### Normal functions and normal families.

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By using an extension of the spherical derivative introduced by Lappan, we obtain some results on normal functions and normal families, which extend Lappan's five-point theorems and Marty's criterion, and improve some previous results due to Li and Xie, and the author. Also, another proof of Lappan's theorem is given.

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 $a\u2080,a\u2081,...,{a}_{k-1}$ be holomorphic functions on ℂ. If all zeros of f have multiplicity at least k except possibly finitely many, and $f=0\iff {f}^{\left(k\right)}+{a}_{k-1}{f}^{(k-1)}+\cdots +a\u2080f=0$, then ${f}^{\left(k\right)}+{a}_{k-1}{f}^{(k-1)}+\cdots +a\u2080f-\phi $ has infinitely many zeros.

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.

In this paper we study the uniqueness for meromorphic functions sharing one value, and obtain some results which improve and generalize the related results due to M. L. Fang, X. Y. Zhang, W. C. Lin, T. D. Zhang, W. R. Lü and others.

The main purpose of this paper is to investigate the uniqueness of meromorphic functions that share two finite sets in the k-punctured complex plane. It is proved that there exist two sets S1, S2 with ♯S1 = 2 and ♯S2 = 5, such that any two admissible meromorphic functions f and g in Ω must be identical if EΩ(Sj, f) = EΩ(Sj, g)(j = 1,2).

In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r→+∞ T(r, f) T(r, f ′ ) <+∞, $$\underset{r\to +\infty}{lim\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}}\frac{T(r,\phantom{\rule{0.166667em}{0ex}}f)}{T(r,\phantom{\rule{0.166667em}{0ex}}{f}^{\text{'}})}<+\infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value...

We deal with the uniqueness problem for analytic functions sharing four distinct values in an angular domain and obtain some theorems which improve the result given by Cao and Yi [J. Math. Anal. Appl. 358 (2009)].

We deal with the uniqueness of analytic functions in the unit disc sharing four distinct values and obtain two theorems improving a previous result given by Mao and Liu (2009).

We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation ${f}^{\left(k\right)}+{a}_{k-1}\left(z\right){f}^{(k-1)}+\cdots +a\u2081\left(z\right){f}^{\text{'}}+D\left(z\right)f=0$, (1) where $D\left(z\right)=Q\u2081\left(z\right){e}^{P\u2081\left(z\right)}+Q\u2082\left(z\right){e}^{P\u2082\left(z\right)}+Q\u2083\left(z\right){e}^{P\u2083\left(z\right)}$, 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.

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