Normality criteria of meromorphic functions with multiple zeros

Take positive integers n,k?2. Let F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k. If, for each pair (f,g) in F, fn(f(k)) and gn(g(k)) share a non-zero complex number a ignoring multiplicity, then F is normal in D.     

Normal families of meromorphic functions with multiple zeros

Let F be a family of meromorphic functions defined in a domain D such that for each f∈F, all zeros of f(z) are of multiplicity at least 3, and all zeros of f^′(z) are of multiplicity at least 2 in D. If for each f∈F, f^′(z)−1 has at most 1 zero in D, ignoring multiplicity, then F is normal in D.     

Derivatives of meromorphic functions with multiple zeros and elliptic functions

Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple except finitely many and T (r, h) = o{T (r, f )} as r →∞, then f' = h has infinitely many solutions (including poles).     

Exceptional functions and normal families of meromorphic functions with multiple zeros

Let k be a positive integer with k?2; let h(?0) be a holomorphic function which has no simple zeros in D; and let F be a family of meromorphic functions defined in D, all of whose poles are multiple, and all of whose zeros have multiplicity at least k+1. If, for each function f∈F, f(k)(z)≠h(z), then F is normal in D.     

Normality of meromorphic functions with multiple zeros and shared values

In this paper, we study the normality of a family of meromorphic functions and general criteria for normality of families of meromorphic functions with multiple zeros concerning shared values are obtained.     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.     

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,f ¹(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. Partially supported by the Shanghai Priority Academic Discipline and by the NNSF of China Approved No. 10271122. Research supported by the German-Israeli Foundation for Scientific Research and Development, G.I.F. Grant No. G-643-117.6/1999.     

Picard values and normal families of meromorphic functions with multiple zeros

In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved. Research supported by the National Science Foundation of China     

On zeros and fixed points of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+1)−f(z). A number of results are proved concerning the existences of zeros and fixed points of g(z) or g(z)/f(z) which expand results of Bergweiler and Langley [W. Bergweiler, J.K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007) 133-147].     

Distribution of zeros of certain meromorphic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1428–1430, October, 1989.     

Non-real zeros of derivatives of meromorphic functions

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane.     

Estimates for the zeros of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.     

Normal families of meromorphic functions with multiple values

Let F be a family of meromorphic functions defined in a domain D, let ψ(?0) be a holomorphic function in D, and k be a positive integer. Suppose that, for every function f∈F, f≠0, f(k)≠0, and all zeros of f(k)−ψ(z) have multiplicities at least (k+2)/k. If, for k=1, ψ has only zeros with multiplicities at most 2, and for k?2, ψ has only simple zeros, then F is normal in D. This improves and generalizes the related results of Gu, Fang and Chang, Yang, Schwick, et al.     

On permutable meromorphic functions

We study the class \({\mathcal{M}}\) of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in \({\mathcal{M}}\), with at least one essential singularity, permutes with a non-constant rational map g, then g is a Möbius map that is not conjugate to an irrational rotation. For a given function \({f \in\mathcal{M}}\) which is not a Möbius map, we show that the set of functions in \({\mathcal{M}}\) that permute with f is countably infinite. Finally, we show that there exist transcendental meromorphic functions \({f : \mathbb{C} \to \mathbb{C}}\) such that, among functions meromorphic in the plane, f permutes only with itself and with the identity map.