Xuecheng Pang

Normal Families and Shared Values

For f a meromorphic function on the plane domain D and a ∈ [Copf ], let Ē f ( a ) = { z ∈ D [ratio ] f ( z ) = a }. Let [Fscr ] be a family of meromorphic functions on D , all of whose zeros are of multiplicity at least k . If there exist b ≠ 0 and h > 0 such that for every f ∈ [Fscr ], Ē f (0) = Ē f ( k ) ( b ) and 0 f ( k +1) ( z )[mid ] [les ] h whenever z ∈ Ē f (0), then [Fscr ] is a normal family on D . The case Ē f (0) = O is a celebrated result of Gu [ 5 ].

Normality and shared values

LetF be a family of meromorphic functions on the unit disc Δ and leta andb be distinct values. If for everyf∈F,f andf′ sharea andb on Δ, thenF is normal on Δ.

On the derivative of meromorphic functions with multiple zeros

Abstract Let f be a transcendental meromorphic function and let R be a rational function, R ≢0. We show that if all zeros and poles of f are multiple, except possibly finitely many, then f ′− R has infinitely many zeros. If f has finite order and R is a polynomial, then the conclusion holds without the hypothesis that poles be multiple.

Normal families of meromorphic functions with multiple zeros and poles

LetF be a family of functions meromorphic in the plane domainD, all of whose zeros and poles are multiple. Leth be a continuous function onD. Suppose that, for eachf ≠F,f1(z) εh(z) forz εD. We show that ifh(z) ≠ 0 for allz εD, or ifh is holomorphic onD but not identically zero there and all zeros of functions inF have multiplicity at least 3, thenF is a normal family onD.

Normal Families of Meromorphic Functions whose Derivatives Omit a Function

Let

Derivatives of Meromorphic Functions with Multiple Zeros and Rational Functions

\cal F

Picard-Hayman behavior of derivatives of meromorphic functions with multiple zeros

be a family of functions meromorphic on the plane domain D, and let h be a holomorphic function on D, h n= 0. Suppose that, for each

Normal families of holomorphic functions with multiple zeros

f \in {\cal F}

Quasinormal Families of Meromorphic Functions

, f(m)(z) ≠ h(z) for z ∈ D. Then

Quasinormality and meromorphic functions with multiple zeros

t\cal F