Related papers: A lemma on the difference quotients

For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min \{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both of which order…

In this article, we deal with the order of growth of solutions of non-homogeneous linear differential-difference equation \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F(z), \end{equation*} where $A_{ij},$ $F\left(…

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = f(z+1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f.

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative…

We give an alternative and simpler method for getting pointwise estimate of meromorphic solutions of homogeneous linear differential equations with coefficients meromorphic in a finite disk or in the open plane originally obtained by Hayman…

In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some…

In this paper, we mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables, and then investigate meromorphic solutions of partial difference equations from the viewpoint of…

The meromorphic solutions $f$ with $\rho_2(f)<1$ of the non-linear difference equation \begin{align*} f^n(z)+P_d(z,f)=p_1e^{{\lambda_1}z}+p_2e^{{\lambda_2}z}+p_3e^{{\lambda_3}z}, \end{align*} are characterized in terms of exponential…

In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)\Delta^mf(z)+\cdots+P_1(z)\Delta f(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for…

In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for $(\varphi,\nabla)$-modules over the bounded Robba ring. The main result is a proof of a conjecture proposed by B.…

In this note it is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.

The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated. The $\varphi$-order is used as a general growth indicator, which covers the growth spectrum between the…

This paper investigates the value distribution and growth properties of linear total differential polynomials $\mathcal{L}_k[D]f$ for meromorphic functions in several complex variables $\mathbb{C}^n$. By extending the classical Milloux…

In this paper we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations with entire or meromorphic coefficients. We further extend and improve few results on the order of meromorphic…

Let $\mu$ be a finite positive Borel measure supported on R, $\LL[f] =xf''+(\alpha+1-x)f'$ with $\alpha>-1$, or $\LL[f] =\frac{1}{2}f''-xf'$, and $m$ a natural number. We study algebraic, analytic and asymptotic properties of the sequence…

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a…

Let $\Lambda = \{\lambda_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{\lambda_{k} \in \Lambda : | \lambda_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an…

For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erd\H{o}s problem 684 asks for bounds on $f(n)$. We resolve the…

Suppose that a function $F$ is meromorphic in the domain $\mathbb H(-m) = \{ z : \mathrm{Im}\, z > -m(\mathrm{Re}\, z) \}$, where $m$ is an even, positive, and continuous function that does not increase on $\mathbb R_{\ge 0}$, and suppose…

It is shown that if 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 $f$ with meromorphic coefficients and $\deg_f(R(z,f))=n$, has an admissible meromorphic…