Hyperbolic Metric and Normal Family

In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.

Slant Submanifolds of an Almost Hyperbolic Contact Metric Manifolds

In the present paper, we introduce the notion of slant submanifolds of an almost hyperbolic contact metric manifolds. We have obtained some results on slant submanifolds of an almost hyperbolic contact metric manifolds. We have given a necessary and sufficient condition for a slant submanifold of an almost hyperbolic contact metric manifolds.

TWO GENERALIZATIONS OF BOHR RADIUS

The purpose of this paper is twofold.First,by using the hyperbolic metric,we establish the Bohr radius for analytic functions from shifted disks containing the unit disk D into convex proper domains of the complex plane.As a consequence,we generalize the Bohr radius of Evdoridis,Ponnusamy and Rasila based on geometric idea.By introducing an alternative multidimensional Bohr radius,the second purpose is to obtain the Bohr radius of higher dimensions for Carathéodory families in the unit ball B of a complex Banach space X.Notice that when B is the unit ball of the complex Hilbert space X,we show that the constant 1/3 is the Bohr radius for normalized convex mappings of B,which generalizes the result of convex functions on D.

LIPSCHITZ TYPE CHARACTERIZATIONS FOR BERGMAN-ORLICZ SPACES AND THEIR APPLICATIONS

We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk.

Extremal Problems of Bloch Function Spaces

Let f be analytic in a hyperbolic region Ω. The Bloch constant β_f of f is defined by β_f=sup z∈Ω|f’(z)|/λ_Ω(z), where λ_Ω(z)|dz|is the Poincare metric in Ω. Suppose △is hyperbolic and lim inf ω→cλ△(w)】λ(△)】0, (?)c∈(?)△ where λ(△)=inf w∈△ λ△(w). Then for all f with f(Ω)(?)△, we have β_f≤1/λ(△). In this paper we study the extremal functions defined by β_f=1/λ(△) and the existence of those functions.

On Equivalence of Simple Closed Curves in Flat Surfaces

By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX（α） = lX（β）,extX（α） = extX（β） and lq（α） = lq（β）,where lX（·）,extX（·） and lq（·） are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.