We investigate the uniqueness problem of entire functions that share two polynomials with their $k$th derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible.

In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f\left(z\right)$ and $g\left(z\right)$ be two transcendental entire functions of finite order, and $\alpha \left(z\right)$ a small function with respect to both $f\left(z\right)$ and $g\left(z\right)$. Suppose that $c$ is a non-zero complex constant and $n\ge 7$ (or $n\ge 10$) is an integer. If $f^n\left(z\right)(f\left(z\right)-1)f(z+c)$ and $g^n\left(z\right)(g\left(z\right)-1)g(z+c)$ share “$\left(\alpha \right(z),2)$” (or ${(\alpha \left(z\right),2)}^{*}$), then $f\left(z\right)\equiv g\left(z\right)$. Our results extend and generalize some well known previous...

Let f and g be entire functions, n, k and m be positive integers, and λ, μ be complex numbers with |λ| + |μ| ≠ 0. We prove that ${\left(fⁿ\left(z\right)(\lambda f^m\left(z\right)+\mu )\right)}^{(}k)$ must have infinitely many fixed points if n ≥ k + 2; furthermore, if ${\left(fⁿ\left(z\right)(\lambda f^m\left(z\right)+\mu )\right)}^{(}k)$ and ${\left(gⁿ\left(z\right)(\lambda g^m\left(z\right)+\mu )\right)}^{(}k)$ have the same fixed points with the same multiplicities, then either f ≡ cg for a constant c, or f and g assume certain forms provided that n > 2k + m* + 4, where m* is an integer that depends only on λ.

We prove some theorems on the hyper-order of solutions of the equation $f^{\left(k\right)}-e^{Q}f=a(1-e^Q)$, where Q is an entire function, which is a polynomial or not, and a is an entire function whose order can be larger than 1. We improve some results by J. Wang and X. M. Li.

Let f be a transcendental meromorphic function of infinite order on ℂ, let k ∈ ℕ and $\phi =R{e}^P$, where R ≢ 0 is a rational function and P is a polynomial, and let $a₀,a₁,...,a_{k-1}$ be holomorphic functions on ℂ. If all zeros of f have multiplicity at least k except possibly finitely many, and $f=0\iff f^{\left(k\right)}+a_{k-1}{f}^{(k-1)}+\cdots +a₀f=0$, then $f^{\left(k\right)}+a_{k-1}{f}^{(k-1)}+\cdots +a₀f-\phi$ has infinitely many zeros.

We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $f^{\left(k\right)}+a_{k-1}\left(z\right)f^{(k-1)}+\cdots +a₁\left(z\right)f^{\text{'}}+D\left(z\right)f=0$, (1) where $D\left(z\right)=Q₁\left(z\right)e^{P₁\left(z\right)}+Q₂\left(z\right)e^{P₂\left(z\right)}+Q₃\left(z\right)e^{P₃\left(z\right)}$, P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z),$a_j\left(z\right)$ (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation $$f^{\left(k\right)}+A_{k-1}\left(z\right)f^{(k-1)}+\cdots +A_1\left(z\right)f^{\text{'}}+A_0\left(z\right)f=0,$$ where $A_i\left(z\right)$ $(i=0,1,\cdots ,k-1)$ are meromorphic functions of finite order in the complex plane.

This paper studies the unicity of meromorphic(resp. entire) functions of the form $f^n{f}^{\text{'}}$ and obtains the following main result: Let $f$ and $g$ be two non-constant meromorphic (resp. entire) functions, and let $a\in ℂ\setminus \left\{0\right\}$ be a non-zero finite value. Then, the condition that $E_{3)}(a,f^n{f}^{\text{'}})=E_{3)}(a,g^n{g}^{\text{'}})$ implies that either $f=dg$ for some $(n+1)$-th root of unity $d$, or $f=c_1{e}^{cz}$ and $g=c_2{e}^{-cz}$ for three non-zero constants $c$, $c_1$ and $c_2$ with ${\left(c_1{c}_2\right)}^{n+1}{c}^2=-a^2$ provided that $n\ge 11$ (resp. $n\ge 6$). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed. ...

Let f be a transcendental entire function of finite lower order, and let $q_{\nu }$ be rational functions. For 0 < γ < ∞ let B(γ):= πγ/sinπγ if γ ≤ 0.5, B(γ):= πγ if γ > 0.5. We estimate the upper and lower logarithmic density of the set $r:{\sum }_{1\le \nu \le k}log⁺ma{x}_{\left|\right|z\left|\right|=r}{\left|f\left(z\right)-q_{\nu }\left(z\right)\right|}^{-1}<B\left(\gamma \right)T(r,f)$.

The main purpose of this paper is to partly answer a question of L. Z. Yang [Israel J. Math. 147 (2005), 359-370] by proving that every entire solution f of the differential equation $f^{\text{'}}-e^{P\left(z\right)}f=1$ has infinite order and its hyperorder is a positive integer or infinity, where P is a nonconstant entire function of order less than 1/2. As an application, we obtain a uniqueness theorem for entire functions related to a conjecture of Brück [Results Math. 30 (1996), 21-24].

Let f be a transcendental meromorphic function and $g\left(z\right)=f(z+c₁)+\cdots +f(z+c_k)-kf\left(z\right)$ and $g_k\left(z\right)=f(z+c₁)\cdots f(z+c_k)-f^k\left(z\right)$. A number of results are obtained concerning the exponents of convergence of the zeros of g(z), $g_k\left(z\right)$, g(z)/f(z), and $g_k\left(z\right)/f^k\left(z\right)$.

Let ℱ be a family of meromorphic functions on a plane domain D, all of whose zeros are of multiplicity at least k ≥ 2. Let a, b, c, and d be complex numbers such that d ≠ b,0 and c ≠ a. If, for each f ∈ ℱ, $f\left(z\right)=a\iff f^{\left(k\right)}\left(z\right)=b$, and $f^{\left(k\right)}\left(z\right)=d\Rightarrow f\left(z\right)=c$, then ℱ is a normal family on D. The same result holds for k=1 so long as b≠(m+1)d, m=1,2,....

Let ℱ be a family of meromorphic functions defined in a domain D, let ψ (≢ 0, ∞) be a meromorphic function in D, and k be a positive integer. If, for every f ∈ ℱ and z ∈ D, (1) f≠ 0, $f^{\left(k\right)}\ne 0$; (2) all zeros of $f^{\left(k\right)}-\psi$ have multiplicities at least (k+2)/k; (3) all poles of ψ have multiplicities at most k, then ℱ is normal in D.