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.

We investigate the uniqueness of a $q$-shift difference polynomial of meromorphic functions sharing a small function which extend the results of N. V. Thin (2017) to $q$-difference operators.

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.

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,....

The purpose of this paper is twofold. The first is to weaken or omit the condition $dim{f}^{-1}\left(H_i\cap H_j\right)\le m-2$ for i ≠ j in some previous uniqueness theorems for meromorphic mappings. The second is to decrease the number q of hyperplanes $H_j$ such that f(z) = g(z) on ${\bigcup }_{j=1}^q{f}^{-1}\left(H_j\right)$, where f,g are meromorphic mappings.

This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $𝔸\left(R_0\right)$ share four distinct values regardless of multiplicity and have the of positive counting function, then $f_1=f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and sharing...

We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P\left(f\right){\prod }_{i=1}^k{\left(f^{\left(i\right)}\right)}^{n_i}$ and $P\left(g\right){\prod }_{i=1}^k{\left(g^{\left(i\right)}\right)}^{n_i}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp.

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 Mₚ denote the class of functions f of the form $f\left(z\right)=1/z^p+{\sum }_{k=0}^{\infty }aₖz^k$, p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let $L_{n,p}\left(\alpha \right)=f:f\in Mₚ,Re-(z^{p+1}/p){\left(Dⁿf\right)}^{\text{'}}>\alpha$, α < 1, where $Dⁿf={\left(z^{n+p}f\left(z\right)\right)}^{\left(n\right)}/(z^{p}n!)$. Results on $L_{n,p}\left(\alpha \right)$ are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.

It is known that if $f$ is holomorphic in the open unit disc $𝔻$ of the complex plane and if, for some $c>0$, $\left|f\left(z\right)\right|\le {1/(1-|z|}^2{)}^c$, $z\in 𝔻$, then $|f^{\text{'}}{\left(z\right)|\le 2(c+1)/(1-\left|z\right|}^2{)}^{c+1}$. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in $𝔻$. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.

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. ...

With the idea of normal family we study the uniqueness of meromorphic functions $f$ and $g$ when $f^n{\left(f^{\left(k\right)}\right)}^m-p$ and $g^n{\left(g^{\left(k\right)}\right)}^m-p$ share two values, where $p$ is any nonzero polynomial. The result of this paper significantly improves and generalizes the result due to A. Banerjee and S. Majumder (2018).

Using the notion of weighted sharing of values which was introduced by Lahiri (2001), we deal with the uniqueness problem for meromorphic functions when two certain types of nonlinear differential monomials namely $h^n{h}^{\left(k\right)}$ $(h=f,g)$ sharing a nonzero polynomial of degree less than or equal to $3$ with finite weight have common poles and obtain two results. The results in this paper significantly rectify, improve and generalize the results due to Cao and Zhang (2012).