Landau Theorem For Certain Biharmonic Mappings And Convexity Radius Estimate For Certain Harmonic Mappings

In the theory of complex analysis, univalent radius estimate such as Landau theorem and Bloch theorem is an important and fundamental problem. In 1926, Landau obtained the classical Landau theorem of analytic functions. In the past two decades, Landau theorem of analytic functions has been generalized to the one of harmonic mappings, biharmonic mappings, P-harmonic mappings, logarithmic harmonic mappings and so on. In 1989, Ruscheweyh and Salinas proved maps a convex region onto a convex region for a harmonic mappingwith. Convexity radius of the subclass of harmonic mappings has also benn studied by many scholars.In this paper, we study the representation theory of harmonic mappings with hyperbolic weights and univalent radius estimate for harmonic mappings or harmonic mappings with hyperbolic weights under the action of differential operator and integral operator.In Chapter I, we first give basic definitions and notations, backgrounds of our research problems and then enumerate our main results in this paper.In Chapter II, we study harmonic mappings with hyperbolic weights, which are solutions of the following quasilinear partial differential equation and utilize harmonic mappings to give an explicit representation of its solutions. By this representation theorem, we obtain two versions of Landau’s theorem of this class of mappings.In Chapter III, we study some properties of given mappings under the action of the differential operator introduced by Abduhadi in 2006. This operator keeps both harmonicity and biharmonicity. We give Landau’s theorem for harmonic mappings with hyperbolic weights under the action of. Moreover, our result is sharp when.In chapter IV, we study the convex hereditary problem of harmonic mappings. The integral operator was introduced by Alexander in 1915. Nash gave the sharp convex radius of analytic functions under the action of. Nagpal and Rovichandran defined the operator of harmonic mappings as, we generalize the result given by Nash to the class of harmonic mappings. Moreover, the sharp examples are also given in this article.