For functions in $C^k(\mathbb {R})$ which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle $T\subseteq \mathbb {C}$ with $1\notin A$ , $1\notin B$ , then there is an analytic function $h\colon \mathbb {C}\setminus \{0\}\to \mathbb {C}$ that restricts to an order isomorphism of the arc $T\setminus \{1\}$ onto itself and satisfies $h(A)=B$ and $h'(z)\not =0$ when $z\in T$ . This answers a question of P. M. Gauthier.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation ${{f}^{2}}\,+\,{{g}^{2}}\,=\,1$ and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51(4) (1999), 781–799].

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)<+\infty$ and hyper-order ${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

In this paper we study uniqueness of entire functions sharing a non-zero finite value with linear differential polynomials and address a result of W.Wang and P. Li.

We study the uniqueness of entire functions sharing a nonzero finite value with linear differential polynomials and improve a result of P. Li.

We investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

Let $f\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be ${{C}^{\infty }}$ and let $h\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be positive and continuous. For any unbounded nondecreasing sequence $\{{{c}_{k}}\}$ of nonnegative real numbers and for any sequence without accumulation points $\{{{x}_{m}}\}$ in ${{\mathbb{R}}^{n}}$, there exists an entire function $g\,:\,{{\mathbb{C}}^{n}}\,\to \,\mathbb{C}$ taking real values on ${{\mathbb{R}}^{n}}$ such that

$$\left| {{g}^{\left( \alpha \right)}}\left( x \right)-{{f}^{\left( \alpha \right)}}\left( x \right) \right|\text{ }<h\left( x \right),\left| x \right|\ge {{c}_{k}},\left| \alpha \right|\le k,k=0,1,2,...,$$

$${{g}^{\left( \alpha \right)}}\left( {{x}_{m}} \right)\,=\,{{f}^{\left( \alpha \right)}}\left( {{x}_{m}} \right),\,\,\,\left| {{x}_{m}} \right|\,\ge \,{{c}_{k}},\,\left| \alpha \right|\,\le \,k,\,m,\,k\,=\,0,\,1,\,2,\,.\,.\,.\,.$$

This is a version for functions of several variables of the case $n\,=\,1$ due to $L$. Hoischen.

We study the rate of growth of entire functions that are frequently hypercyclic for the differentiation operator or the translation operator. Moreover, we prove the existence of frequently hypercyclic harmonic functions for the translation operator and we study the rate of growth of harmonic functions that are frequently hypercyclic for partial differentiation operators.

We investigate the growth and the frequency of zeros of the solutions of the differential equation f(n) + Pn–1 (z) f(n–1) + … + P0 (z) f = H (z), where P0 (z), P1(z), …, Pn–1(z) are polynomials with P0 (z) ≢ 0, and H (z) ≢ 0 is an entire function of finite order.

In this paper, we improve the Rosenbloom's fixed-point theorem and prove a related normality criterion. We also consider the corresponding unicity theorem for transcendental entire functions.

Let f be a transcendental entire function and denote the n-th iterate fn. For n ≥ 2, we give an explict estimate of the number of periodic points of f with period n, that is, fix-points of fn which are not fix-points of fk for 1 ≤ k <n.