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On the five-point theorems due to Lappan

Yan Xu — 2011

Annales Polonici Mathematici

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.

On the value distribution of differential polynomials of meromorphic functions

Yan XuHuiling Qiu — 2010

Annales Polonici Mathematici

Let f be a transcendental meromorphic function of infinite order on ℂ, let k ∈ ℕ and φ = R e P , where R ≢ 0 is a rational function and P is a polynomial, and let a , a , . . . , 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 f ( k ) + a k - 1 f ( k - 1 ) + + a f = 0 , then f ( k ) + a k - 1 f ( k - 1 ) + + a f - φ has infinitely many zeros.

Normality criteria and multiple values II

Yan XuJianming Chang — 2011

Annales Polonici Mathematici

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 ( k ) 0 ; (2) all zeros of f ( k ) - ψ have multiplicities at least (k+2)/k; (3) all poles of ψ have multiplicities at most k, then ℱ is normal in D.

Weighted sharing and uniqueness of entire functions

Fengqin WuYan Xu — 2010

Czechoslovak Mathematical Journal

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 uniqueness of meromorphic functions ink-punctured complex plane

Hong Yan XuSan Yang Liu — 2017

Open Mathematics

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).

Results on the deficiencies of some differential-difference polynomials of meromorphic functions

Xiu-Min ZhengHong-Yan Xu — 2016

Open Mathematics

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 ′ ) <+∞, lim sup r + T ( r , f ) T ( r , f ' ) < + , 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...

Zeros of solutions of certain higher order linear differential equations

Hong-Yan XuCai-Feng Yi — 2010

Annales Polonici Mathematici

We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation f ( k ) + a k - 1 ( z ) f ( k - 1 ) + + a ( z ) f ' + D ( z ) f = 0 , (1) where D ( z ) = Q ( z ) e P ( z ) + Q ( z ) e P ( z ) + Q ( z ) e P ( z ) , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), a j ( z ) (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

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