterates over the rational numbers. In each case, low complexity impl...

It is shown that the difference equation \begin{equation}\label{abseq} (\Delta f(z))^2=A(z)(f(z)f(z+1)-B(z)) \qquad\qquad (1) \end{equation} possesses a continuous limit to the differential equation \begin{equation}\label{abseq2} (w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions in certain cases. In addition, if (1) poss...

Let f(z) be a transcendental meromorphic function in the complex plane of hyper-order strictly less than 1. It is shown that if f(z) and its nth exact difference Δⁿf(z) (≢ 0) share three distinct periodic functions \({\rm{a, b, c}} \in \mathcal{\hat{S}}(f)\) with period 1 CM, where \(\mathcal{\hat{S}}(f) = \mathcal{S}(f)\cup\{{\infty}\}\) and \(\ma...

Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\log T(r,f)\leq r/(\log r)^{2+\nu}$, $\nu>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on dif...

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a $\delta$-subharmonic function is established allowing the hyper-order equal to one case with...

Differential calculus is not a unique way to observe polynomial equations such as $a+b=c$. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials $a$, $b$ and $c$ satisfying the equation above. Then a difference $abc$ theorem for polynomials is proved using a new notion of a radical of a polynomi...

In this paper, we investigate shared value problems related to an entire function f(z) of hyper-order less than one and its linear difference polynomial L(f) = where ai; ci 2 C. We give sufficient conditions in terms of weighted value sharing and truncated deficiencies, which imply that L(f) = f.

We extend the difference analogue of Cartan's second main theorem for the case of slowly moving periodic hyperplanes, and introduce two different natural ways to find a difference analogue of the truncated second main theorem. As applications, we obtain a new Picard type theorem and difference analogues of the deficiency relation for holomorphic cu...

Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We pr...

The existence of zero-order meromorphic solutions is used as a sufficient condition in detecting q-difference equations of Painlevé type. It is shown that demanding the existence of at least one non-rational zeroorder meromorphic solution w(z) is sufficient to reduce a canonical class of q-difference equations with rational coefficients into a shor...

We introduce the notion of partial value sharing as follows. Let E[U+203E](a,f) be the set of zeros of f(z)-a(z), where each zero is counted only once and a is a meromorphic function, small with respect to f. A meromorphic function f is said to share a partially with a meromorphic function g if E[U+203E](a,f)⊆E[U+203E](a,g). We show that partial va...

Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, -difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differe...

Nevanlinna's second main theorem is a far-reaching generalisation of Picard's Theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f' appear. In this paper we show that a similar result holds for spe...

An order reduction method for homogeneous linear difference equations, analogous to the standard order reduction of linear differential equa-tions, is introduced, and this method is applied to study the Nevanlinna growth relations between meromorphic coefficients and solutions of linear difference equations.

Shared value problems related to a meromorphic function f (z) and its shift f (z + c), where c  , are studied. It is shown, for instance, that if f (z) is of finite order and shares at least three values counting multiplicities with its shift f (z + c), then f is a periodic function with period c. The assumption on the order of f can be dropped if...

In this paper, we investigate the relation of the Nevanlinna characteristic functions T(r,f(qz)) and T(r,f(z)) for a zero-order meromorphic function f and a non-zero constant q. It is shown that T(r,f(qz))=(1+o(1))T(r,f(z)) for all r on a set of lower logarithmic density 1. This estimate is sharp in the sense that for any q∈C such that |q|≠1, and ρ...

This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f(z) and its shift f(z+c), where c∈C, are studied. It is shown, for instance, that if f(z) is of finite order and shares two values CM and one value IM with its shift f(z+c), then f is a periodic function with per...

If f: C -> P^n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation t(z)=z+c, then f is periodic with period c. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follow...

It is shown that if w(z) is a finite-order meromorphic solution of the equation H(z,w) P(z,w) = Q(z,w), where P(z,w) = P(z,w(z),w(z+c_1),...,w(z+c_n)) is a homogeneous difference polynomial with meromorphic coefficients, and H(z,w) = H(z,w(z)) and Q(z,w) = Q(z,w(z)) are polynomials in w(z) with meromorphic coefficients having no common factors such...

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if n∈N and three distinct values of a meromorphic function f of hyper-order less than 1/n2 h...

It is shown that if three distinct values of a meromorphic function f:C^n -> P^1 of hyper-order strictly less than 2/3 have forward invariant pre-images with respect to a translation t:C^n -> C^n, t(z)=z+c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green's Picard-type theorem in the special case...

Necessary and sufficient conditions for the analytic coefficients of the complex linear differential equation(†)f(k)+ak−1(z)f(k−1)+⋯+a1(z)f′+a0(z)f=ak(z) are found such that all solutions satisfy σ(f):=lim supr→1−log+T(r,f)−log(1−r)⩽α. Moreover, estimates for the number of linearly independent solutions of maximal growth are found in terms of the g...

