On the Nevanlinna's theory for vector-valued mappings.
In the paper we discuss the uniqueness of the -th power of a meromorphic function sharing a small function with the power of its -th derivative and improve and supplement a result of Zhang-Lü [Complex Var. Elliptic Equ. 53 (2008), no. 9, 857–867]. We also rectify one recent result obtained by Chen and Zhang in [Kyungpook Math. J. 50 (2010), no. 1, 71–80] dealing with a question posed by T.D. Zhang and W.R. Lü in [Complex Var. Elliptic Equ. 53 (2008), no. 9, 857–867].
We prove a theorem on the growth of nonconstant solutions of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its linear differential polynomial share a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.
With the aid of the notion of weighted sharing and pseudo sharing of sets we prove three uniqueness results on meromorphic functions sharing three sets, all of which will improve a result of Lin-Yi in Complex Var. Theory Appl. (2003).
Let f be a transcendental meromorphic function of infinite order on ℂ, let k ∈ ℕ and , where R ≢ 0 is a rational function and P is a polynomial, and let be holomorphic functions on ℂ. If all zeros of f have multiplicity at least k except possibly finitely many, and , then has infinitely many zeros.