Search

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = (−1, 1), holds for z ∈ D, where Λ( z) is the maximum dilation of U at z. The inequality is sharp for any z ∈ D and any value of U( z), and the equality occurs for some point in D if and only if $$U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}$$, z ∈ D, with a Möbius transformation φ of D onto itself. [ABSTRACT FROM AUTHOR]

New function spaces, which generalize the classical Dirichlet space, BMOA or also the recently defined Qp space, are introduced on Riemann surfaces. Except inclusions between these generalized spaces it is shown that the capacity Bloch space is a maximal space for them. [ABSTRACT FROM AUTHOR]

In this paper we prove a high order Schwarz-Pick lemma for holomorphic mappings between unit balls in complex Hilbert spaces. In addition, a Schwarz-Pick estimate for high order Fréchet derivatives of a holomorphic function f of a Hilbert ball into the right half-plane is obtained. [ABSTRACT FROM AUTHOR]

In this paper we establish characterizations of α-Bloch functions on the unit ball without use of derivative, which are stronger, more precise and general than those obtained by Nowak and Zhao. [ABSTRACT FROM AUTHOR]

We give a new lower bound of the locally univalent Bloch constant. [ABSTRACT FROM AUTHOR]

This paper proves that a necessary condition for a composition operator on the Bloch space of the unit disk to be bounded below given by Ghatage, Yan and Zheng is also sufficient. Given furthermore are a sufficient condition and a necessary condition of the boundedness from below for a composition operator on the Bloch space of a unit ball with dimensions bigger than one. [ABSTRACT FROM AUTHOR]

The classical Julia’s lemma is improved, branch points of Bloch functions are investigated, and new lower bounds of Bloch constants for functions with branch points are obtained. [ABSTRACT FROM AUTHOR]

An upper bound is given for the error term S( r, | a j |, f) in Nevanlinna’s inequality. For given positive increasing functions p and $ with ∫ d r/ p( r) = ∫ d r/ r ϕ( r) = ∞, set P( r) = ∫ d t/ p, Ψ( r) = ∫ d t/ t ϕ( t) We prove that holds, with a small exceptional set of r, for any finite set of points | a j | in the extended plane and any meromorphic function f such that Ψ( T( r, f)) = O( P( r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered. [ABSTRACT FROM AUTHOR]

Let ℑ be a family of holomorphic functions in the unit disk D. Suppose that there exists a nonzero and finite value a such that for each function ℑ, f, f′ and f″ share the value a IM in, D. Then the family ℑ is normal in D. An example shows that a cannot be zero. [ABSTRACT FROM AUTHOR]

The well-known lower estimate for Bloch's constant B is that B > √3/4. Recently, M. Bonk has improved this estimate by showing that B > √3/4 + 10. We give a further improvement by showing that B ≥√3/4 + 2 × 10. [ABSTRACT FROM AUTHOR]