Let α∈ℝ, and let C>max{1,α}. It is shown that if {p n /q n } is a sequence formed out of all rational numbers p/q such that α-p q≤1 Cq 2 , where p∈ℤ and q∈ℕ are relatively prime numbers, then either {p n /q n } has finitely many elements or limsup n→∞loglogq n logn≥1, where the points {q n } n∈ℕ are ordered by increasing modulus. This implies that...

Complex linear differential equations of the form (†) $f^{(k)}+a_{k-1}(z)f^{(k-1)}+...+a_{1}(z)f^{\prime}+a_{0}(z)f=0$ with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient $a_{j}(z)$ of (†) belongs to the weighted Bergman space $A_{\alpha}^{\frac{1}{k-j}}$ , where α ≥ 0, for all j = 0,...

The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry an...

Sufficient conditions for the analytic coefficients of the linear differential equationare found such that all solutions belong to a given -space, or to the Dirichlet type subspace Dp of the classical Hardy space Hp , where 0 < p ≤ 2. For 0 < q < ∞, the space consists of those functions f, analytic in the unit disc D, for which |f(z)|(1 – |z|2) q i...

It is shown that, if f is a meromorphic function of order zero and q ∈ ℂ, then m(r,f(qz)/f(z)) = o(T(r, f)) (‡) for all r on a set of logarithmic density 1. The remainder of the paper consists of applications of identity (‡) to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions...

A function g, analytic in the unit disc D, belongs to the weighted Hardy space H ∞ q if supo 0, then it is said to be an H-function. Heittokangas has shown that all solutions of the linear differential equation (t) f (k) + A k-1 (z)f (k-1) +... + A 1 (z)f' + A 0 (z)f = 0, where Aj(z) is analytic in D for all j = 0,..., k - 1, are of finite order of...

The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé prop...

For n Î \mathbbNn \in \mathbb{N} , the n-order of an analytic function f in the unit disc D is defined by sM,n (f) = limsupr ® 1 - \fraclog + n + 1 M(r,f) - log(1 - r),\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}}, where log+ x = max{log x, 0}, log+1 x = log+ x,...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/\%7Eschief/side/side.html). Participants at that meeting, as well...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/%7Eschief/side/side.html). Participants at that meeting, as well a...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/%7Eschief/side/side.html). Participants at that meeting, as well a...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/%7Eschief/side/side.html). Participants at that meeting, as well a...

Sharp versions of some classical results in differential equations are given. Main results consists of a Clunie and a Mohon'ko type theorems, both with sharp forms of error terms. The sharpness of these results is discussed and some applications to nonlinear differential equations are given in the conluding remarks. Moreover, a short introduction o...

The existence of sufficiently many finite-order (in the sense of Nevanlinna) meromorphic solutions of a difference equation appears to be a good indicator of integrability. It is shown that, out of a large class of second-order difference equations, the only equation that can admit a sufficiently general finite-order meromorphic solution is the dif...

Certain estimates involving the derivative $f\mapsto f'$ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna theory to a theory for the exact difference $f\mapsto \Delta f=f(z+c)-f(z)$. An $a$-point of a meromorphic function $f$ is...

Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation (1) can be transformed to one of a list of canonical difference equations. This list c...

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equation...

Certain intersection and unions with respect to dierent param- eters of the weighted Bergman spaces are shown to be equal. These results are extended to a more general class of function spaces. Some of the results proved here play an important role in the study of complex linear dierential equations in the unit disc.

A concrete presentation of Nevanlinna theory in a domain z: ¦z¦≥R has been offered by Bieberbach. He applied Green’s formula to prove the first main theorem and the lemma of the logarithmic derivative for meromorphic functions outside a disc of radius R. Apart from this work, Nevanlinna theory outside a disc has been considered in the form of brief...

Two methods are used to nd growth estimates (in terms of the p-characteristic) for the analytic solutions of f (k) + Ak 1(z)f (k 1) + + A1(z)f0 + A0(z)f = 0 in the disc fz 2 C : jzj < Rg, 0 < R 1. By restricting to special cases, these estimates yield known results in the complex plane without appealing to the commonly used Wiman{Valiron theory.

In this paper, we consider the growth of meromorphic solutions of nonlinear differential equations of the form L (f) + P (z, f) = h (z), where L (f) denotes a linear differential polynomial in f, P (z, f) is a polynomial in f, both with small meromorphic coefficients, and h (z) is a meromorphic function. Specialising to L (f) − p (z) fn = h (z), wh...

We will prove that the inequality m µ r; f 0

In a recent paper [1], Ablowitz, Halburd and Herbst applied Nevan-linna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typ-ical example of their results tells us that if a complex difference equation y(z + 1) + y(z − 1) = R(z, y) with R(z, y) rational...