Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Real Schwarz Lemma: The Sequel

A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to have a real analytic automorphism $f$ of $(-1,1)$ with $f'(0)$ arbitrarily large. In this note we provide other families with this property, and moreover show that we can always find such a function so that $f'(0)$ equals any desired real number. We end with some questions on related problems.

The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.

On the growth of logarithmic difference of meromorphic functions and a Wiman-Valiron estimate

The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.

Kobayashi双曲黎曼流形的性质与例子

Properties and examples of Kobayashi hyperbolic Riemannian manifolds

本文在黎曼流形中推广了Brody引理，并构造了新的Kobayashi双曲流形例子。

Some Normality Criteria for Families of Holomorphic Functions of Several Complex Variables

We prove a Zalcman-Pang lemma in several complex variables and apply it to obtain several complex variables analogues of the known normality criteria like Lappan's five-point theorem and Schwick's theorem.

On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen \cite{AHH}, \cite {HK-2} that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $\log^+|f(z+1)/f(z)|$ on $|z|=r$. We obtained essentially the same estimate in our previous paper \cite{CF} independent of Halburd et al \cite{HK-1}. We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic…

函数方程的非阿基米德亚纯解

Non-Archimedean meromorphic solutions of functional equations

本文研究非阿基米德域上函数方程的亚纯解，建立了Clunie引理、Malmquist型定理和Mokhon'ko定理的对应版本。

Nevanlinna theory for holomorphic curves from annuli into semi Abelian varieties

In this paper, we prove a lemma on logarithmic derivative for holomorphic curves from annuli into Kähler compact manifold and. As its application, a second main theorem for holomophic curves from annuli into semi abelian varieties intersecting with only one divisor is given.