| Hyperbolic geometry was created by Gauss, Bolyai and Lobachevskii. It has lots of applications in conformal mapping theory, topology and group theory. The estimate of the hyperbolic areas of hyperbolic polygons is a basic problem in hyperbolic geometry which is closely related with quasiconformal mapping and univalent functions, and harmonic mappings. Function space theory is another important branch of complex analysis, and the estimation of integral growth for the derivatives of given class of functions is also an important issue in the theory of function space.In this article we first focus on the study of hyperbolic areas of hyperbolic triangles and hyperbolic polygons and hyperbolic geometric characterizations about Lambert quadrilaterals and Saccheri quadrilaterals. After achieving a new hyperbolic area formula of a hyperbolic triangle by using the coordinates of its vertices instead of its interior angles, then we give the hyperbolic area formulas of Lambert quadrilaterals and Saccheri quadrilaterals and some geometric characterizations.Second, Fricain and Mashreghi study the estimation of integrable growth of the derivative of a Blaschke product in the Hardy space. Aleman and Vukoti study the same question in the Bergman space with normal weights. We continue to study this question and give the estimate of integral growth of the derivatives of a Blaschke product in the weighted Bergman space. Then we generalized it to the case of inner functions by the Frostman theorem.Third, the coefficient conjecture of harmonic mappings is an important issue in the theory of harmonic mappings. Some good works have been done for the case of bounded harmonic mappings and the question in weighted Bergman spaces also draw people’s attention. After giving a necessary and sufficient condition for the area integral means belonging to the harmonic weighted Bergman space, we obtain a coefficient estimate of harmonic mappings in the harmonic weighted Bergman space, which generalize the result of Chen, Ponnusamy and Wang. Moreover, this estimate is sharp for. |
