Related papers: Rational functions that share finite values with t…

We consider uniqueness results for meromorphic functions $f:{\mathbb C} \to \widehat{\mathbb C}$ such that for certain values $a\in {\mathbb C}$ the implication $f(z)=a \Rightarrow f'(z)=a$ holds, i.e. that $f$ and $f'$ share values {\it…

We improve well-known results concerning normal families and shared values of meromorphic functions in the plane. In particular, we obtain two corollaries concerning meromorphic functions $f \colon {\mathbb C} \to {\widehat{\mathbb C}}$: i)…

The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results have been obtained. In this paper, we study a…

For R(z, w) rational with complex coefficients, of degree at least 2 in w, we show that the number of rational functions f(z) solving the difference equation f(z+1)=R(z, f(z)) is finite and bounded just in terms of the degrees of R in the…

In the paper, we investigate the uniqueness problem of a power of an entire function that share one value partially with it's linear differential polynomial and obtain a result, which improves several previous results in a large scale. Also…

Let $f$ be an entire function and $L(f)$ a linear differential polynomial in $f$ with constant coefficients. Suppose that $f$, $f'$, and $L(f)$ share a meromorphic function $\alpha(z)$ that is a small function with respect to $f$. A…

We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.

Two meromorphic functions $ f $ and $ g $ are said to share a value $ s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $ g(z)-s $ have the same set of zeros counting multiplicities (ignoring multiplicities). We…

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of…

Consider the space $R_{\Delta}$ of rational functions of several variables with poles on a fixed arrangement $\Delta$ of hyperplanes. We obtain a decomposition of $R_{\Delta}$ as a module over the ring of differential operators with…

In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in…

In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively.…

Let $r(z)$ be a rational function with at most $n$ poles, $a_1, a_2, \ldots, a_n,$ where $|a_j| > 1,$ $1\leq j\leq n.$ This paper investigates the estimate of the modulus of the derivative of a rational function $r(z)$ on the unit circle.…

In this paper, we shall study the uniqueness problems on meromorphic functions sharing a polynomial. We give a complete answer to a problem posed by Fang Mingliang. Our results improve or generalize those given by Fang and Hua, Yang and…

The objective of the paper is twofold. The first objective is to study the uniqueness problem of meromorphic function $f(z)$ when $f^{(1)}(z)$ shares two distinct finite values $a_1$, $a_2$ and $\infty$ CM with $\Delta_cf(z)$. In this…

We establish a general result on the existence of partially defined semiconjugacies between rational functions acting on the Riemann sphere. The semiconjugacies are defined on the complements to at most one-dimensional sets. They are…

In this article, by comparing the characteristic functions, we prove that for any $\nu$-valued algebroid function $w(z)$ defined in the unit disk with $\limsup_{r\to1-}T(r,w)/\log\frac{1}{1-r}=\infty$ and the hyper order $\rho_2(w)=0$, the…

The existence of meromorphic solutions to various difference equations has been extensively studied in recent years, the precise functional forms of such solutions -- particularly when the function and its difference operators share values…

In the existing literature, many researchers consider the uniqueness of the power of a meromorphic function with its derivative counterpart share certain values or small functions. Here we consider the same problem under the aegis of a more…

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results…