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.     

The support points of several classes of analytic functions with fixed coefficients

Let B denote the set of functions ?(z) that are analytic in the unit disk D and satisfy |?(z)|?1(|z|<1). Let P denote the set of functions p(z) that are analytic in D and satisfy p(0)=1 and Rep(z)>0(|z|<1). Let T denote the set of functions f(z) that are analytic in D, normalized by f(0)=0 and f^′(0)=1 and satisfy that f(z) is real if and only if z is real (|z|<1). In this article we investigate the support points of the subclasses of B, P and T of functions with fixed coefficients.     

An improvement of Montel's criterion

Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a^′(z) or b(z)?b^′(z). In this paper, we prove that: if, for each f∈F, f(z)−a(z) and f(z)−b(z) have no common zeros, f^′(z)=a(z) whenever f(z)=a(z), and f^′(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.     

Univalent Functions, <Emphasis Type="Italic">VMOA</Emphasis> and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬ_log which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup_{|z| < 1}(1-|z|)log\frac11-|z||f^￠(z)| < ￥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B ^p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of C n

Let B denote the unit ball in [(?)\tilde] \widetilde{\nabla\hskip-4pt}\hskip4pt denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces D_g{{\cal D}_{\gamma}} , $\gamma > (n-1)$\gamma > (n-1) , and weighted Bergman spaces A^p_a{A^p_{\alpha}} , 0 < p < ￥0 < p < \infty , $\alpha > n$\alpha > n , of holomorphic functions f on B for which D_g( f)D_{\gamma}(\,f) and || f||_A^pa\Vert\, f\Vert_{A^p_{\alpha}} respectively are finite, where D_g( f)=ò_B (1-|z|²)^g|[(?)\tilde]  f(z)|²dt(z),D_{\gamma}(\,f)=\int_B (1-\vert z\vert^2)^{\gamma}\vert\widetilde{\nabla\hskip-4pt}\hskip4pt f(z)\vert^2d\tau(z), and || f||^p_A^pa=ò_B(1-|z|²)^a| f(z)|^pdt(z).\Vert\, f\Vert^p_{A^p_{\alpha}}=\int_B(1-\vert z\vert^2)^{\alpha}\vert\, f(z)\vert^pd\tau(z). The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and $\alpha > n$\alpha > n .     

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.     

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

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.     

Dirichlet and VMOA domains via Schwarzian derivative

Short proofs of the following results concerning a bounded conformal map g of the unit disc D are presented: (1) logg^′ belongs to the Dirichlet space if and only if the Schwarzian derivative Sg of g satisfies Sg(z)(1−²|z|)∈L²(D); (2) logg^′∈VMOA if and only if ²|Sg(z)|³(1−²|z|) is a vanishing Carleson measure on D. Analogous results for Besov and Qp,0 spaces are also given.     

Об оценках проиэведений модулей нулей функций иэ пространств Бергмана и более общих пространств

We consider functions f(z) (f(0) = 1, |z| < 1) from spaces endowed with mixed norm; in particular, the Bergmann-Dzharbashian space, with zeroes z _k (f) (|z ₁(f)| ≤ |z ₂(f)| ≤ ...). We construct examples of such functions f that the products $$\pi n(f) = \left( {\left| {z_1 \left( f \right)} \right|} \right) \cdots \left( {\left| {z_n \left( f \right)} \right|} \right)^{ - 1}$$ have a well defined order of magnitude as n→∞ with respect to certain subsequences of n. We also establish necessary and sufficient conditions for the existence of such subsequences. These results are applied to study a number of spaces under consideration.     

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.     

The generalized Roper-Suffridge extension operator

Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z₁,z₂,…,z_n), z₁∈D, , p_i?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z₁),f′(z₁)^1/p₂z₂,…,f′(z₁)^1/p_nz_n) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z₁))^1/p_i|_z1=0=1, i=2,…,n. Some other related results are proved.     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

A slight improvement to Korenblum's constant

Let A²(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c∈(0,1) such that whenever |f(z)|?|g(z)| in the annulus c<|z|<1, then ‖f(z)‖?‖g(z)‖. This conjecture had been solved by Hayman [W.K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19 (1999) 195-205. [1]], but the constant c in that paper is not optimal. Since then, there are many papers dealing with improving the upper and lower bounds for the best constant c. For example, in 2004 C. Wang gave an upper bound on c, that is, c<0.67795, and in 2006 A. Schuster gave a lower bound, c>0.21. In this paper we slightly improve the upper bound for c.     

Local analytic solutions of the generalized Dhombres functional equation II

We study local analytic solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)), where φ is holomorphic at w₀≠0, f is holomorphic in some open neighborhood of 0, depending on f, and f(0)=w₀. After deriving necessary conditions on φ for the existence of nonconstant solutions f with f(0)=w₀ we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w₀ is not a root of 1. If |w₀|≠1 or if w₀ is a Siegel number we show that all formal solutions yield local analytic ones. For w₀ with 0<|w₀|<1 we give representations of these solutions involving infinite products.     

Extension of holomorphic and pluriharmonic functions with thin singularities on parallel sections

The paper is of survey character. We present and discuss recent results concerning the extension of functions that admit holomorphic or plurisubharmonic extension in a fixed direction. These results are closely related to Hartogs’ fundamental theorem, which states that if a function f(z), z = (z ₁, z ₂, ..., z _z ), is holomorphic in a domain D ? ?ⁿ in each variable z _j , then it is holomorphic in D in the n-variable sense.     

Distortion Properties of Holomorphic Functions of Several Complex Variables

Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C ⁿ, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff S^c(a, b; D) — the class of functions q such that q ? S^c(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z₁,z₂,…z_n. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and S^c(a, b; D).     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.