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1

On the five-point theorems due to Lappan

100%

Xu Y.

Annales Polonici Mathematici

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2011

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tom 101

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nr 3

227-235

EN

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.

2

Normality criteria and multiple values II

63%

Xu Y., Chang J.

Annales Polonici Mathematici

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2011

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tom 102

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nr 1

91-99

EN

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.

3

On the value distribution of differential polynomials of meromorphic functions

63%

Xu Y., Qiu H.

Annales Polonici Mathematici

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2010

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tom 98

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nr 3

283-289

EN

Let f be a transcendental meromorphic function of infinite order on ℂ, let k ∈ ℕ and $φ = Re^P$, 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 $f = 0 ⇔ f^{(k)} + a_{k-1}f^{(k-1)} + ⋯ + a₀f = 0$, then $f^{(k)} + a_{k-1}f^{(k-1)} + ⋯ + a₀f - φ$ has infinitely many zeros.

Ograniczanie wyników

3 Annales Polonici Mathematici

3 Xu Y.

1 Chang J.

1 Qiu H.

2 2011

1 2010

On the five-point theorems due to Lappan

Normality criteria and multiple values II

On the value distribution of differential polynomials of meromorphic functions