A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

A criterion of normality based on a single holomorphic function

Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f ^(k)(z)| ≤ c, where c is a constant. Then F is normal on D.     

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.     

Normal families of meromorphic functions and shared values

Let k be a positive integer, b ≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let F{\mathcal{F}} be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in F, P(f)f^(k){\mathcal{F}, P(f)f^{(k)}} and P(g)g ^(k) share b in D, then F{\mathcal{F}} is normal in 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.     

Share fix-points and normal families of holomorphic functions

Let be a family of holomorphic functions in a domain D, let k 2 be a positive integer, and let K be a positive number. In this paper, we prove that: if, for each f and f share fix-points CM, and |f^(k) (z)| K whenever f(z) = z in D, then is normal in D. Some examples are given to show the sharpness of our result.Received: 17 June 2004     

Connected components of spaces of Morse functions with fixed critical points

Let M be a smooth closed orientable surface and F = F _p,q,r be the space of Morse functions on M having exactly p points of local minimum, q ≥ 1 saddle critical points, and r points of local maximum, moreover, all the points are fixed. Let F _f be a connected component of a function f ∈ F in F.We construct a surjection π ₀(F) → ℤ ^p+r−1 by means of the winding number introduced by Reinhart (1960). In particular, |π₀(F)| = ∞, and the component F _f is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D ⁰ be the connected component of id _M in D, and D _f ⊂ D be the set of diffeomorphisms preserving F _f . Let H _f be the subgroup of D _f generated by D ⁰ and all diffeomorphisms h ∈ D preserving some function f ₁ ∈ F _f , and let H _f ^abs be its subgroup generated by D ⁰ and the Dehn twists about the components of level curves of the functions f ₁ ∈ F _f . We prove thatH _f ^abs ⊊ D _f for q ≥ 2 and construct an epimorphism D _f /H _f ^abs → ℤ₂ ^q−1 by means of the winding number. A finite polyhedral complex K = K _p,q,r associated with the space F is defined. An epimorphism μ: π ₁(K) → D _f /H _f and finite generating sets for the groups D _f /D ⁰ and D _f /H _f in terms of the 2-skeleton of the complex K are constructed.     

On normal forms of singular Levi-flat real analytic hypersurfaces

Let F(z) = Re(P(z)) + h.o.t be such that M = (F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂ ⁿ , n ≥ 2, where P (z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂ ⁿ and Milnor number μ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ(M) = (Re(h) = 0), where h(z) is a polynomial of degree μ + 1 and j ₀ ^k (h) = P.     

Properties of generalised juxtapolynomials

GivenF(z),f ₁(z), ..,f _n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ _k ^=1/n a _kf_k(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case. This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this paper.     

Normality of meromorphic functions whose derivatives have 1-points

Let k be a positive integer and let ${\mathcal F}Let k be a positive integer and let F{\mathcal F} be a family of functions meromorphic in a plane domain D, all of whose zeros have multiplicity at least k + 3. If there exists a subset E of D which has no accumulation points in D such that for each function f ? F{f\in\mathcal F}, f ^(k)(z) − 1 has no zeros in D\E{D\setminus E}, then F{\mathcal F} is normal. The number k + 3 is sharp. The proof uses complex dynamics.     

Entire functions that share a polynomial with their derivatives

Let f be a nonconstant entire function, k and q be positive integers satisfying k>q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f^′ CM, and if f^(k)(z)−Q(z)=0 whenever f(z)−Q(z)=0, then f≡f^′. We give two examples to show that the hypothesis k>q is necessary.     

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy–Carleman–Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M(r, f) ≥ (N/2) log r − O(1), where M(r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Carleman and Tsuji related to the Denjoy–Carleman–Ahlfors theorem.     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

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.     

Quelques théorèmes sur la structure complexe

We prove the following result: If the function Max (log|ω -f ₁(z)|, ..., log|ω -f _k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ ⁿ ), thenf ₁,...,f _k are analytic functions iff ₁,...,f _k are continuous functions onD(k≥2). We prove also some other results.     

Normal Families and Uniqueness of Entire Functions and Their Derivatives

Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f（z） = a→f′（z） = a, and f′（z） = a →f^（k）（z） = a, then either f = Ce^λz ＋ a or f = Ce^λz ＋ a（λ - 1）/）λ, where C and ）, are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

Normal families and shared values of meromorphic functions

Let be a positive integer, let F be a family of meromorphic functions in a domain D, all of whose zeros have multiplicity at least k+1, and let , be two holomorphic functions on D. If, for each f∈F, f=a(z)⇔f(k)=h(z), then F is normal in D.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.